Lectures on noise sensitivity and percolation

0. Without this assumption of “p olynomial decay” on H ( f n ), the pro of is more technical and relies on estimates obtained b y T alagrand. See the remark at the end of this pro of. F or our application to the noise sensitivity of p er c olation (see Chapter VI), this stronger assumption will b e satisfied and hence we stic k to this simpler case in these notes. The assumption of p olynomial decay in fact enables us to prov e the following more quan titativ e result. Prop osition V.5 ([BKS99]) . F or any δ > 0 , ther e exists a c onstant M = M ( δ ) > 0 such that if f n : Ω m n → { 0 , 1 } is any se quenc e of Bo ole an functions satisfying H ( f n ) ≤ ( m n ) − δ , then X 1 ≤| S |≤ M log ( m n ) b f n ( S ) 2 → 0 . Using Prop osition IV.1, this prop osition ob viously implies Theorem I.5 when H ( f n ) deca ys as assumed. F urthermore, this gives a quantitativ e “logarithmic” control on the noise sensitivit y of such functions. Pro of. The strategy will b e very similar to the one used in the KKL Theorems (ev en though the goal is v ery different). The main difference here is that the regularization term ρ used in the hypercontractiv e estimate m ust b e chosen in a more delicate w a y than in the pro ofs of KKL results (where w e simply to ok ρ = 1 / 2). Let M > 0 b e a constant whose v alue will b e chosen later. 50 CHAPTER V. HYPERCONTRA CTIVITY AND ITS APPLICA TIONS X 1 ≤| S |≤ M log( m n ) b f n ( S ) 2 ≤ 4 X 1 ≤| S |≤ M log( m n ) | S | b f n ( S ) 2 = X k X 1 ≤| S |≤ M log( m n ) [ ∇ k f n ( S ) 2 ≤ X k ( 1 ρ 2 ) M log( m n ) k T ρ ( ∇ k f n ) k 2 2 ≤ X k ( 1 ρ 2 ) M log( m n ) k∇ k f n k 2 1+ ρ 2 . b y Theorem V.2. No w, since f n is Bo olean, one has k∇ k f n k 1+ ρ 2 = k∇ k f n k 2 / (1+ ρ 2 ) 2 , hence X 0 < | S | 0. Dividing b oth sides by a 1+ ρ 2 , w e need to show that (1 + ρ 2 y 2 ) (1+ ρ 2 ) / 2 ≤ | 1 + y | 1+ ρ 2 + | 1 − y | 1+ ρ 2 2 (V.6) 5. THE NOISE SENSITIVITY THEOREM 53 for all y and clearly it suffices to assume y ≥ 0. W e first do the case that y ∈ [0 , 1). By the generalized Binomial formula, the right hand side of (V.6) is 1 2 " ∞ X k =0  1 + ρ 2 k  y k + ∞ X k =0  1 + ρ 2 k  ( − y ) k # = ∞ X k =0  1 + ρ 2 2 k  y 2 k . F or the left hand side of (V.6), we first note the following. F or 0 < λ < 1, a simple calculation sho ws that the function g ( x ) = (1 + x ) λ − 1 − λx has a negative deriv ativ e on [0 , ∞ ) and hence g ( x ) ≤ 0 on [0 , ∞ ). This yields that the left hand side of (V.6) is at most 1 +  1 + ρ 2 2  ρ 2 y 2 whic h is precisely the first tw o terms of the righ t hand side of (V.6). On the other hand, the binomial co efficients app earing in the other terms are nonnegativ e, since in the n umerator there are an even num ber of terms with the first t w o terms being p ositiv e and all the other terms b eing negativ e. This v erifies the desired inequalit y for y ∈ [0 , 1). The case y = 1 for (V.6) follo ws by contin uity . F or y > 1, we let z = 1 /y and note, by multiplying b oth sides of (V.6) b y z 1+ ρ 2 , we need to sho w ( z 2 + ρ 2 ) (1+ ρ 2 ) / 2 ≤ | 1 + z | 1+ ρ 2 + | 1 − z | 1+ ρ 2 2 . (V.7) No w, expanding (1 − z 2 )(1 − ρ 2 ), one sees that z 2 + ρ 2 ≤ 1 + z 2 ρ 2 and hence the desired inequality follo ws precisely from (V.6) for the case y ∈ (0 , 1) already prov ed. This completes the n = 1 case and thereby the pro of. 54 CHAPTER V. HYPERCONTRA CTIVITY AND ITS APPLICA TIONS Exercise sheet of Chapter V Exercise V.1. Find a direct pro of that Theorem I.3 implies Theorem I.2. Exercise V.2. Is it true that the smaller the influences are, the more noise sensitiv e the function is? Exercise V.3. Prov e that Theorem V.3 indeed implies Theorem I I I.2. Hin t: use the natural pro jection. Problem V.4. In this problem, we prov e Theorems I I I.2 and I I I.3 for the monotone case. 1. Sho w that Theorem II I.3 implies II I.2 and hence one needs to pro v e only Theorem I I I.3 (This is the basically the same as Exercise V.1). 2. Sho w that it suffices to prov e the result when p = k / 2 ` for in tegers k and ` . 3. Let Π : { 0 , 1 } ` → { 0 , 1 / 2 ` , . . . , (2 ` − 1) / 2 ` } by Π( x 1 , . . . , x ` ) = P ` i =1 x i / 2 i . Ob- serv e that if x is uniform, then Π( x ) is uniform on its range and that P (Π( x ) ≥ i/ 2 ` ) = (2 ` − i ) / 2 ` . 4. Define g : { 0 , 1 } ` → { 0 , 1 } by g ( x 1 , . . . , x ` ) := I { Π( x ) ≥ 1 − p } . Note that P ( g ( x ) = 1) = p . 5. Define ˜ f : { 0 , 1 } n` → { 0 , 1 } b y ˜ f ( x 1 1 , . . . , x 1 ` , x 2 1 , . . . , x 2 ` , . . . , x n 1 , . . . , x n ` ) = f ( g ( x 1 1 , . . . , x 1 ` ) , g ( x 2 1 , . . . , x 2 ` ) , . . . , g ( x n 1 , . . . , x n ` )) . Observ e that ˜ f (defined on ( { 0 , 1 } n` , π 1 / 2 )) and f (defined on ( { 0 , 1 } n , π p )) ha v e the same distribution and hence the same v ariance. 6. Sho w (or observ e) that I ( r,j ) ( ˜ f ) ≤ I p r ( f ) for eac h r = 1 , . . . , n and j = 1 , . . . , ` . Deduce from Theorem I.3 that X r,j I ( r,j ) ( ˜ f ) ≥ c V ar( f ) log (1 /δ p ) where δ p := max i I p i ( f ) where c comes from Theorem I.3. 55 56 CHAPTER V. HYPERCONTRA CTIVITY AND ITS APPLICA TIONS 7. (Key step). Sho w that for each r = 1 , . . . , n and j = 1 , . . . , ` , I ( r,j ) ( ˜ f ) ≤ I p r ( f ) / 2 j − 1 . 8. Com bine parts 6 and 7 to complete the pro of. Chapter VI First evidence of noise sensitivit y of p ercolation In this lecture, our goal is to collect some of the facts and theorems we hav e seen so far in order to conclude that p ercolation crossings are indeed noise sensitive. Recall from the “BKS” Theorem (Theorem I.5) that it is enough for this purp ose to prov e that influences are “small” in the sense that P k I k ( f n ) 2 go es to zero. In the first section, we will deal with a careful study of influences in the case of p ercolation crossings on the triangular lattice. Then, we will treat the case of Z 2 , where conformal in v ariance is not known. Finally , w e will sp eculate to what “exten t” p ercolation is noise sensitiv e. This whole chapter should b e c onsider e d somewhat of a “p ause” in our pr o gr am, wher e we take the time to summarize what we have achieve d so far in our understanding of the noise sensitivity of p er c olation, and what r emains to b e done if one wishes to pr ove things such as the existenc e of exc eptional times in dynamic al p er c olation. 1 Bounds on influences for crossing ev en ts in criti- cal p ercolation on the triangular lattice 1.1 Setup Fix a, b > 0, let us consider some rectangle [0 , a · n ] × [0 , b · n ], and let R n b e the set of of hexagons in T whic h intersect [0 , a · n ] × [0 , b · n ]. Let f n b e the ev en t that there is a left to righ t crossing ev en t in R n . (This is the same even t as in Example 7 in c hapter I, but with Z 2 replaced b y T ). By the RSW Theorem I I.1, we know that { f n } is non-degenerate. Conformal inv ariance tells us that E  f n  = P  f n = 1  con v erges as n → ∞ . The limit is given by the so-called Cardy’s formula . In order to pro v e that this sequence of Boolean functions { f n } is noise sensitiv e, 57 58 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION w e wish to study its influence vector Inf ( f n ) and we w ould like to pro v e that H ( f n ) = k Inf ( f n ) k 2 2 = P I k ( f n ) 2 deca ys p olynomially fast tow ards 0. (Recall that in these notes, w e gav e a complete pro of of Theorem I.5 only in the case where H ( f n ) decreases as an in v erse p olynomial of the n um ber of v ariables.) 1.2 Study of the set of influences Let x b e a site (i.e. a hexagon) in the rectangle R n . One needs to understand I x ( f n ) := P  x is pivotal for f n  It is easy but crucial to note that if x is at distance d from the b oundary of R n , in order for x to be piv otal, the four-arm even t describ ed in Chapter II (see Figure II.2) has to b e satisfied in the ball B ( x, d ) of radius d around the hexagon x . See the figure on the righ t. d x In particular, this implies (still under the assumption that dist( x, ∂ R n ) = d ) that I x ( f n ) ≤ α 4 ( d ) = d − 5 4 + o (1) , where α 4 ( d ) denotes the probabilit y of the four-arm even t up to di stance d . See Chapter I I. The statemen t α 4 ( R ) = R − 5 / 4+ o (1) implies that for any  > 0, there exists a constan t C = C  , suc h that for all R ≥ 1, α 4 ( R ) ≤ C R − 5 / 4+  . The ab o v e b ound gives us a v ery go o d control on the influences of the p oin ts in the bulk of the domain (i.e. the p oints far from the b oundary). Indeed, for any fixed δ > 0, let ∆ δ n b e the set of hexagons in R n whic h are at distance at least δ n from ∂ R n . Most of the p oin ts in R n (except a prop ortion O ( δ ) of these) lie in ∆ δ n , and for any suc h point x ∈ ∆ δ n , one has by the ab ov e argumen t I x ( f n ) ≤ α 4 ( δ n ) ≤ C ( δ n ) − 5 / 4+  ≤ C δ − 5 / 4 n − 5 / 4+  . (VI.1) Therefore, the con tribution of these p oints to H ( f n ) = P k I k ( f n ) 2 is b ounded b y O ( n 2 )( C δ − 5 / 4 n − 5 / 4+  ) 2 = O ( δ − 5 / 2 n − 1 / 2+2  ). As n → ∞ , this go es to zero p olynomially fast. Since this estimate concerns “almost” all p oints in R n , it seems w e are close to pro ving the BKS criterion. 1. INFLUENCES OF CR OSSING EVENTS 59 1.3 Influence of the b oundary Still, in order to complete the ab o v e analysis, one has to estimate what the influence of the p oin ts near the b oundary is. The main difficult y here is that if x is close to the b oundary , the probability for x to b e piv otal is not related anymore to the ab o v e four-arm even t. Think of the ab ov e figure when d gets very small compared to n . One has to distinguish tw o cases: • x is close to a c orner . This will corresp ond to a two-arm even t in a quarter-plane. • x is close to an e dge . This inv olv es the thr e e-arm ev en t in the half-plane H . Before detailing how to estimate the influence of p oin ts near the b oundary , let us start b y giving the necessary background on the inv olv ed critical exp onents. The t w o-arm and three-arm even ts in H . F or these particular ev en ts, it turns out that the critical exp onents are kno wn to b e universal : they are tw o of the very few crit- ical exp onents whic h are known also on the square lattice Z 2 . The deriv ations of these t yp es of exp onen ts do not rely on SLE technology but are “elemen tary”. Therefore, in this discussion, w e will consider b oth lattices T and Z 2 . The thr e e-arm even t in H corre- sp onds to the even t that there are three arms (t w o op en arms and one ‘closed’ arm in the dual) go- ing from 0 to distance R and suc h that they remain in the upper half- plane. See the figure for a self- explanatory definition. The two- arm even t corresp onds to just hav- ing one op en and one closed arm. R T or Z 2 Let α + 2 ( R ) and α + 3 ( R ) denote the probabilities of these even ts. As in c hapter I I, let α + 2 ( r , R ) and α + 3 ( r , R ) b e the natural extensions to the ann ulus case (i.e. the probabilit y that these ev en ts are satisfied in the ann ulus b et w een radii r and R in the upp er half- plane). W e will rely on the following result, which go es back as far as w e know to M. Aizenman. See [W er07] for a pro of of this result. Prop osition VI.1. Both on the triangular lattic e T and on Z 2 , one has that α + 2 ( r , R )  ( r /R ) and α + 3 ( r , R )  ( r /R ) 2 . Note that, in these sp e cial c ases, ther e ar e no o (1) c orr e ction terms in the exp onent. The pr ob abilities ar e in this c ase known up to c onstants. 60 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION The t w o-arm even t in the quarter-plane. In this case, the corresp onding exp onen t is unfortunately not kno wn on Z 2 , so w e will need to do some work here in the next section, where w e will prov e noise sensitivity of p ercolation crossings on Z 2 . The two-arm even t in a corner corresp onds to the ev en t illustrated on the follo wing picture. W e will use the following prop osition: Prop osition VI.2 ([SW01]) . If α ++ 2 ( R ) de- notes the pr ob ability of this event, then α ++ 2 ( R ) = R − 2+ o (1) , and with the obvious notations α ++ 2 ( r , R ) = ( r /R ) 2+ o (1) . R T No w, bac k to our study of influences, w e are in go o d shap e (at least for the triangular lattice) since the t w o critical exp onen ts arising from the b oundary effects are larger than the bulk exp onent 5 / 4. This means that it is less likely for a p oin t near the b oundary to b e pivotal than for a p oint in the bulk. Therefore in some sense the b oundary helps us here. More formally , summarizing the ab o v e facts, for any  > 0, there is a constant C = C (  ) such that for any 1 ≤ r ≤ R , max { α 4 ( r , R ) , α + 3 ( r , R ) , α ++ 2 ( r , R ) } ≤ C ( r /R ) 5 4 −  . (VI.2) No w, if x is some hexagon in R n , let n 0 b e the distance to the closest edge of ∂ R n and let x 0 b e the p oin t on ∂ R n suc h that dist( x, x 0 ) = n 0 . Next, let n 1 ≥ n 0 b e the distance from x 0 to the closest corner and let x 1 b e this closest corner. It is easy to see that for x to b e pivotal for f n , the follo wing ev en ts all ha v e to b e satisfied: • The four-arm even t in the ball of radius n 0 around x . • The H -three-arm even t in the annulus centered at x 0 of radii 2 n 0 and n 1 . • The corner-t w o-arm ev en t in the annulus centered at x 1 of radii 2 n 1 and n . By indep endence on disjoin t sets, one thus concludes that I x ( f n ) ≤ α 4 ( n 0 ) α + 3 (2 n 0 , n 1 ) α ++ 2 (2 n 1 , n ) ≤ O (1) n − 5 / 4+  . 2. THE CASE OF Z 2 PER COLA TION 61 1.4 Noise sensitivit y of crossing ev en ts This uniform b ound on the influences ov er the whole domain R n enables us to conclude that the BKS criterion is indeed verified. Indeed, H ( f n ) = X x ∈ R n I x ( f n ) 2 ≤ C n 2 ( n − 5 / 4+  ) 2 = C n − 1 / 2+2  , (VI.3) where C = C ( a, b,  ) is a univ ersal constant. By taking  < 1 / 4, this giv es us the desired p olynomial deca y on H ( f n ), whic h b y Prop osition V.5) implies Theorem VI.3 ([BKS99]) . The se quenc e of p er c olation cr ossing events { f n } on T is noise sensitive. W e will give some other consequences (for example, to sharp thresholds) of the ab o v e analysis on the influences of the crossing even ts in a later section. 2 The case of Z 2 p ercolation Let R n denote similarly the Z 2 rectangle closest to [0 , a · n ] × [0 , b · n ] and let f n b e the corresp onding left-righ t crossing even t (so here this corresp onds exactly to example 7). Here one has to face tw o main difficulties: • The main one is that due to the missing ingredien t of c onformal invarianc e , one do es not ha v e at our disp osal the v alue of the four-arm critical exp onen t (whic h is of course b elieved to b e 5 / 4). In fact, ev en the existenc e of a critical exp onen t is an op en problem. • The second difficulty (also due to the lack of conformal in v ariance) is that it is no w slightly harder to deal with b oundary issues. Indeed, one can still use the ab o v e b ounds on α + 3 whic h are universal , but the exp onen t 2 for α ++ 2 is not known for Z 2 . So this requires some more analysis. Let us start by taking care of the boundary effects. 2.1 Handling the b oundary effect What w e need to do in order to carry through the ab ov e analysis for Z 2 is to ob- tain a reasonable estimate on α ++ 2 . F ortunately , the follo wing b ound, which follo ws immediately from Prop osition VI.1, is sufficient. α ++ 2 ( r , R ) ≤ O (1) r R . (VI.4) No w let e b e an edge in R n . W e wish to b ound from ab o v e I e ( f n ). W e will use the same notation as in the case of the triangular lattice: recall the definitions of n 0 , x 0 , n 1 , x 1 there. 62 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION W e obtain in the same w a y I e ( f n ) ≤ α 4 ( n 0 ) α + 3 (2 n 0 , n 1 ) α ++ 2 (2 n 1 , n ) . (VI.5) A t this p oin t, w e need another universal exp onent, which go es back also to M. Aizenman: Theorem VI.4 (M. Aizenman, see [W er07]) . L et α 5 ( r , R ) denote the pr ob ability that ther e ar e 5 arms (with four of them b eing of ‘alternate c olors’). Then ther e ar e some universal c onstants c, C > 0 such that b oth for T and Z 2 , one has for al l 1 ≤ r ≤ R , c  r R  2 ≤ α 5 ( r , R ) ≤ C  r R  2 . This result allo ws us to get a low er b ound on α 4 ( r , R ). Indeed, it is clear that α 4 ( r , R ) ≥ α 5 ( r , R ) ≥ Ω(1) α + 3 ( r , R ) . (VI.6) In fact, one can obtain the follo wing b etter lo w er bound on α 4 ( r , R ) whic h we will need later. Lemma VI.5. Ther e exists some  > 0 and some c onstant c > 0 such that for any 1 ≤ r ≤ R , α 4 ( r , R ) ≥ c ( r /R ) 2 −  . Pro of. There are several wa ys to see wh y this holds, none of them b eing either very hard or v ery easy . One of them is to use Reimer’s inequalit y (see [Gri99]) which in this case w ould imply that α 5 ( r , R ) ≤ α 1 ( r , R ) α 4 ( r , R ) . (VI.7) The RSW Theorem I I.1 can b e used to show that α 1 ( r , R ) ≤ ( r /R ) α for some p ositive α . By Theorem VI.4, we are done. [See [[GPS10], Section 2.2 as well as the app endix] for more on these b ounds.] Com bining (VI.5) with (VI.6), one obtains I e ( f n ) ≤ O (1) α 4 ( n 0 ) α 4 (2 n 0 , n 1 ) α ++ 2 (2 n 1 , n ) ≤ O (1) α 4 ( n 1 ) n 1 n , where in the last inequality w e used quasi-multiplicativit y (Prop osition I I.3) as w ell as the b ound giv en b y (VI.4). 2. THE CASE OF Z 2 PER COLA TION 63 Recall that we w an t an upp er b ound on H ( f n ) = P I e ( f n ) 2 . In this sum o v er edges e ∈ R n , let us divide the set of edges into dyadic ann uli centered around the 4 cor- ners as in the next picture. bn 2 k 2 k +1 Notice that there are O (1)2 2 k edges in an annulus of radius 2 k . This enables us to b ound H ( f n ) as follo ws: X e ∈ R n I e ( f n ) 2 ≤ O (1) log 2 n + O (1) X k =1 2 2 k  α 4 (2 k ) 2 k n  2 ≤ O (1) 1 n 2 X k ≤ log 2 n + O (1) 2 4 k α 4 (2 k ) 2 . (VI.8) It no w remains to obtain a go o d upp er b ound on α 4 ( R ), for all R ≥ 1. 2.2 An upp er b ound on the four-arm ev en t in Z 2 This turns out to b e a rather non-trivial problem. Recall that w e obtained an easy lo w er b ound on α 4 using α 5 (and Lemma VI.5 strengthens this low er b ound). F or an upp er b ound, completely differen t ideas are required. On Z 2 , the follo wing estimate is a v ailable for the 4-arm even t. Prop osition VI.6. F or critic al p er c olation on Z 2 , ther e exists c onstants , C > 0 such that for any R ≥ 1 , one has α 4 (1 , R ) ≤ C  1 R  1+  . Before discussing where such an estimate comes from, let us see that it indeed implies a p olynomial deca y for H ( f n ). Recall equation (VI.8). Plugging in the ab ov e estimate, this gives us X e ∈ R n I e ( f n ) 2 ≤ O (1) 1 n 2 X k ≤ log 2 n + O (1) 2 4 k (2 k ) − 2 − 2  ≤ O (1) 1 n 2 n 2 − 2  = O (1) n − 2  , 64 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION whic h implies the desired polynomial deca y and th us the fact that { f n } is noise sensitiv e b y Prop osition V.5). Let us now discuss differen t approac hes whic h enable one to pro v e Prop osition VI.6. (a) Kesten prov ed implicitly this estimate in his celebrated pap er [Kes87]. His main motiv ation for such an estimate was to obtain b ounds on the corresp onding critical exp onen t which gov erns the so-called critic al length . (b) In [BKS99], in order to prov e noise sensitivity of p ercolation using their criterion on H ( f n ), the authors referred to [Kes87], but they also gav e a completely different approac h whic h also yields this estimate. Their alternativ e approac h is v ery nice: finding an upp er b ound for α 4 ( R ) is related to finding an upp er b ound for the influences for crossings of an R × R b o x. F or this, they noticed the follo wing nice phenomenon: if a monotone function f happ ens to b e v ery little correlated with ma jorit y , then its influences ha v e to be small. The pro of of this phenomenon uses for the first time in this con text the concept of “randomized algorithms”. F or more on this approach, see Chapter VI I I, which is devoted to these types of ideas. (c) In [SS10b], the concept of randomized algorithms is used in a more p ow erful wa y . See again Chapter VI I I. In this chapter, w e provide a pro of of this estimate in Prop osition VI I I.8. R emark VI.1 . It turns out that that a multi-scale v ersion of Prop osition VI.6 stating that α 4 ( r , R ) ≤ C  r R  1+  is also true. How ev er, none of the three arguments given ab o v e seem to prov e this stronger version. A proof of this stronger v ersion is giv en in the appendix of [SS10a]. Since this multi-scale version is not needed until Chapter X, w e stated here only the weak er version. 3 Some other consequences of our study of influ- ences In the previous sections, we handled the b oundary effects in order to chec k that H ( f n ) indeed deca ys p olynomially fast. Let us list some related results implied b y this analysis. 3.1 Energy sp ectrum of f n W e start by a straigh tforw ard observ ation: since the f n are monotone, we ha v e by Prop osition IV.4 that b f n ( { x } ) = 1 2 I x ( f n ) , 3. SOME OTHER CONSEQUENCES OF OUR STUDY OF INFLUENCES 65 for any site x (or edge e ) in R n . Therefore, the b ounds we obtained on H ( f n ) imply the follo wing con trol on the first la y er of the energy sp ectrum of the crossing ev en ts { f n } . Corollary VI.7. L et { f n } b e the cr ossing events of the r e ctangles R n . • If we ar e on the triangular lattic e T , then we have the b ound E f n (1) = X | S | =1 b f n ( S ) 2 ≤ n − 1 / 2+ o (1) . • On the squar e lattic e Z 2 , we end up with the we aker estimate E f n (1) ≤ C n −  , for some , C > 0 . 3.2 Sharp threshold of p ercolation The ab ov e analysis gav e an upp er b ound on P k I k ( f n ) 2 . As we ha v e seen in the first c hapters, the total influence I ( f n ) = P k I k ( f n ) is also a very in teresting quantit y . Recall that, by Russo’s formula, this is the quan tit y which sho ws “ho w sharp” the threshold is for p 7→ P p [ f n = 1]. The ab o v e analysis allows us to prov e the following. Prop osition VI.8. Both on T and Z 2 , one has I ( f n )  n 2 α 4 ( n ) . In p articular, this shows that on T that I ( f n )  n 3 / 4+ o (1) . R emark VI.2 . Since f n is defined on {− 1 , 1 } O ( n 2 ) , note that the Ma jority function defined on the same hypercub e has a muc h sharp er threshold than the p ercolation crossings f n . Pro of. W e first deriv e an upp er b ound on the total influence. In the same vein (i.e., using dyadic annuli and quasi-multiplicativit y) as w e derived (VI.8) and with the same notation one has I ( f n ) = X e I e ( f n ) ≤ X e O (1) α 4 ( n 1 ) n 1 n ≤ O (1) 1 n X k ≤ log 2 n + O (1) 2 3 k α 4 (2 k ) . 66 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION No w, and this is the main step here, using quasi-m ultiplicativit y one has α 4 (2 k ) ≤ O (1) α 4 ( n ) α 4 (2 k ,n ) , whic h giv es us I ( f n ) ≤ O (1) α 4 ( n ) n X k ≤ log 2 n + O (1) 2 3 k 1 α 4 (2 k , n ) ≤ O (1) α 4 ( n ) n X k ≤ log 2 n + O (1) 2 3 k n 2 2 2 k since α 4 ( r , R ) ≥ α 5 ( r , R )  ( r /R ) − 2 ≤ O (1) n α 4 ( n ) X k ≤ log 2 n + O (1) 2 k ≤ O (1) n 2 α 4 ( n ) as desired. F or the low er bound on the total influence, we pro ceed as follo ws. One obtains a lo w er bound by just summing o v er the influences of p oints whose distance to the b oundary is at least n/ 4. It w ould suffice if we knew that for such edges or hexagons, the influence is at least a constant times α 4 ( n ). This is in fact kno wn to b e true. It is not very inv olv ed and is part of the folklor e results in p ercolation. Ho w ev er, it still w ould lead us to o far from our topic. The needed techn ique is kno wn under the name of separation of arms and is clearly related to the statemen t of quasi-m ultiplicativit y . See [W er07] for more details. 4 Quan titativ e noise sensitivit y In this c hapter, w e hav e prov ed that the sequence of crossing ev en ts { f n } is noise sensitiv e. This can b e roughly translated as follows: for an y fixed level of noise  > 0, as n → ∞ , the large scale clusters of ω in the windo w [0 , n ] 2 are asymptotically indep enden t of the large clusters of ω  . R emark VI.3 . Note that this picture is correct, but in order to mak e it rigorous, this w ould require some work, since so far w e only work ed with left-righ t crossing even ts. The non-trivial step here is to pro ve that in some sense, in the scaling limit n → ∞ , an y macroscopic prop erty concerning percolation (e.g., diameter of clusters) is measurable with resp ect to the σ -algebra generated by the crossing ev en ts. This is a rather subtle problem since we need to mak e precise what kind of information we k eep in what w e call the “scaling limit” of p ercolation (or subsequential scaling limits in the case of Z 2 ). An example of something which is not present in the scaling limit is whether one has more open sites than closed ones since by noise sensitivity w e kno w that this is asymptotically uncorrelated with crossing ev en ts. W e will not need to discuss these notions of scaling limits more in these lecture notes, since the fo cus is mainly on the discrete mo del itself including the mo del of dynamical p ercolation whic h is presented at the end of these lecture notes. 4. QUANTIT A TIVE NOISE SENSITIVITY 67 A t this stage, a natural question to ask is to what extent the p ercolation picture is sensitiv e to noise. In other words, can we let the noise  =  n go to zero with the “size of the system” n , and yet keep this indep endence of large scale structures b etw een ω and ω  n ? If y es, can we give quantitative estimates on how fast the noise  =  n ma y go to zero? One can state this question more precisely as follows. Question VI.1. If { f n } denote our left-right cr ossing events, for which se quenc es of noise-levels {  n } do we have lim n →∞ Co v[ f n ( ω ) , f n ( ω  n )] = 0 ? The purp ose of this section is to briefly discuss this question based on the results w e ha v e obtained so far. 4.1 Link with the energy sp ectrum of { f n } It is an exercise to show that Question VI.1 is essentially equiv alent to the following one. Question VI.2. F or which se quenc es { k n } going to infinity do we have k n X m =1 E f n ( m ) = X 1 ≤| S |≤ k n b f n ( S ) 2 − → n →∞ 0 ? Recall that w e hav e already obtained some relev an t information on this question. Indeed, we ha v e pro v ed in this chapter that H ( f n ) = P x I x ( f n ) 2 deca ys p olynomially fast to w ards 0 (b oth on Z 2 and T ). Therefore Prop osition V.5 tells us that for some constan t c > 0, one has for b oth T and Z 2 that X 1 ≤| S |≤ c log n b f n ( S ) 2 → 0 . (VI.9) Therefore, back to our original question VI.1, this gives us the following quan titativ e statemen t: if the noise  n satisfies  n  1 log n , then f n ( ω ) and f n ( ω  n ) are asymptotically indep enden t. 4.2 Noise stabilit y regime Of course, one cannot b e to o demanding on the rate of decay of {  n } . F or example if  n  1 n 2 , then in the window [0 , n ] 2 , with high probability , the configurations ω and ω  n are iden tical. This brings us to the next natural question concerning the noise stability r e gime of crossing even ts. 68 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION Question VI.3. L et { f n } b e our se quenc e of cr ossing events. F or which se quenc es {  n } do we have P  f n ( ω ) 6 = f n ( ω  n )  − → n →∞ 0 ? It is an exercise to sho w that this question is essen tially equiv alen t to the following one. F or which sequences { k n } do we ha v e X | S | >k n b f n ( S ) 2 → 0 ? Using the estimates of the presen t chapter, one can give the follo wing non-trivial b ound on the noise stability regime of { f n } . Prop osition VI.9. Both on Z 2 and T , if  n = o  1 n 2 α 4 ( n )  , then P  f n ( ω ) 6 = f n ( ω  n )  − → n →∞ 0 On the triangular grid, using the critic al exp onent, this gives us a b ound of n − 3 / 4 on the noise stability r e gime of p er c olation. Pro of. Let {  n } b e a sequence satisfying the ab o v e assumption. There are O ( n 2 ) bits concerned. F or simplicity , assume that there are exactly n 2 bits. Let us order these in some arbitrary w a y: { x 1 , . . . , x n 2 } (or on Z 2 , { e 1 , . . . , e n 2 } ). Let ω = ω 0 = ( x 1 , . . . , x n 2 ) b e sampled according to the uniform measure. Recall that the noised configuration ω  n is pro duced as follows: for eac h i ∈ [ n 2 ], resample the bit x i with probability  n , indep enden tly of ev erything else, obtaining the bit y i . (In particular y i 6 = x i with probabilit y  n / 2). No w for eac h i ∈ [ n 2 ] define the intermediate configuration ω i := ( y 1 , . . . , y i , x i +1 , . . . , x n 2 ) Notice that for each i ∈ [ n 2 ], ω i is also sampled according to the uniform measure and one has for each i ∈ { 1 , . . . , n 2 } that P  f n ( ω i − 1 ) 6 = f n ( ω i )  = (  n / 2) I x i ( f n ) . 4. QUANTIT A TIVE NOISE SENSITIVITY 69 Summing o v er all i , one obtains P  f n ( ω ) 6 = f n ( ω  n )  = P  f n ( ω 0 ) 6 = f n ( ω n 2 )  ≤ n 2 − 1 X i =0 P  f n ( ω i ) 6 = f n ( ω i +1 )  = (  n / 2) n 2 X i =1 I x i ( f n ) = (  n / 2) I ( f n ) ≤  n O (1) n 2 α 4 ( n ) b y Prop osition VI.8, whic h concludes the pro of. 4.3 Where do es the sp ectral mass lies? Prop osition VI.9 (together with Exercise IX.2 in Chapter IX) implies that the F ourier co efficien ts of { f n } satisfy X | S | n 2 α 4 ( n ) b f n ( S ) 2 − → n →∞ 0 . (VI.10) F rom Lemma VI.5, w e know that ev en on Z 2 , n 2 α 4 ( n ) is larger than n  for some exp onen t  > 0. Combining the estimates on the sp ectrum that we achiev ed so far (equations (VI.9) and (VI.10)), w e see that in order to localize the sp ectral mass of { f n } , there is still a missing gap. See Figure VI.1. F or our later applications to the mo del of dynamical p ercolation (in the last chapter of these lecture notes), a b etter understanding of the noise sensitivit y of p ercolation than the “logarithmic” control we ac hiev ed so far will b e needed. 70 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION k E f n ( k ) := P | S | = k c f n ( S ) 2 c log n n 3 / 4+ o (1) Where is the Sp ec- tral mass of f n ? . . . Figure VI.1: This picture summarizes our presen t knowledge of the energy spectrum of { f n } on the triangular lattice T . Muc h remains to b e understo o d to know where, in the range [Ω(log n ) , n 3 / 4+ o (1) ], the sp ectral mass lies. This question will b e analyzed in the follo wing c hapters. Exercise sheet on c hapter VI Instead of b eing the usual exercise sheet, this page will b e devoted to a single Problem whose goal will b e to do “hands-on” computations of the first la y ers of the energy sp ectrum of the p ercolation crossing ev en ts f n . Recall from Prop osition IV.1 that a sequence of Bo olean functions { f n } is noise sensitiv e if and only if for an y fixed k ≥ 1, k X m =1 X | S | = m ˆ f n ( S ) 2 = k X m =1 E f n ( m ) − → n →∞ 0 . In the present chapter, w e obtained (using Prop osition IV.4) that this is indeed the case for k = 1. The purp ose here is to c hec k b y simple combinatorial arguments (without relying on hypercontractivit y) that it is still the case for k = 2 and to con vince ourselv es that it work s for all lay ers k ≥ 3. T o start with, w e will simplify our task by working on the torus Z 2 /n Z 2 . This has the v ery nice adv antage that there are no b oundary issues here. Energy sp ectrum of crossing ev en ts on the torus (study of the first la y ers) Let T n b e either the square grid torus Z 2 /n Z 2 or the triangular grid torus T /n T . Let f n b e the indicator of the even t that there is an op en circuit along the first co ordinate of T n . 1. Using RSW, prov e that there is a constant c > 0 such that for all n ≥ 1, c ≤ P  f n = 1  ≤ 1 − c . (In other w ords, { f n } is non-degenerate.) 2. Sho w that for all edges e (or sites x ) in T n I e ( f n ) ≤ α 4 ( n 2 ) . 71 72 CHAPTER VI. FIRST EVIDENCE OF NOISE SENSITIVITY OF PERCOLA TION 3. Chec k that the BKS criterion (ab out H ( f n )) is satisfied. Therefore { f n } is noise- sensitiv e F rom now on, one w ould like to forget ab out the BKS Theorem and try to do some hands-on computations in order to get a feeling wh y most frequencies should b e large. 4. Sho w that if x, y are t w o sites of T n (or similarly if e, e 0 are tw o edges of T n ), then | ˆ f ( { x, y } ) | ≤ 2 P  x and y are pivotal p oints  . Do es this result hold for general Bo olean functions? 5. Sho w that if d := | x − y | , then P  x and y are pivotal p oints  ≤ O (1) α 4 ( n/ 2) 2 α 4 ( d 2 , n 2 ) . (Hin t: use Prop osition I I.3.) 6. On the square lattice Z 2 , b y carefully summing o v er all edges e, e 0 ∈ T n × T n , sho w that E f n (2) = X | S | =2 b f n ( S ) 2 ≤ O (1) n −  , for some exp onen t  > 0. Hin t: you migh t decomp ose the sum in a dyadic w a y (as we did many times in the presen t section) dep ending on the mutual distance d ( e, e 0 ). 7. On the triangular grid, what exp onen t do es it giv e for the decay of E f n (2)? Com- pare with the decay w e found in Corollary VI.7 ab out the decay of the first lay er E f n (1) (i.e. k = 1). See also Lemma V.6 in this regard. Discuss this. 8. F or T , what do you exp ect for higher (fixed) v alues of k ? (I.e. for E f n ( k ), k ≥ 3) ? 9. (Quite har d) T ry to obtain a nonrigorous combinatorial argument similar to the one ab o v e in the particular case k = 2, that for any fixed lay er k ≥ 1, E f n ( k ) − → n →∞ 0 . This w ould give us an alternativ e pro of of noise sensitivity of p ercolation (at least in the case of the torus T n ) not relying on Theorem I.5. Observ e that one can do similar things for rectangles but then one has to deal with b oundary issues. Chapter VI I Anomalous fluctuations In this lecture, our goal is to extend the tec hnology we used to prov e the KKL Theorems on influences and the BKS Theorem on noise sensitivity to a slightly differen t context: the study of fluctuations in first passage p ercolation . 1 The mo del of first passage p ercolation Let us first explain what the mo del is. Let 0 < a < b b e tw o p ositiv e n um b ers. W e define a random metric on the graph Z d , d ≥ 2 as follo ws. Indep endently for eac h edge e ∈ E d , fix its length τ e to b e a with probabilit y 1/2 and b with probability 1 / 2. This is represen ted by a uniform configuration ω ∈ {− 1 , 1 } E d . This pro cedure induces a well-defined (random) metric dist ω on Z d in the usual fashion. F or any vertices x, y ∈ Z d , let dist ω ( x, y ) := inf paths γ = { e 1 , . . . , e k } connecting x → y n X τ e i ( ω ) o . R emark VI I.1 . In greater generality , the lengths of the edges are i.i.d. non-negative random v ariables, but here, following [BKS03], we will restrict ourselves to the ab ov e uniform distribution on { a, b } to simplify the exp osition; see [BR08] for an extension to more general laws. One of the main goals in first passage p ercolation is to understand the large-scale prop erties of this random metric space. F or example, for an y T ≥ 1, one ma y consider the (random) ball B ω ( x, T ) := { y ∈ Z d : dist ω ( x, y ) ≤ T } . T o understand the name first p assage p er c olation , one can think of this mo del as follo ws. Imagine that water is pump ed in at v ertex x , and that for each edge e , it takes τ e ( ω ) units of time for the w ater to trav el across the edge e . Then, B ω ( x, T ) represen ts the region of space that has b een wetted by time T . 73 74 CHAPTER VI I. ANOMALOUS FLUCTUA TIONS Figure VI I.1: A sample of a wetted region at time T , i.e. B ω ( x, T ), in first passage p ercolation. An application of subadditivity shows that the renormalized ball 1 T B ω (0 , T ) con- v erges as T → ∞ tow ards a deterministic shap e which can in certain cases b e com- puted explicitly . This is a kind of “geometric law of large num bers”. Whence the natural question: Question VI I.1. Describ e the fluctuations of B ω (0 , T ) ar ound its asymptotic deter- ministic shap e. This question has received tremendous interest in the last 15 years or so. It is widely b elieved that these fluctuations should b e in some sense “universal”. More precisely , the b ehavior of B ω (0 , T ) around its limiting shap e should not dep end on the “microscopic” particularities of the model such as the la w on the edges lengths but only on the dimension d of the underlying graph. The shape itself depends on the other hand of course on the microscopic parameters, in the same w a y as the critical p oin t dep ends on the graph in p ercolation. In the t w o-dimensional case, using very b eautiful combinatorial bijections with ran- dom matrices, certain cases of dir e cte d last passage p ercolation (where the law on the edges is taken to be geometric or exp onential) ha v e b een understo o d very deeply . F or example, it is kno wn (see [Joh00]) that the fluctuations of the ball of radius n (i.e. the p oin ts whose last passage times are b elow n ) around n times its asymptotic deter- ministic shap e are of order n 1 / 3 and the law of these fluctuations prop erly renormalized follo ws the T racy-Widom distribution. V ery in terestingly , the fluctuations of the largest eigen v alue of GUE ensembles also follow this distribution. 2. ST A TE OF THE AR T 75 2 State of the art Returning to our initial mo del of (non-directed) first passage p ercolation, it is thus conjectured that, for dimension d = 2, fluctuations are of order n 1 / 3 follo wing a T racy- Widom Law. Still, the curren t state of understanding of this mo del is far from this conjecture. Kesten first prov ed that the fluctuations of the ball of radius n are at most √ n (this did not y et exclude a p ossible Gaussian b ehavior with Gaussian scaling). Benjamini, Kalai and Schramm then strengthened this result b y showing that the fluctuations are sub-Gaussian. This is still far from the conjectured n 1 / 3 -fluctuations, but their approac h has the great adv an tage of b eing very general; in particular their result holds in an y dimension d ≥ 2. Let us no w state their main theorem concerning the fluctuations of the metric dist. Theorem VI I.1 ([BKS03]) . F or al l a, b, d , ther e exists an absolute c onstant C = C ( a, b, d ) such that in Z d , V ar(dist ω (0 , v )) ≤ C | v | log | v | for any v ∈ Z d , | v | ≥ 2 . T o k eep things simple in these notes, we will only pro v e the analogous statement on the torus where one has more symmetries and inv ariance to play with. 3 The case of the torus Let T d m b e the d -dimensional torus ( Z /m Z ) d . As in the ab o v e lattice mo del, indep en- den tly for each edge of T d m , w e c ho ose its length to b e either a or b equally lik ely . W e are in terested here in the smallest length among all closed paths γ “winding” around the torus along the first co ordinate Z /m Z (i.e. those paths γ whic h when pro jected onto the first co ordinate ha v e winding num b er one). In [BKS03], this is called the shortest cir cumfer enc e . F or any configuration ω ∈ { a, b } E ( T d m ) , this shortest circumference is denoted b y Circ m ( ω ). Theorem VI I.2 ([BKS03]) . Ther e is a c onstant C = C ( a, b ) (which do es not dep end on the dimension d ), such that v ar(Circ m ( ω )) ≤ C m log m . R emark VI I.2 . A similar analysis as the one carried out b elo w w orks in greater gen- eralit y: if G = ( V , E ) is some finite connected graph endow ed with a random metric d ω with ω ∈ { a, b } ⊗ E , then one can obtain b ounds on the fluctuations of the random diameter D = D ω of ( G, d ω ). See [BKS03, Theorem 2] for a precise statemen t in this more general con text. 76 CHAPTER VI I. ANOMALOUS FLUCTUA TIONS ( Z /m Z ) 2 γ m Figure VI I.2: The shortest geo desic along the first co ordinate for the random metric dist ω on ( Z /m Z ) 2 . Pro of. F or an y edge e , let us consider the gradient along the edge e : ∇ e Circ m . These gradien t functions hav e v alues in [ − ( b − a ) , b − a ]. By dividing our distances b y the constan t factor b − a , we can ev en assume without loss of generality that our gradien t functions ha v e v alues in [ − 1 , 1]. Doing so, w e end up b eing in a setup similar to the one we had in Chapter V. The influence of an edge e corresp onds here to I e (Circ m ) := P  ∇ e Circ m ( ω ) 6 = 0  . W e will pro v e later on that Circ m has very small influences. In other w ords, we will show that the ab ov e gradient functions hav e small supp ort, and h yp ercon tractivit y will imply the desired bound. W e ha v e thus reduced the problem to the follo wing general framework. Consider a real-v alued function f : {− 1 , 1 } n → R , such that for any v ariable k , ∇ k f ∈ [ − 1 , 1]. W e are interested in V ar( f ) and w e wan t to sho w that if “influences are small” then V ar( f ) is small. It is easy to chec k that the v ariance can b e written V ar( f ) = 1 4 X k X ∅6 = S ⊆ [ n ] 1 | S | d ∇ k f ( S ) 2 . If all the v ariables ha v e v ery small influence, then, as previously , ∇ k f should b e of high frequency . Heuristically , this should then imply that V ar( f )  X k X S 6 = ∅ d ∇ k f ( S ) 2 = X k I k ( f ) . 3. THE CASE OF THE TOR US 77 This intuition is quantified b y the follo wing lemma on the link b etw een the fluctu- ations of a real-v alued function f on Ω n and its influence vector. Lemma VI I.3. L et f : Ω n → R b e a (r e al-value d) function such that e ach of its discr ete derivatives ∇ k f , k ∈ [ n ] have their values in [ − 1 , 1] . L et I k ( f ) := P  ∇ k f 6 = 0  b e the influenc e of the k th bit. Assume that the influenc es of f ar e smal l in the sense that ther e exists some α > 0 such that for any k ∈ { 1 , . . . , n } , I k ( f ) ≤ n − α . Then ther e is some c onstant C = C ( α ) such that V ar( f ) ≤ C log n X k I k ( f ) . R emark VI I.3 . If f is Bo olean, then this follo ws from Theorem I.3 with C ( α ) = c/α with c universal. The pro of of this lemma is postp oned to the next section. In the mean time, let us sho w that in our sp ecial case of first passage percolation on the torus, the assumption on small influences is indeed v erified. Since the edge lengths are in { a, b } , the smallest con tour Circ m ( ω ) in T d m around the first co ordinate lies somewhere in [ am, bm ]. Hence, if γ is a geo desic (a path in the torus with the required winding n um b er) satisfying length( γ ) = Circ m ( ω ), then γ uses at most b a m edges. There might b e several different geo desics minimizing the circumference. Let us choose randomly one of these in an “in v arian t” wa y and call it ˜ γ . F or any edge e ∈ E ( T d m ), if, b y changing the length of e , the circumference increases, then e has to b e contained in any geo desic γ , and in particular in ˜ γ . This implies that P  ∇ e Circ m ( ω ) > 0  ≤ P  e ∈ ˜ γ  . By symmetry w e obtain that I e (Circ m ) = P  ∇ e Circ m ( ω ) 6 = 0  ≤ 2 P  e ∈ ˜ γ  . No w using the symmetries b oth of the torus T d m and of our observ able Circ m , if ˜ γ is c hosen in an appropriate in v arian t wa y (uniformly among all geo desics for instance), then it is clear that all the “v ertical” edges (meaning those edges which, when pro jected on to the first co ordinate, pro ject on to a single v ertex) ha v e the same probability to lie in ˜ γ . The same is true for the “horizontal” edges. In particular we hav e that X “vertical” edges e P  e ∈ ˜ γ  ≤ E  | ˜ γ |  ≤ b a m . Since there are at least order m d v ertical edges, the influence of eac h of these is bounded b y O (1) m 1 − d . The same is true for the horizontal edges. All together this giv es the desired assumption needed in Lemma VI I.3. Applying this lemma, w e indeed obtain that V ar(Circ m ( ω )) ≤ O (1) m log m , where the constant do es not dep end on the dimension d ; the dimension in fact helps us here, since it makes the influences smaller. 78 CHAPTER VI I. ANOMALOUS FLUCTUA TIONS R emark VI I.4 . At this p oint, w e kno w that for any edge e , I e (Circ m ) = O ( m m d ). Hence, at least in the case of the torus, one easily deduces from P oincar ´ e’s inequality the theorem b y Kesten whic h sa ys that V ar(Circ m ) = O ( m ). 4 Upp er b ounds on fluctuations in the spirit of KKL In this section, we prov e Lemma VI I.3. Pro of. Similarly as in the pro ofs of Chapter V, the pro of relies on implementing h yp ercon tractivit y in the right wa y . W e hav e that for any c , v ar( f ) = 1 4 X k X S 6 = ∅ 1 | S | d ∇ k f ( S ) 2 ≤ 1 4 X k X 0 < | S | 0 ˆ g J ( S ) 2 . T aking the exp ectation ov er ˜ ω ∈ ˜ Ω, this leads to E  ˆ g J ( ∅ ) 2  = E  k g J k 2 2  − X | S | > 0 E  ˆ g J ( S ) 2  = k g k 2 2 − X | S | > 0 E  ˆ g J ( S ) 2  b y (VI I I.2) = X | S | = k ˆ g ( S ) 2 − X | S | > 0 E  ˆ g J ( S ) 2   since g is supp orted on lev el- k co efficien ts ≤ X | S | = k E  ˆ g ( S ) 2 − ˆ g J ( S ) 2   b y restricting to lev el- k co efficien ts No w, since g J is built randomly from g b y fixing the v ariables in J = J ( ˜ ω ), and since g by definition do es not ha v e frequencies larger than k , it is clear that for any S with | S | = k we hav e ˆ g J ( S ) =  ˆ g ( S ) = ˆ f ( S ) , if S ∩ J ( ˜ ω ) = ∅ 0 , otherwise. 3. AN APPLICA TION TO NOISE SENSITIVITY OF PERCOLA TION 87 Therefore, w e obtain k E  g   J  k 2 2 = E  ˆ g J ( ∅ ) 2  ≤ X | S | = k ˆ g ( S ) 2 P  S ∩ J 6 = ∅  ≤ k g k 2 2 k δ . Com bining with (VI I I.3) completes the pro of. Prop osition IV.1 and Theorem VI I I.1 immediately imply the following corollary . Corollary VI I I.2. If the r eve alments satisfy lim n →∞ δ f n = 0 , then { f n } is noise sensitive. In the exercises, one is asked to sho w that certain sequences of Bo olean functions are noise sensitiv e b y applying the ab o v e corollary . 3 An application to noise sensitivit y of p ercolation In this section, we apply Corollary VI I I.2 to pro v e noise sensitivit y of p ercolation cross- ings. The following result giv es the necessary assumption that the revealmen ts approac h 0. Theorem VI I I.3 ([SS10b]) . L et f = f n b e the indic ator function for the event that critic al site p er c olation on the triangular grid c ontains a left to right cr ossing of our n × n b ox. Then δ f n ≤ n − 1 / 4+ o (1) as n → ∞ . F or critic al b ond p er c olation on the squar e grid, this holds with 1 / 4 r eplac e d by some p ositive c onstant a > 0 . Outline of Pro of. W e outline the argumen t only for the triangular lattice; the argu- men t for the square lattice is similar. W e first give a first attempt at a go o d algorithm. W e consider from Chapter I I the exploration path or interface from the b ottom right of the square to the top left used to detect a left right crossing. This (deterministic) algorithm simply asks the bits that it needs to kno w in order to con tin ue the interface. Observ e that if a bit is queried, it is necessarily the case that there is b oth a black and white path from next to the hexagon to the b oundary . It follo ws, from the exp onen t of 1 / 4 for the 2-arm even t in Chapter I I, that, for hexagons far from the b oundary , the probabilit y that they are rev ealed is at most R − 1 / 4+ o (1) as desired. How ever, one cannot conclude that p oints near the b oundary hav e small rev ealmen t and of course the right b ottom p oin t is alwa ys revealed. The w a y that w e mo dify the ab o v e algorithm so that all p oin ts ha v e small rev ealmen t is as follows. W e first choose a p oin t x at random from the middle third of the right 88 CHAPTER VI I I. RANDOMIZED ALGORITHMS AND NOISE SENSITIVITY side. W e then run tw o algorithms, the first one which chec ks whether there is a left righ t path from the righ t side ab ove x to the left side and the second one which chec ks whether there is a left right path from the right side b elow x to the left side. The first part is done b y lo oking at an in terface from x to the top left corner as abov e. The second part is done by lo oking at an interfac e from x to the b ottom left corner as ab ov e (but where the colors on the tw o sides of the interface need to b e swapped.) It can then b e shown with a little work (but no new conceptual ideas) that this mo dified algorithm has the desired revealmen t of at most R − 1 / 4+ o (1) as desired. One of the things that one needs to use in this analysis is the so-called one-arm half-plane exp onen t, which has a kno wn v alue of 1 / 3. See [SS10b] for details. 3.1 First quan titativ e noise sensitivity result In this subsection, we giv e our first “p olynomial b ound” on the noise sensitivity of p er- colation. This is an imp ortant step in our understanding of quantitativ e noise sensitivity of p ercolation initiated in Chapter VI. Recall that in the definition of noise s ensitivit y ,  is held fixed. Ho w ev er, as w e hav e seen in Chapter VI, it is of interest to ask if the correlations can still go to 0 when  =  n go es to 0 with n but not so fast. The techniques of the present c hapter imply the follo wing result. Theorem VI I I.4 ([SS10b]) . L et { f n } b e as in The or em VIII.3. Then, for the triangular lattic e, for al l γ < 1 / 8 , lim n →∞ E [ f n ( ω ) f n ( ω 1 /n γ )] − E [ f n ( ω )] 2 = 0 . (VI I I.4) On the squar e lattic e, ther e exists some γ > 0 with the ab ove pr op erty. Pro of. W e pro v e only the first statement; the square lattice case is handled similarly . First, (IV.3) giv es us that every n and γ , E [ f n ( ω ) f n ( ω 1 /n γ )] − E [ f n ( ω )] 2 = X k =1 E f n ( k )(1 − 1 /n γ ) k . (VI I I.5) 4. LOWER BOUNDS ON REVEALMENTS 89 Note that there are order n 2 terms in the sum. Fix γ < 1 / 8. Cho ose  > 0 so that γ +  < 1 / 8. F or large n , w e hav e that δ f n ≤ 1 /n 1 / 4 −  . The right hand side of (VI I I.5) is at most n γ + / 2 X k =1 k /n 1 / 4 −  + (1 − 1 /n γ ) n γ + / 2 b y breaking up the sum at n γ + / 2 and applying Theorems VI I I.1 and VI I I.3 to b ound the E f n ( k ) terms in the first part. The second term clearly go es to 0 while the first part also go es to 0 by the wa y  was chosen. Observ e that the pr o of of Theorem VI I I.4 immediately yields the following general result. Corollary VI I I.5. L et { f n } b e a se quenc e of Bo ole an functions on m n bits with δ ( f n ) ≤ O (1) /n β for al l n . Then for al l γ < β / 2 , we have that lim n →∞ E [ f n ( ω ) f n ( ω 1 /n γ )] − E [ f n ( ω )] 2 = 0 . (VI I I.6) 4 Lo w er b ounds on rev ealmen ts One of the goals of the present section is to sho w that one cannot hop e to reac h the conjectured 3 / 4-sensitivity exp onen t with Theorem VI I I.1. Theorem VI I I.4 told us that w e obtain asymptotic decorrelation if the noise is 1 /n γ for γ < 1 / 8. Note that this differs from the conjectured “critical exp onent” of 3 / 4 b y a factor of 6. In this section, we inv estigate the degree to whic h the 1 / 8 could p oten tially b e impro v ed and in the discussion, we will bring up an interesting op en problem. W e will also derive an in teresting general theorem giving a non trivial lo w er b ound on the revealmen t for monotone functions. W e start with the follo wing definition. Definition VI I I.2. Given a r andomize d algorithm A for a Bo ole an function f , let C ( A ) (the c ost of A ) b e the exp e cte d numb er of queries that the algorithm A makes. L et C ( f ) (the c ost of f ) b e the infimum of C ( A ) over al l r andomize d algorithms A for f . R emark VI I I.1 . (i). It is easy to see that C ( f ) is unc hanged if w e tak e the infim um o v er deterministic algorithms. (ii). Clearly nδ A ≥ C ( A ) and hence nδ f ≥ C ( f ). (iii). C ( f ) is at least the total influence I ( f ) since for any algorithm A and an y i , the ev en t that i is pivotal necessarily implies that the bit i is queried by A . The following result due to O’Donnell and Servedio ([OS07])is an essen tial impro v e- men t on the third part of the last remark. Theorem VI I I.6. L et f b e a monotone Bo ole an function mapping Ω n into {− 1 , 1 } . Then C ( f ) ≥ I ( f ) 2 and henc e δ f ≥ I ( f ) 2 /n . 90 CHAPTER VI I I. RANDOMIZED ALGORITHMS AND NOISE SENSITIVITY Pro of. Fix an y randomized algorithm A for f . Let J = J A b e the random set of bits queried b y A . W e then ha v e I ( f ) = E [ X i f ( ω ) ω i ] = E [ f ( ω ) X i ω i I { i ∈ J } ] ≤ p E [ f ( ω ) 2 ] s E [( X i ω i I { i ∈ J } ) 2 ] where the first equalit y uses monotonicity (recall Prop osition IV.4) and then the Cauc h y- Sc h w arz inequality is used. W e now b ound the first term by 1. F or the second moment inside the second square ro ot, the sum of the diagonal terms yields E [ | J | ] while the cross terms are all 0 since for i 6 = j , E [ ω i I { i ∈ J } ω j I { j ∈ J } ] = 0 as can b e seen b y breaking up the sum dep ending on whether i or j is queried first. This yields the result. Returning to our even t f n of p ercolation crossings, since the sum of the influences is n 3 / 4+ o (1) , Theorem VI I I.6 tells us that δ f n ≥ n − 1 / 2+ o (1) . It follows from the metho d of pro of in Theorem VI I I.4 that Theorem VI I I.1 cannot impro v e the result of Theorem VI I I.4 past γ = 1 / 4 whic h is still a factor of 3 from the critical v alue 3 / 4. Of course, one could in v estigate the degree to which Theorem VI I I.1 itself could b e improv ed. Theorem VI I I.3 tells us that there are algorithms A n for f n suc h that C ( A n ) ≤ n 7 / 4+ o (1) . On the other hand, Theorem VI I I.6 tell us that it is necessarily the case that C ( A ) ≥ n 6 / 4+ o (1) . Op en Question: Find the smallest σ suc h that there are algorithms A n for f n with C ( A n ) ≤ n σ . (W e know σ ∈ [6 / 4 , 7 / 4].) W e mention another inequalit y relating revealmen t with influences whic h is a con- sequence of the results in [OSSS05]. Theorem VI I I.7. L et f b e a Bo ole an function mapping Ω n into {− 1 , 1 } . Then δ f ≥ V ar( f ) / ( n max i I i ( f )) It is interesting to compare Theorems VI I I.6 and VI I I.7. Assuming V ar( f ) is of order 1, and all the influences are of order 1 /n α , then it is easy to c hec k that Theorem VI I I.6 gives a b etter b ound when α < 2 / 3 and Theorem VI I I.7 gives a b etter b ound when α > 2 / 3. F or crossings of p ercolation, where α should b e 5 / 8, it is better to use Theorem VI I I.6 rather than VI I I.7. Finally , there are a num b er of interesting results concerning rev ealmen t obtained in the pap er [BSW05]. F our results are as follo ws. 1. If f is reasonably balanced on n bits, then the rev ealmen t is at least of order 1 /n 1 / 2 . 2. There is a reasonably balanced function on n bits whose revealmen t is at most O (1)(log n ) /n 1 / 2 . 3. If f is reasonably balanced on n bits and is monotone, then the revealmen t is at least of order 1 /n 1 / 3 . 4. There is a reasonably balanced monotone function on n bits whose rev ealmen t is at most O (1)(log n ) /n 1 / 3 . W e finally end this section b y giving one more reference whic h gives an interesting connection b et w een p ercolation, algorithms and game theory; see [PSSW07]. 5. AN APPLICA TION TO A CRITICAL EXPONENT 91 5 An application to a critical exp onen t In this section, we show ho w Theorem VI I I.1 or in fact Theorem VI I I.6 can b e used to sho w that the 4-arm exp onent is strictly larger than 1; recall that with SLE technology , this can b e sho wn for the triangular lattice. Prop osition VI I I.8. Both on the triangular lattic e T and on Z 2 , ther e exists  0 > 0 such that α 4 ( R ) ≤ 1 /R 1+  0 W e will assume the separation of arms result mentioned earlier in Chapter VI whic h sa ys that for the ev en t f R , the influence of an y v ariable further than distance R / 10 from the b oundary , a set of v ariables that we will denote by B for bulk, is  α 4 ( R ). Pro of. Theorems VI I I.3 and VI I I.1 imply that for some a > 0, X i ˆ f R ( { i } ) 2 ≤ 1 /R a . Next, using the separation of arms as explained ab ov e, we hav e R 2 α 2 4 ( R ) ≤ O (1) X i ∈ B I 2 i . (VI I I.7) Prop osition IV.4 then yields R 2 α 2 4 ( R ) ≤ O (1 /R a ) and the result follows. Observ e that Theorem VI I I.6 could also b e used as follows. Theorem VI I I.3 implies that C ( f R ) ≤ R 2 − a for some a > 0 and then Theorem VI I I.6 yields I ( f R ) 2 ≤ R 2 − a . Exactly as in (VI I I.7), one has, again using separation of arms, that R 2 α 4 ( R ) ≤ O (1) X i ∈ B I i ≤ O (1) I ( f R ) . (VI I I.8) Altogether this giv es us R 4 α 2 4 ( R ) ≤ O (1) R 2 − a , again yielding the result. W e finally mention that it is not so strange that either of Theorems VI I I.1 or VI I I.6 can b e used here since, as the reader can easily v erify , for the case of monotone functions all of whose v ariables hav e the same influence, the case k = 1 in Theorem VI I I.1 is equiv alen t to Theorem VI I I.6. R emark VI I I.2 . W e now men tion that the pro of for the multi-scale v ersion of Prop osi- tion VI.6 is an extension of the approac h of O’Donnell and Servedio ab ov e. 92 CHAPTER VI I I. RANDOMIZED ALGORITHMS AND NOISE SENSITIVITY 6 Do es noise sensitivit y imply lo w rev ealmen t? As far as these lectures are concerned, this subsection will not connect to anything that follo ws and hence can b e viewed as tangential. It is natural to ask if the conv erse of Corollary VI I I.2 migh t b e true. A moment’s though t rev eals that example 2, P arit y , provides a coun terexample. How ev er, it is more in teresting perhaps that there is a monotone coun terexample to the conv erse which is pro vided b y example 5, Clique con tainmen t. Prop osition VI I I.9. Clique c ontainment pr ovides an example showing that the c on- verse of Cor ol lary VIII.2 is false for monotone functions. Outline of Pro of. W e first explain more precisely the size of the clique that we are lo oking for. Given n and k , let f ( n, k ) :=  n k  2 − ( k 2 ) , which is just the exp ected num ber of cliques of size k in a random graph. When k is around 2 log 2 ( n ), it is easy to chec k that f ( n, k + 1) /f ( n, k ) is o (1) as n → ∞ . F or suc h k , clearly if f ( n, k ) is small, then with high probability there is no k -clique while it can b e shown, via a second momen t t yp e argument, that if f ( n, k ) is large, then with high probability there is a k -clique. One now takes k n to b e around 2 log 2 ( n ) such that f ( n, k n ) ≥ 1 and f ( n, k n + 1) < 1. Since f ( n, k + 1) /f ( n, k ) is o (1), it follo ws with some though t from the ab o v e that the clique n um b er is concen trated on at most 2 p oin ts. F urthermore, if f ( n, k n ) is v ery large and f ( n, k n + 1) very small, then it is concentrated on one p oint. Again, see [AS00] for details. Finally , we denote the even t that the random graph on n vertices contains a clique of size k n b y A n . W e hav e already seen in one of the exercises that this example is noise sensitiv e. W e will only consider a sequence of n ’s so that A n is nondegenerate in the sense that the probabilities of this sequence stay b ounded a w a y from 0 and 1. An in teresting p oint is that there is such a sequence. Again, see [AS00] for this. T o show that the rev ealmen ts do not go to 0, it suffices to sho w that the sequence of costs (see Definition VI I I.2 and the remarks afterw ards) is Ω( n 2 ). W e prov e something stronger but, to do this, we must first give a few more definitions. Definition VI I I.3. F or a given Bo ole an function f , a witness for ω is any subset W of the variables such that the elements of ω in W determine f in the sense that for every ω 0 which agr e es with ω on W , we have that f ( ω ) = f ( ω 0 ) . The witness size of ω , denote d w ( ω ) , is the size of the smal lest witness for ω . The exp ected witness size , denote d by w ( f ) , is E ( w ( ω )) . Observ e that, for any Bo olean function f , the bits revealed b y any algorithm A for f and for any ω is alwa ys a witness for ω . It easily follows that the cost C ( f ) satisfies C ( f ) ≥ w ( f ). Therefore, in order to prov e the prop osition, it suffices to sho w that w ( f n ) = Ω( n 2 ) . (VI I I.9) 6. DOES NOISE SENSITIVITY IMPL Y LO W REVEALMENT? 93 R emark VI I I.3 . (i). The ab ov e also implies that with a fixed uniform probability , w ( ω ) is Ω( n 2 ). (ii). Of course when f n is 1, there is alw a ys a (small) witness of size  k n 2   n and so the large a v erage witness size comes from when f n is − 1. (iii). How ev er, it is not deterministically true that when f n is − 1, w ( ω ) is necessarily of size Ω( n 2 ). F or example, for ω ≡ − 1 (corresp onding to the empty graph), the witness size is o ( n 2 ) as is easily c hec k ed. Clearly the empt y graph has the smallest witness size among ω with f n = − 1. Lemma VI I I.10. L et E n b e the event that al l sets of vertic es of size at le ast . 97 n c ontains C k n − 3 . Then lim n →∞ P ( E n ) = 1 . Pro of. This follo ws, after some w ork, from the Janson inequalities. See [AS00] for details concerning these inequalities. Lemma VI I I.11. L et U b e any c ol le ction of at most n 2 / 1000 e dges in C n . Then ther e exist distinct v 1 , v 2 , v 3 such that no e dge in U go es b etwe en any v i and v j and |{ e ∈ U : e is an e dge b etwe en { v 1 , v 2 , v 3 } and { v 1 , v 2 , v 3 } c }| ≤ n/ 50 . (VI I I.10) Pro of. W e use the probabilistic metho d where we choose { v 1 , v 2 , v 3 } to b e a uniformly c hosen 3-set. It is immediate that the probabilit y that the first condition fails is at most 3 | U | /  n 2  ≤ 1 / 100. Letting Y be the n um b er of edges in the set appearing in (VI I I.10) and Y 0 b e the n um b er of U edges touching v 1 , it is easy to see that E ( Y ) ≤ 3 E ( Y 0 ) = 6 | U | /n ≤ n/ 100 where the equalit y follows from the fact that, for an y graph, the n um b er of edges is half the total degree. By Marko v’s inequality , the probability of the even t in (VI I I.10) holds with probably at least 1/2. This shows that the random 3-set { v 1 , v 2 , v 3 } satisfies the t w o stated conditions with positive probability and hence suc h a 3-set exists. By Lemma VII I.10, we hav e P ( A c n ∩ E n ) ≥ c > 0 for all large n . T o pro v e the theorem, it therefore suffices to sho w that if A c n ∩ E n o ccurs, there is no witness of size smaller than n 2 / 1000. Assume U to b e an y set of edges of size smaller than n 2 / 1000. Cho ose { v 1 , v 2 , v 3 } from Lemma VI I I.11. By the second condition in this lemma, there exists a set S of size at least . 97 n which is disjoin t from { v 1 , v 2 , v 3 } which has no U -edge to { v 1 , v 2 , v 3 } . Since E n o ccurs, S con tains a C k n − 3 , whose v ertices w e denote b y T . Since there are no U -edges b etw een T and { v 1 , v 2 , v 3 } or within { v 1 , v 2 , v 3 } (by the first condition in Lemma VI I I.11) and T is the complete graph, U cannot b e a witness since A c n o ccured. The k ey step in the pro of of Prop osition VI I I.9 is (VI I I.9). This is stated without pro of in [FKW02]; ho w ev er, E. F riedgut provided us with the ab ov e pro of. 94 CHAPTER VI I I. RANDOMIZED ALGORITHMS AND NOISE SENSITIVITY Exercise sheet of Chapter VI I I Exercise VI I I.1. Compute the rev ealmen t for Ma jorit y function on 3 bits. Exercise VI I I.2. Use Corollary VII I.2 to sho w that Examples 4 and 6, Iterated 3- Ma jorit y function and trib es, are noise sensitive. Exercise VI I I.3. F or transitive monotone functions, is there a relationship b et w een rev ealmen t and the minimal cost o v er all algorithms? Exercise VI I I.4. Sho w that for transitiv e monotone functions, Theorem VI I I.6 yields the same result as Theorem VI I I.1 do es for the case k = 1. Exercise VI I I.5. What can you say ab out the sequence of rev ealmen ts for the Iterated 3-Ma jorit y function? [It can b e sho wn that the sequence of rev ealmen ts deca ys lik e 1 /n σ for some σ but it is an open question what σ is.] Exercise VI I I.6. Y ou are giv en a sequence of Bo olean functions and told that it is not noise sensitiv e using noise  n = 1 /n 1 / 5 . What, if an ything, can y ou conclude ab out the sequence of revealmen ts δ n ? Exercise VI I I.7. Note that a consequence of Corollary VI I I.2 and the last line in Remark IV.2 is that if { f n } is a sequence of monotone functions, then, if the rev eal- men ts of { f n } go to 0, the sums of the squared influences approach 0. Show that this implication is false without the monotonicity assumption. 95 96 CHAPTER VI I I. RANDOMIZED ALGORITHMS AND NOISE SENSITIVITY Chapter IX The sp ectral sample It turns out that it is very useful to view the F ourier co efficien ts of a Bo olean function as a random subset of the input bits where the “weigh t” or “probability” of a subset is its squared F ourier co efficient. It is our understanding that it w as Gil Kalai who suggested that thinking of the sp ectrum as a random set could shed some light on the t yp es of questions we are lo oking at here. The following is the crucial definition in this c hapter. 1 Definition of the sp ectral sample Definition IX.1. Given a Bo ole an function f : Ω n → {± 1 } or { 0 , 1 } , we let the sp ectral measure ˆ Q = ˆ Q f of f b e the me asur e on subsets { 1 , . . . , n } given by ˆ Q f ( S ) := ˆ f ( S ) 2 , S ⊂ { 1 , . . . , n } . We let S f = S denote a subset of { 1 , . . . , n } chosen ac c or ding to this me asur e and c al l this the sp ectral sample . We let ˆ Q also denote the c orr esp onding exp e ctation (even when ˆ Q is not a pr ob ability me asur e). By P arsev al’s formula, the total mass of the so-defined spectral measure is X S ⊂{ 1 ,...,n } ˆ f ( S ) 2 = E  f 2  . This mak es the follo wing definition natural. Definition IX.2. Given a Bo ole an function f : Ω n → {± 1 } or { 0 , 1 } , we let the sp ectral probabilit y measure ˆ P = ˆ P f of f b e the pr ob ability me asur e on subsets of { 1 , . . . , n } given by ˆ P f ( S ) := ˆ f ( S ) 2 E [ f 2 ] , S ⊂ { 1 , . . . , n } . 97 98 CHAPTER IX. THE SPECTRAL SAMPLE Sinc e ˆ P f is just ˆ Q f up to a r enormalization factor, the sp ectral sample S f = S wil l denote as wel l a r andom subset of [ n ] sample d ac c or ding to ˆ P f . We let ˆ E f = ˆ E denote its c orr esp onding exp e ctation. R emark IX.1 . (i) Note that if f maps in to {± 1 } , then, by Parsev al’s form ula, ˆ Q f = ˆ P f while if it maps in to { 0 , 1 } , ˆ Q f will b e a subprobabilit y measure. (ii) Observ e that if ( f n ) n is a sequence of non-degenerate Bo olean functions in to { 0 , 1 } , then ˆ P f n  ˆ Q f n . (iii) There is no statistical relationship b et w een ω and S f as they are defined on differen t probabilit y spaces. The sp ectral sample will just b e a con v enien t p oint of view in order to understand the questions we are studying. Some of the form ulas and results w e hav e previously derived in these notes ha v e v ery simple formulations in terms of the sp ectral sample. F or example, it is immediate to c hec k that (IV.2) simply becomes E [ f ( ω ) f ( ω  )] = ˆ Q f [(1 −  ) | S | ] (IX.1) or E [ f ( ω ) f ( ω  )] − E [ f ( ω )] 2 = ˆ Q f [(1 −  ) | S | I S 6 = ∅ ] . (IX.2) Next, in terms of the sp ectral sample, Prop ositions IV.1 and IV.2 simply b ecome the follo wing prop osition. Prop osition IX.1. If { f n } is a se quenc e of Bo ole an functions mapping into {± 1 } , then we have the fol lowing. 1. { f n } is noise sensitive if and only if | S f n | → ∞ in pr ob ability on the set {| S f n | 6 = 0 } . 2. { f n } is noise stable if and only if the r andom variables {| S f n |} ar e tight. There is also a nice relationship b et w een the piv otal set P and the sp ectral sample. The following result, which is simply Prop osition IV.3 (see also the remark after this prop osition), tells us that the tw o random sets P and S ha v e the same 1-dimensional marginals. Prop osition IX.2. If f is a Bo ole an function mapping into {± 1 } , then for al l i ∈ [ n ] we have that P ( i ∈ P ) = ˆ Q ( i ∈ S ) and henc e E ( |P | ) = ˆ Q ( | S | ) . (This prop osition is stated with ˆ Q instead of ˆ P since if f maps in to { 0 , 1 } instead, then the reader can chec k that the ab o v e holds with an extra factor of 4 on the righ t hand side while if ˆ P were used instead, then this w ould not b e true for an y constant.) 2. A W A Y TO SAMPLE THE SPECTRAL SAMPLE IN A SUB-DOMAIN 99 Ev en though S and P hav e the same “1-dimensional” marginals, it is not how ev er true that these tw o random sets hav e the same distribution. F or example, it is easily c hec k ed that for MAJ 3 , these tw o distributions are differen t. Interestingly , as w e will see in the next section, S and P also alwa ys hav e the same “2-dimensional” marginals. This will pro v e useful when applying second moment metho d arguments. Before ending this section, let us give an alternativ e pro of of Prop osition VI.9 using this p oin t of view of thinking of S as a random set. Alternativ e pro of of Prop osition VI.9 The statement of the prop osition when con v erted to the sp ectrum states (see the exercises in this chapter if this is not clear) that for an y a n → ∞ , lim n →∞ ˆ P ( | S n | ≥ a n n 2 α 4 ( n )) = 0 . Ho w ev er this immediately follo ws from Mark ov’s inequalit y using Propositions VI.8 and IX.2. 2 A w a y to sample the sp ectral sample in a sub- domain In this section, w e describ e a metho d of “sampling” the sp ectral measure restricted to a subset of the bits. As an application of this, w e show that S and P in fact hav e the same 2-dimensional marginals, namely that for all i and j , P ( i, j ∈ P ) = ˆ Q ( i, j ∈ S ). In order to first get a little in tuition ab out the sp ectral measure, w e start with an easy prop osition. Prop osition IX.3 ([GPS10]) . F or a Bo ole an function f and A ⊆ { 1 , 2 , . . . , n } , we have ˆ Q ( S f ⊆ A ) = E [ | E ( f | A ) | 2 ] wher e c onditioning on A me ans c onditioning on the bits in A . Pro of. Noting that E ( χ S | A ) is χ S if S ⊆ A and 0 otherwise, we obtain b y expanding that E ( f | A ) = X S ⊆ A ˆ f ( S ) χ S . No w apply P arsev al’s formula. If we hav e a subset A ⊆ { 1 , 2 , . . . , n } , how do we “sample” from A ∩ S ? A nice w a y to pro ceed is as follows: choose a random configuration outside of A , then lo ok at the induced function on A and sample from the induced function’s sp ectral measure. The follo wing prop osition justifies in precise terms this wa y of sampling. Its pro of is just an extension of the pro of of Prop osition IX.3. 100 CHAPTER IX. THE SPECTRAL SAMPLE Prop osition IX.4 ([GPS10]) . Fix a Bo ole an function f on Ω n . F or A ⊆ { 1 , 2 , . . . , n } and y ∈ {± 1 } A c , that is a c onfigur ation on A c , let g y b e the function define d on {± 1 } A obtaine d by using f but fixing the c onfigur ation to b e y outside of A . Then for any S ⊆ A , we have ˆ Q ( S f ∩ A = S ) = E [ ˆ Q ( S g y = S )] = E [ ˆ g 2 y ( S )] . Pro of. Using the first line of the pro of of Prop osition IX.3, it is easy to chec k that for an y S ⊆ A , we hav e that E  f χ S   F A c  = X S 0 ⊆ A c b f ( S ∪ S 0 ) χ S 0 . This giv es E h E  f χ S   F A c  2 i = X S 0 ⊆ A c b f ( S ∪ S 0 ) 2 = ˆ Q [ S ∩ A = S ] whic h is precisely the claim. R emark IX.2 . Observe that Prop osition IX.3 is a sp ecial case of Prop osition IX.4 when S is taken to b e ∅ and A is replaced b y A c . The follo wing corollary w as first observed by Gil Kalai. Corollary IX.5 ([GPS10]) . If f is a Bo ole an function mapping into {± 1 } , then for al l i and j , P ( i, j ∈ P ) = ˆ Q ( i, j ∈ S ) . (The commen t immediately follo wing Prop osition IX.2 holds here as well.) Pro of. Although it has already b een established that P and S ha v e the same 1- dimensional marginals, w e first show ho w Proposition IX.4 can b e used to establish this. This latter prop osition yields, with A = S = { i } , that ˆ Q ( i ∈ S ) = ˆ Q ( S ∩ { i } = { i } ) = E [ ˆ g 2 y ( { i } )] . Note that g y is ± ω i if i is pivotal and constant if i is not piv otal. Hence the last term is P ( i ∈ P ). F or the 2-dimensional marginals, one first chec ks this b y hand when n = 2. F or general n , taking A = S = { i, j } in Prop osition IX.4, w e ha v e ˆ Q ( i, j ∈ S ) = P ( S ∩ { i, j } = { i, j } ) = E [ ˆ g 2 y ( { i, j } )] . F or fixed y , the n = 2 case tells us that ˆ g 2 y ( { i, j } ) = P ( i, j ∈ P g y ). Finally , a little though t sho ws that E [ P ( i, j ∈ P g y )] = P ( i, j ∈ P ), completing the pro of. 3. NONTRIVIAL SPECTR UM NEAR THE UPPER BOUND FOR PER COLA TION 101 3 Non trivial sp ectrum near the upp er b ound for p ercolation W e no w return to our cen tral ev en t of percolation crossings of the rectangle R n where f n denotes this ev en t. At this p oint, we kno w that for Z 2 , (most of ) the sp ectrum lies b et w een n  0 (for some  0 > 0) and n 2 α 4 ( n ) while for T it sits b et w een n 1 / 8+ o (1) and n 3 / 4+ o (1) . In this section, we sho w that there is a non trivial amount of sp ectrum near the upp er b ound n 2 α 4 ( n ). F or T , in terms of quantitativ e noise sensitivity , this tells us that if our noise sequence  n is equal to 1 /n 3 / 4 − δ for fixed δ > 0, then in the limit, the t w o v ariables f ( ω ) and f ( ω  n ) are not p erfectly correlated; i.e., there is some degree of indep endence. (See the exercises for understanding such argumen ts.) How ev er, we cannot conclude that there is full indep endence since we don’t kno w that “all” of the sp ectrum is near n 3 / 4+ o (1) (y et!). Theorem IX.6 ([GPS10]) . Consider our p er c olation cr ossing functions { f n } (with values into {± 1 } ) of the r e ctangles R n for Z 2 or T . Ther e exists c > 0 such that for al l n , ˆ P  | S n | ≥ cn 2 α 4 ( n )  ≥ c. The k ey lemma for proving this is the following second moment b ound on the n um ber of pivotals which w e prov e afterw ards. It has a similar fla v or to Exercise 6 in Chapter VI. Lemma IX.7 ([GPS10]) . Consider our p er c olation cr ossing functions { f n } ab ove and let R 0 n b e the b ox c onc entric with R n with half the r adius. If X n = |P n ∩ R 0 n | is the c ar dinality of the set of pivotal p oints in R 0 n , then ther e exists a c onstant C such that for al l n we have that E [ | X n | 2 ] ≤ C E [ | X n | ] 2 . Pro of of Theorem IX.6. Since P n and S n ha v e the same 1 and 2-dimensional marginals, it follows fairly straightforw ard from Lemma IX.7 that we also hav e that for all n ˆ P  | S n ∩ R 0 n | 2  ≤ C ˆ P  | S n ∩ R 0 n |  2 . Recall now the Paley-Zygm und inequality whic h states that if Z ≥ 0, then for all θ ∈ (0 , 1), P ( Z ≥ θ E [ Z ]) ≥ (1 − θ ) 2 E [ Z ] 2 E [ Z 2 ] . (IX.3) The t w o ab ov e inequalities (with Z = | S n ∩ R 0 n | and θ = 1 / 2) imply that for all n , ˆ P  | S n ∩ R 0 n | ≥ ˆ E  | S n ∩ R 0 n |  2  ≥ 1 4 C . 102 CHAPTER IX. THE SPECTRAL SAMPLE No w, by Prop osition IX.2, one has that ˆ E  | S n ∩ R 0 n |  = E [ X n ]. F urthermore (a trivial mo dification of ) Prop osition VI.8 yields E [ X n ]  n 2 α 4 ( n ) whic h th us completes the pro of. W e are now left with Pro of of Lemma IX.7. As indicated at the end of the pro of of Theorem IX.6, w e ha v e that E ( X n )  n 2 α 4 ( n ). Next, for x, y ∈ R 0 n , a picture shows that P ( x, y ∈ P n ) ≤ α 2 4 ( | x − y | / 2) α 4 (2 | x − y | , n/ 2) since w e need to hav e the 4-arm ev en t around x to distance | x − y | / 2, the same for y , and the 4-arm ev en t in the ann ulus cen tered at ( x + y ) / 2 from distance 2 | x − y | to distance n/ 2 and finally these three ev en ts are indep endent. This is b y quasi-multiplicit y at most O (1) α 2 4 ( n ) /α 4 ( | x − y | , n ) and hence E [ | X n | 2 ] ≤ O (1) α 2 4 ( n ) X x,y 1 α 4 ( | x − y | , n ) . Since, for a given x , there are at most O (1)2 2 k y ’s with | x − y | ∈ [2 k , 2 k +1 ], using quasi-m ultiplicit y , the ab ov e s um is at most O (1) n 2 α 2 4 ( n ) log 2 ( n ) X k =0 2 2 k α 4 (2 k , n ) . Using 1 α 4 ( r , R ) ≤ ( R/r ) 2 −  (this is the fact that the four-arm exp onent is strictly less than 2), the sum b ecomes at most O (1) n 4 −  α 2 4 ( n ) log 2 ( n ) X k =0 2 k . Since the last sum is at most O (1) n  , w e are done. In terms of the consequences for quan titativ e noise sensitivity , Theorem IX.6 implies the following corollary; see the exercises for similar implications. W e state this only for the triangular lattice. An analogous result holds for Z 2 . Corollary IX.8. F or T , ther e exists c > 0 so that if  n = 1 / ( n 2 α 4 ( n )) , then for al l n , P ( f n ( ω ) 6 = f n ( ω  n )) ≥ c. 3. NONTRIVIAL SPECTR UM NEAR THE UPPER BOUND FOR PER COLA TION 103 Note, imp ortantly , this do es not say that f n ( ω ) and f n ( ω  n ) b ecome asymptotically uncorrelated, only that they are not asymptotically completely correlated. T o ensure that they are asymptotically uncorrelated is significan tly more difficult and requires sho wing that “all” of the sp ectrum is near n 3 / 4 . This muc h more difficult task is the sub ject of the next chapter. 104 CHAPTER IX. THE SPECTRAL SAMPLE Exercise sheet on c hapter IX Exercise IX.1. Let { f n } b e an arbitrary sequence of Bo olean functions mapping into {± 1 } with corresp onding sp ectral samples { S n } . (i). Sho w that ˆ P  0 < | S n | ≤ A n  → 0 implies that ˆ E  (1 −  n ) | S n | I S n 6 = ∅  → 0 if  n A n → ∞ . (ii). Show that ˆ E  (1 −  n ) | S n | I S n 6 = ∅  → 0 implies that ˆ P  0 < | S n | ≤ A n  → 0 if  n A n = O (1). Exercise IX.2. Let { f n } b e an arbitrary sequence of Bo olean functions mapping into {± 1 } with corresp onding sp ectral samples { S n } . (i). Sho w that P  f ( ω ) 6 = f ( ω  n )  → 0 and A n  n = Ω(1) imply that ˆ P  | S n | ≥ A n  → 0. (ii). Sho w that ˆ P  | S n | ≥ A n  → 0 and A n  n = o (1) imply that P  f ( ω ) 6 = f ( ω  n )  → 0. Exercise IX.3. Prov e Corollary IX.8. Exercise IX.4. F or the iterated 3-Ma jorit y sequence, recall that the total influence is n α where α = 1 − log 2 / log 3. Sho w that for  n = 1 /n α , P ( f n ( ω ) 6 = f n ( ω  n )) do es not tend to 0. Exercise IX.5. Assume that { f n } is a sequence of monotone Bo olean functions on n bits with total influence equal to n 1 / 2 up to constan ts. Sho w that the sequence cannot b e noise sensitiv e. Is it necessarily noise stable as the Ma jority function is? Exercise IX.6. Assume that { f n } is a sequence of monotone Bo olean functions with mean 0 on n bits. Show that one cannot hav e noise sensitivity when using noise level  n = 1 /n 1 / 2 . Exercise IX.7. Sho w that P and S hav e the same 2-dimensional marginals using only Prop osition IX.3 rather than Prop osition IX.4. Hin t: It suffices to show that P ( { i, j } ∩ P = ∅ ) = ˆ Q ( { i, j } ∩ S = ∅ ). Exercise IX.8. (Challenging problem) Do y ou exp ect that exercise IX.5 is sharp, meaning that, if 1 / 2 is replaced b y α < 1 / 2, then one can find noise sensitiv e examples? 105 106 CHAPTER IX. THE SPECTRAL SAMPLE Chapter X Sharp noise sensitivit y of p ercolation W e will explain in this chapter the main ideas of the pro of in [GPS10] that most of the “sp ectral mass” lies near n 2 α 4 ( n ) ≈ n 3 / 4+ o (1) . This pro of b eing rather long and in v olv ed, the con ten t of this c hapter will b e far from a formal pro of. Rather it should b e considered as a (hop efully convincing) heuristic explanation of the main results, and p ossibly for the in terested readers as a “reading guide” for the pap er [GPS10]. V ery briefly sp eaking, the idea b ehind the pro of is to identify prop erties of the geom- etry of S f n whic h are reminiscen t of a self-similar fractal structure. Ideally , S f n w ould b eha v e like a spatial branc hing tree (or in other w ords a fractal p ercolation pro cess), where distinct branches evolv e indep endently of eac h other. This is conjecturally the case, but it turns out that it is v ery hard to control the dep endency structure within S f n . In [GPS10], only a tiny hin t of spatial indep endenc e within S f n is prov ed. One of the main difficulties of the pro of is to o v ercome the fact that one has v ery little indep endence to pla y with. A substan tial part of this c hapter fo cuses on the m uc h simpler case of fractal p er- colation. Indeed, this pro cess can b e seen as the simplest toy mo del for the sp ectral sample S f n . Explaining the simplified pro of adapted to this setting already enables us to con v ey some of the main ideas for handling S f n . 1 State of the art and main statemen t See Figure X.1 where we summarize what we hav e learned so far ab out the sp ectral sample S f n of a left to right crossing even t f n . F rom this table, w e see that the main question no w is to prov e that all the sp ectral mass indeed div erges at sp eed n 2 α 4 ( n ) which is n 3 / 4+ o (1) for the triangular lattice. This is the con ten t of the following theorem. 107 108 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION on the square lattice Z 2 on the triangular lattice T The sp ectral mass div erges at p olynomial sp eed There is a p ositiv e ex- p onen t  > 0, s.t. ˆ P  0 < | S f n | < n   → 0 The same holds for all  < 1 / 8 Lo w er tail esti- mates On b oth lattices, Theorem VI I I.1 enables to obtain (non-sharp) lo w er tail estimates A p ositiv e fraction of the sp ectral mass lies “where it should” There is some univ ersal c > 0 s.t. ˆ P  | S f n | > c n 2 α 4 ( n )  > c ˆ P  | S f n | > c n 3 / 4+ o (1)  > c Ma y b e summa- rized by the fol- lo wing picture k E f n ( k ) := P | S | = k c f n ( S ) 2 n 1 / 8 n 3 / 4+ o (1) . . . A smaller “bump” of pos- itiv e Spectral mass ?? A t least a postive fraction of the spec- tral mass lies here Figure X.1: A summary of some of the results obtained so far for S f n . 2. OVERALL STRA TEGY 109 Theorem X.1 ([GPS10]) . lim sup n →∞ ˆ P  0 < | S f n | < λ n 2 α 4 ( n )  − → λ → 0 0 . On the triangular lattice T , the rate of decay in λ is known explicitly . Namely: Theorem X.2 ([GPS10]) . On the triangular grid T , the lower tail of | S f n | satisfies lim sup n →∞ ˆ P  0 < | S f n | < λ ˆ E  | S f n |  )   λ → 0 λ 2 / 3 . This result deals with what one migh t call the “macroscopic” lo w er tail, i.e. with quan tities whic h asymptotically are still of order ˆ E  | S f n |  (since λ remains fixed in the ab o v e lim sup). It turns out that in our later study of dynamical p ercolation in Chapter XI, w e will need a sharp con trol on the full low er tail. This is the conten t of the follo wing stronger theorem: Theorem X.3 ([GPS10]) . On Z 2 and on the triangular grid T , for al l 1 ≤ r ≤ n , one has ˆ P  0 < | S f n | < r 2 α 4 ( r )   n 2 r 2 α 4 ( r , n ) 2 . On the triangular grid, this tr anslates into ˆ P  0 < | S f n | < u  ≈ n − 1 2 u 2 3 , wher e we write ≈ to avoid r elying on o (1) terms in the exp onents. 2 Ov erall strategy In the ab o v e theorems, it is clear that we are mostly in terested in the cardinalit y of S f n . Ho w ev er, our strategy will consist in understanding as muc h as we can ab out the t ypical ge ometry of the random set S f n sampled according to the sp ectral probabilit y measure ˆ P f n . As w e ha v e seen so far, the random set S f n shares man y prop erties with the set of piv otal p oints P f n . A first p ossibilit y would b e that they are asymptotically similar. After all, noise sensitivity is intimately related with pivotal points, so it is not unrea- sonable to hope for such a b ehavior. This scenario would b e v ery conv enien t for us since the geometry of P f n is now w ell understo o d (at least on T ) thanks to the SLE pro cesses. In particular, in the case of P f n , one can “explore” P f n in a Marko vian wa y b y relying on exploration pro cesses. Unfortunately , based on v ery con vincing heuristics, it is conjectured that the scaling limits of 1 n S f n and 1 n P f n are singular random compact sets of the square. See Figure X.2 for a quic k o v erview of the similarities and differences b et w een these tw o random sets. 110 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION Sp ectral set S f n Piv otal set P f n E  |P f n |  = ˆ E  | S f n |  = E  |P f n | 2  ˆ E  | S f n | 2  In general, they differ ! Second moment Higher momen ts ( k ≥ 3) First moment Metho ds for sam- pling these ran- dom sets Easy (and fast) using tw o explo- ration paths: The sp ectral sample S f n is m uc h harder to sample. In fact, the only known w a y to pro ceed is to compute the w eigh ts ˆ f n ( S ) 2 , one at a time ... Spatial correla- tion structure Distan t regions in P f n b eha v e more or less indep enden tly of eac h other. F urthermore, one can use the v ery con v enien t spatial Mark o v prop ert y due to the i.i.d struc- ture of the p ercolation picture. Lo w er T ail b e- ha vior P  |P f n | = 1   n − 11 12 ˆ P  | S f n | = 1   n − 1 2 Muc h of the picture here re- mains unclear ? Figure X.2: Similarities and differences b etw een S f n and P f n . The conclusion of this table is that they indeed share many prop erties, but one cannot deduce lo w er tail estimates on | S f n | out of lo w er tail estimates on |P f n | . Also, ev en w orse, we will not b e allow ed to rely on spatial Marko v prop erties for S f n . Ho w ev er, ev en though P f n and S f n differ in many wa ys, they share at least one essen- tial prop ert y: a seemingly self-similar fr actal b ehavior . The main strategy in [GPS10] to con trol the lo w er-tail b eha vior of | S f n | is to prov e that in some very weak sense, S f n b eha v es lik e the simplest mo del among self-similar fractal pro cesses in [0 , n ] 2 : i.e. 3. TOY MODEL: THE CASE OF FRACT AL PERCOLA TION 111 a sup er-critical spatial Galton-W atson tree em b edded in [0 , n ] 2 , also called a fr actal p er c olation pr o c ess . The lo w er tail of this very simple toy mo del will b e in v estigated in detail in the next section with a techniq ue whic h will b e suitable for S f n . The main difficult y which arises in this program is the lac k of kno wledge of the indep endency structure within S f n . In other words, when we try to compare S f n with a fractal p er- colation pro cess, the self-similarity already requires some w ork, but the hardest part is to deal with the fact that distinct “branches” (or rather their analogues) are not known to b ehav e ev en slightly indep enden tly of eac h other. W e will discuss these issues in Section 4 but will not give a complete proof. 3 T o y mo del: the case of fractal p ercolation As w e explained ab o v e, our main strategy is to exploit the fact that S f n has a certain self-similar fractal structure. Along this section, we will consider the simplest case of suc h a self-similar fractal ob ject: namely fr actal p er c olation , and we will detail in this simple setting what our later strategy will b e. Deliberately , this strategy will not b e optimal in this simplified case. In particular, we will not rely on the martingale tec hniques that one can use with fractal p ercolation or Galton-W atson trees, since suc h metho ds w ould not b e a v ailable for our sp ectral sample S f n . 3.1 Definition of the mo del and first prop erties T o make the analogy with S f n easier let n := 2 h , h ≥ 1 , and let’s fix a parameter p ∈ (0 , 1). No w, fr actal p er c olation on [0 , n ] 2 is defined inductively as follo ws: divide [0 , 2 h ] 2 in to 4 squares and retain each of them indep enden tly with probability p . Let T 1 b e the union of the retained 2 h − 1 -squares. The second-level tree T 2 is obtained by reiterating the same pro cedure indep endently for each 2 h − 1 -square in T 1 . Con tin uing in the same fashion all the wa y to the squares of unit size, one obtains T n = T := T h whic h is a random subset of [0 , n ] 2 . See [LyP11] for more on the definition of fr actal p er c olation . See also Figure X.3 for an example of T 5 . R emark X.1 . W e thus introduced tw o different notations for the same random set ( T n =2 h ≡ T h ). The reason for this is that on the one hand the notation T n defined on [0 , n ] 2 = [0 , 2 h ] 2 mak es the analogy with S f n (also defined on [0 , n ] 2 ) easier, while on the other hand inductive pro ofs will b e more conv enien t with the notation T h . In order to hav e a sup ercritical Galton-W atson tree, one has to choose p ∈ (1 / 4 , 1). F urthermore, one can easily chec k the following easy prop osition. Prop osition X.4. L et p ∈ (1 / 4 , 1) . Then 112 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION Figure X.3: A realization of a fractal p ercolation T 2 5 = T 5 E  |T n |  = n 2 p h = n 2+log 2 p , and E  |T n | 2  ≤ O (1) E  |T n |  2 . In p articular, by the se c ond moment metho d (e.g. the Paley-Zygmund ine quality), with p ositive pr ob ability, T n is of or der n 2+log 2 p . Let α := 2 + log 2 p . This parameter α corresp onds to the “fractal dimension” of T n . T o mak e the analogy with S f n ev en clearer, one could c ho ose p in such a w a y that α = 2 + log 2 p = 3 / 4, but w e will not need to. The ab o v e prop osition implies that on the ev en t T n 6 = ∅ , with p ositiv e conditional probabilit y |T n | is large (of order n α ). This is the exact analogue of Theorem IX.6 for the sp ectral sample S f n . Let us first analyze what w ould b e the analogue of Theorem X.1 in the case of our to y mo del T n . W e hav e the following. 3. TOY MODEL: THE CASE OF FRACT AL PERCOLA TION 113 Prop osition X.5. lim sup n →∞ P  0 < |T n | < λ n α )  − → λ → 0 0 . R emark X.2 . If one could rely on martingale techniques, then this prop osition is a corollary of standard results. Indeed, as is well-kno wn M i := |T i | (4 p ) i , is a p ositiv e martingale. Therefore it conv erges, as n → ∞ , to a non-negativ e random v ariable W ≥ 0. F urthermore, the conditions of the Kesten-Stigum Theorem are ful- filled (see for example Section 5.1 in [LyP11]) and therefore W is p ositive on the ev en t that there is no extinction. This implies the abov e prop osition. As we claimed ab o v e, w e will inten tionally follow a more hands-on approach in this section whic h will b e more suitable to the random set S f n whic h w e hav e in mind. F urthermore this approac h will hav e the great adv an tage to provide the follo wing muc h more precise result, which is the analogue of Theorem X.3 for T n . Prop osition X.6. F or any 1 ≤ r ≤ n , P  0 < |T n | < r α   ( r n ) log 2 1 /µ , wher e µ is an explicit c onstant in (0 , 1) c ompute d in Exer cise X.2. 3.2 Strategy and heuristics Letting u  n α , w e wish to estimate P  0 < |T n | < u  . Even though we are only in- terested in the size of T n , w e will try to estimate this quantit y by understanding the ge ometry of the conditional set: T | u n := L  T n    0 < |T n | < u  . The first natural question to ask is whether this conditional random set is typically lo c alize d or not. See Figure X.4. In tuitiv ely , it is quite clear that the set T n conditioned to b e v ery small will tend to b e lo calized. So it is the picture on the right in Figure X.4 whic h is more likely . This w ould deserv e a pro of of course, but we will come back to this later. The fact that it should lo ok more and more lo calized tells us that as one shrinks u , this should make our conditional T | u n more and more singular with resp ect to the unconditional one. But ho w m uch lo calization should we see? This is again fairly easy to answ er, at least on the in tuitiv e lev el. Indeed, T | u n should tend to lo calize un til it reaches a certain mesoscopic scale r such that 1  r  n . One can compute how muc h it costs to maintain a single 114 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION Ho w do es L  T n    0 < |T n | < u  lo ok ? < u < u < u More En trop y ( in V ol 3 ) but costs more to maintain these 3 “islands” alive. OR? < u Less Entrop y ( in V ol 1 ) but only one island to maintain aliv e. Figure X.4: En trop y v.s. Clustering effect branc h (or O (1) branches) alive until scale r , but once this is ac hiev ed, one should let the system evolv e in a “natural” w a y . In particular, once the tree surviv es all the wa y to a mesoscopic square of size r , it will (by the second moment metho d) pro duce Ω( r α ) lea v es there with p ositiv e probability . T o summarize, typically T | u n will maintain O (1) man y branc hes alive at scale 1  r  n , and then it will let the branching structure ev olv e in a basically unconditional w a y . The intermediate scale r is chosen so that r α  u . Definition X.1. If 1 ≤ r ≤ n = 2 h is such that r = 2 l , 0 ≤ l ≤ h , let T ( r ) denote the set of br anches that wer e stil l alive at sc ale r = 2 l in the iter ative c onstruction of T n . In other wor ds, T ( r ) ≡ T h − l and T n ⊂ S T ( r ) . This r andom set T ( r ) wil l b e the analo gue of the “ r -smo othing” S ( r ) of the sp e ctr al sample S f n define d later in Definition X.2. Returning to our problem, the ab ov e heuristics say that one exp ects to hav e for any 1  u  n α . P  0 < |T n | < u   P  0 < |T ( r ) | ≤ O (1)   P  |T ( r ) | = 1  , where r is a dy adic integer chosen such that r α  u . Or in other words, we exp ect that P  0 < |T n | < r α   P  |T ( r ) | = 1  . (X.1) In the next subsection, we briefly explain ho w this heuristic can b e implemen ted in to a pro of in the case of the tree T n in a wa y which will b e suitable to the study of S f n . W e will only skim through the main ideas for this tree case. 3. TOY MODEL: THE CASE OF FRACT AL PERCOLA TION 115 3.3 Setup of a pro of for T n Motiv ated by the ab o v e heuristics, w e divide our system in to t w o scales: ab o v e and b elo w the mesoscopic scale r . One can write the lo w er tail ev en t as follows (let 1  r  n ): P  0 < |T n | < r α  = X k ≥ 1 P  |T ( r ) | = k  P  0 < |T n | < r α   |T ( r ) | = k  . (X.2) It is not hard to estimate the second term P  0 < |T n | < r α   |T ( r ) | = k  . Indeed, in this term w e are conditioning on having exactly k branches alive at scale r . Indep en- den tly of where they are, “b elo w” r , these k branches evolv e indep enden tly of eac h other. F urthermore, b y the second momen t metho d, there is a universal constant c > 0 suc h that eac h of them exceeds the fatal amoun t of r α lea v es with probabilit y at least c (note that in the opp osite direction, eac h branc h could also go extinct with p ositiv e probabilit y). This implies that P  0 < |T n | < r α   |T ( r ) | = k  ≤ (1 − c ) k . R emark X.3 . Note that one makes hea vy use of the indep endence structure within T n here. This asp ect is m uc h more non trivial for the sp ectral sample S f n . F ortunately it turns out, and this is a k ey fact, that in [GPS10] one can pro v e a w eak indep endence statemen t whic h in some sense mak es it p ossible to follow this route. W e are left with the follo wing upp er b ound: P  0 < |T n | < r α  ≤ X k ≥ 1 P  |T ( r ) | = k  (1 − c ) k . (X.3) In order to prov e our goal of (X.1), by exploiting the exp onen tial decay given b y (1 − c ) k (whic h follo w ed from indep endence), it is enough to prov e the follo wing b ound on the mesoscopic b eha vior of T : Lemma X.7. Ther e is a sub-exp onential function k 7→ g ( k ) such that for al l 1 ≤ r ≤ n , P  |T ( r ) | = k  ≤ g ( k ) P  |T ( r ) | = 1  . Notice as w e did in Definition X.1 that since T ( r ) has the same law as T h − l , this is a purely Galton-W atson tree type of question. The big adv antage of our strategy so far is that initially w e were lo oking for a sharp con trol on P  0 < |T n | < u  and now, using this “tw o-scales” argument, it only remains to prov e a crude upp er b ound on the lo w er tail of |T ( r ) | . By scale in v ariance this is nothing else than obtaining a crude upp er b ound on the lo w er tail of |T n | . Hence this division in to t w o scales greatly simplified our task. 116 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION 3.4 Sub-exp onen tial estimate on the lo w er-tail (Lemma X.7) The first step tow ards pro ving and understanding Lemma X.7 is to understand the term P  |T ( r ) | = 1  . F rom now on, it will b e easier to work with the “dyadic” notations instead, i.e. with T i ≡ T 2 i (see remark X.1). With these notations, the first step is equiv alent to understanding the probabilities p i := P  |T i | = 1  . This asp ect of the problem is very sp ecific to the case of Galton-W atson trees and gives very little insigh t in to the later study of the spectrum S f n . Therefore we postp one the details to Exercise X.2. The conclusion of this (straigh tforw ard) exercise is that p i b eha v es as i → ∞ like p i ∼ c µ i , for an explicit exp onen t µ ∈ (0 , 1) (see Exercise X.2). In particular, in order to pro v e Prop osition X.6, it is no w enough to find a sub-exp onential function k 7→ g ( k ) suc h that for an y i, k ≥ 1, P  |T i | = k  ≤ g ( k ) µ i . (X.4) More precisely , we will prov e the following lemma. Lemma X.8. L et g ( k ) := 2 θ log 2 2 ( k +2) , wher e θ is a fixe d c onstant to b e chosen later. Then for al l i, k ≥ 1 , one has P  |T i | = k  ≤ g ( k ) µ i . (X.5) W e provide the pro of of this lemma here, since it can b e seen as a “toy pro of ” of the corresp onding sub-exp onential estimate needed for the r -smo othed sp ectral samples S ( r ) , stated in the coming Theorem X.13. The pro of of this latter theorem shares some similarities with the pro of b elow but is muc h more tec hnical since in the case of S ( r ) one has to deal with a more complex structure than the branc hing structure of a Galton-W atson tree. Pro of. W e pro ceed b y double induction. Let k ≥ 2 b e fixed and assume that equa- tion (X.5) is already satisfied for all pair ( i 0 , k 0 ) such that k 0 < k . Based on this assumption, let us prov e by induction on i that all pairs ( i, k ) satisfy equation (X.5) as w ell. First of all, if i is small enough, this is obvious by the definition of g ( k ). Let J = J k := sup { i ≥ 1 : g ( k ) µ i > 10 } . Then, it is clear that equation (X.5) is satisfied for all ( i, k ) with i ≤ J k . No w let i > J k . If T i is suc h that |T i | = k ≥ 1, let L = L ( T i ) ≥ 0 b e the largest in teger such that T i in tersects only one square of size 2 i − L . This means that below scale 2 i − L , the tree T i splits into at least 2 live branches in distinct dy adic squares of size 2 i − L − 1 . Let 3. TOY MODEL: THE CASE OF FRACT AL PERCOLA TION 117 d ∈ { 2 , 3 , 4 } b e the n um ber of such live branches. By decomp osing on the v alue of L , and using the ab o v e assumption, w e get P  |T i | = k  ≤ P  L ( T i ) > i − J k  + 1 1 − q i − J k X l =0 P  L ( T i ) = l  4 X d =2  4 d  ( µ i − l − 1 ) d X ( k j ) 1 ≤ j ≤ d k j ≥ 1 , P k j = k Y j g ( k j ) where q is the probability that our Galton-W atson tree go es extinct. Let us first estimate what P  L ( T i ) ≥ m  is for m ≥ 0. If m ≥ 1, this means that among the 2 2 m dy adic squares of size 2 i − m , only one will remain alive all the wa y to scale 1. Y et, it might b e that some other suc h squares are still alive at scale 2 i − m but will go extinct by the time they reac h scale 1. Let p m,b b e the probabilit y that the pro cess T m + b , whic h lives in [0 , 2 m + b ] 2 , is en tirely contained in a dyadic square of size 2 b . With suc h notations, one has P  L ( T i ) ≥ m  = p m,i − m . F urthermore, if i = m , one has p i, 0 = p i ∼ cµ i . It is not hard to pro v e (see Exercise X.2) the follo wing lemma. Lemma X.9. F or any value of m, b ≥ 0 , one has p m,b ≤ µ m . In p articular, one has a universal upp er b ound in b ≥ 0 . It follo ws from the lemma that P  L ( T i ) = l  ≤ P  L ( T i ) ≥ l  ≤ µ l and P  L ( T i ) > i − J k  ≤ µ i − J k (X.6) ≤ 1 10 g ( k ) µ i b y the definition of J k . (X.7) This giv es us that for some constant C P  |T i | = k  ≤ µ i 10 g ( k ) + C i − J k X l =0 µ l 4 X d =2 ( µ i − l ) d X ( k j ) 1 ≤ j ≤ d k j ≥ 1 , P k j = k Y j g ( k j ) = µ i 10 g ( k ) + C µ i 4 X d =2 i − J k X l =0 ( µ i − l ) d − 1 X ( k j ) 1 ≤ j ≤ d k j ≥ 1 , P k j = k Y j g ( k j ) . 118 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION Let us deal with the d = 2 sum (the contributions coming from d > 2 b eing even smaller). By concavit y of k 7→ θ log 2 2 ( k + 2), one obtains that for any ( k 1 , k 2 ) such that k 1 + k 2 = k : g ( k 1 ) g ( k 2 ) ≤ g ( k / 2) 2 . Since there are at most k 2 suc h pairs, this giv es us the follo wing b ound on the d = 2 sum. i − J k X l =0 ( µ i − l ) 2 − 1 X ( k j ) 1 ≤ j ≤ 2 k j ≥ 1 , P k j = k Y j g ( k j ) ≤ i − J k X l =0 µ i − l k 2 g ( k / 2) 2 ≤ 1 1 − µ µ J k k 2 g ( k / 2) 2 ≤ 10 1 1 − µ k 2 g ( k / 2) 2 ( µg ( k )) − 1 , b y definition of J k . No w, some easy analysis implies that if one c ho oses the constan t θ > 0 large enough, then for any k ≥ 2, one has C 10 1 1 − µ k 2 g ( k / 2) 2 ( µg ( k )) − 1 ≤ 1 10 g ( k ). Altogether (and taking in to consideration the d > 2 contributions), this implies that P  |T i | = k  ≤ 2 5 g ( k ) µ i ≤ g ( k ) µ i , as desired. R emark X.4 . Recall the initial question from Figure X.4 which ask ed whether the clus- tering effect wins ov er the en trop y effect or not. This question enabled us to motiv ate the setup of the pro of but in the end, we did not sp ecifically address it. Notice that the ab o v e pro of in fact solv es the problem (see Exercise X.3). 4 Bac k to the sp ectrum: an exp osition of the pro of 4.1 Heuristic explanation Let us no w apply the strategy we developed for T n to the case of the sp ectral sample S f n . Our goal is to prov e Theorem X.3 (of which Theorems X.1 and X.2 are straigh tforw ard corollaries). Let S f n ⊂ [0 , n ] 2 b e our sp ectral sample. W e hav e seen (Theorem IX.6) that with p ositiv e probability | S f n |  n 2 α 4 ( n ). F or all 1 < u < n 2 α 4 ( n ), we wish to understand the probability ˆ P  0 < | S f n | < u  . F ollo wing the notations w e used for T n , let S | u f n b e the sp ectral sample conditioned on the even t { 0 < | S f n | < u } . Question: Ho w do es S | u f n t ypically lo ok? T o answ er this question, one has to understand whether S | u f n tends to b e lo c alize d or not. Recall from Figure X.4 the illustration of the comp etition b etw een entrop y and 4. BACK TO THE SPECTRUM: AN EXPOSITION OF THE PROOF 119 clustering effects in the case of T n . The same figure applies to the sp ectral sample S f n . W e will later state a clustering lemma (Lemma X.14) which will strongly supp ort the lo calized b eha vior describ ed in the next proposition. Therefore w e are guessing that our conditional set S | u f n will tend to lo calize in to O (1) man y squares of a certain scale r and will hav e a “normal” size within these r -squares. It remains to understand what this mesoscopic scale r as a function of u is. By “scale in v ariance”, one exp ects that if S f n is conditioned to live in a square of size r , then | S f n | will b e of order r 2 α 4 ( r ) with p ositive conditional probabilit y . More precisely , the following lemma will b e prov ed in Problem X.6. Lemma X.10. Ther e is a universal c ∈ (0 , 1) such that for any n and for any r -squar e B ⊂ [ n/ 4 , 3 n/ 4] 2 in the “bulk” of [0 , n ] 2 , one has ˆ P  | S f n | r 2 α 4 ( r ) ∈ ( c, 1 /c )   S f n 6 = ∅ and S f n ⊂ B  > c . (X.8) In fact this lemma holds uniformly in the p osition of the r -square B inside [0 , n ] 2 , but w e will not discuss this here. What this lemma tells us is that for an y 1 < u < n 2 α 4 ( n ), if one chooses r = r u in suc h a w a y that r 2 α 4 ( r )  u , then we exp ect to hav e the following estimate: ˆ P  0 < | S f n | < u   ˆ P  S f n in tersects O (1) r -squares in [0 , n ] 2   ˆ P  S f n in tersects a single r -square in [0 , n ] 2  A t this p oint, let us in tro duce a concept which will b e very helpful in what follows. Definition X.2 (“ r -smoothing”) . L et 1 ≤ r ≤ n . Consider the domain [0 , n ] 2 and divide it into a grid of squar es of e dge-length r . (If 1  r  n , one c an view this grid as a mesosc opic grid). If n is not divisible by r , write n = mr + q and c onsider the grid of r -squar es c overing [0 , ( m + 1) r ] 2 . Now, for e ach subset S ⊂ [0 , n ] 2 , define S ( r ) to b e the set of r × r squar es in the ab ove grid which interse ct S . In p articular | S ( r ) | wil l c orr esp ond to the numb er of such r -squar es which interse ct S . With a slight abuse of notation, S ( r ) wil l sometimes also denote the actual subset of [0 , n ] 2 c onsisting of the union of these r -squar es. One c an view the applic ation S 7→ S ( r ) as an r -smo othing sinc e al l the details b elow the sc ale r ar e lost. R emark X.5 . Note that in Definition X.1, w e relied on a slightly different notion of “ r -smo othing” since in that case, T ( r ) could also include r -branc hes whic h might go extinct by the time they reached scale one. The adv antage of this choice w as that there w as an exact scale-in v ariance from T to T ( r ) while in the case of S f n , there is no suc h exact scale-in v ariance from S to S ( r ) . 120 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION With these notations, the ab o v e discussion leads us to believe that the follo wing prop osition should hold. Prop osition X.11. F or al l 1 ≤ r ≤ n , one has ˆ P  0 < | S f n | < r 2 α 4 ( r )   ˆ P f n  | S ( r ) | = 1  . Before explaining the setup used in [GPS10] to prov e suc h a result, let us chec k that it indeed implies Theorem X.3. By neglecting the boundary issues, one has ˆ P f n  | S ( r ) | = 1   X r -squares B ⊂ [ n/ 4 , 3 n/ 4] 2 ˆ P  S f n 6 = ∅ and S f n ⊂ B  . (X.9) There are O ( n 2 r 2 ) suc h B squares, and for each of these, one can chec k (see Exercise X.5) that ˆ P  S f n 6 = ∅ and S f n ⊂ B   α 4 ( r , n ) 2 . (X.10) Therefore, Prop osition X.11 indeed implies Theorem X.3. 4.2 Setup and organization of the pro of of Prop osition X.11 T o start with, assume w e knew that disjoin t regions in the sp ectral sample S f n b eha v e more or less indep enden tly of eac h other in the following (v ague) sense. F or any k ≥ 1 and an y mesoscopic scale 1 ≤ r ≤ n , if one conditions on S ( r ) to b e equal to B 1 ∪ · · · ∪ B k for k disjoint r -squares, then the conditional la w of S | S B i should b e “similar” to an indep endent pro duct of L  S | B i   S ∩ B i 6 = ∅  , i ∈ { 1 , . . . , k } . Similarly as in the tree case (where the analogous prop erty for T n w as an exact indep endence factorization), and assuming that the ab ov e comparison with an indep enden t product could b e made quan titativ e, this would potentially imply the follo wing upp er b ound for a certain absolute constant c > 0: ˆ P  0 < | S f n | < r 2 α 4 ( r )  ≤ X k ≥ 1 ˆ P  | S ( r ) | = k  (1 − c ) k . (X.11) This means that even if one managed to obtain a go o d control on the dep endency structure within S f n (in the ab o v e sense), one w ould still need to hav e a go o d estimate on ˆ P  | S ( r ) | = k  in order to deduce Prop osition X.11. This part of the program is ac hiev ed in [GPS10] without requiring any information on the dep endency structure of S f n . More precisely , the follo wing result is prov ed: Theorem X.12 ([GPS10]) . Ther e is a sub-exp onential function g 7→ g ( k ) , such that for any 1 ≤ r ≤ n and any k ≥ 1 , ˆ P  | S ( r ) | = k  ≤ g ( k ) ˆ P  | S ( r ) | = 1  . 4. BACK TO THE SPECTRUM: AN EXPOSITION OF THE PROOF 121 The pro of of this result will b e describ ed briefly in the next subsection. One can no w describ e ho w the pro of of Theorem X.3 is organized in [GPS10]. It is divided in to three main parts: 1. The first part deals with proving the m ulti-scale sub-exp onential b ound on the lo w er-tail of | S ( r ) | giv en b y Theorem X.12. 2. The second part consists in proving as muc h as we can on the dep endency struc- ture of S f n . Unfortunately here, it seems to b e v ery c hallenging to ac hiev e a go o d understanding of all the “indep endence” that should b e presen t within S f n . The only hint of indep endence which was finally pro v ed in [GPS10] is a v ery we ak one (see subsection 4.4). In particular, it is to o weak to readily imply a b ound lik e (X.11). 3. Since disjoin t regions of the sp ectral sample S f n are not kno wn to b eha v e inde- p enden tly of each other, the third part of the pro of consists in adapting the setup w e used for the tree (where distinct branc hes ev olv e exactly indep enden tly of eac h other) in to a setup where the w eak hin t of indep endence obtained in the second part of the program turns out to b e enough to imply the b ound given b y (X.11) for an appropriate absolute constant c > 0. This final part of the pro of will b e discussed in subsection 4.5. The next three subsections will b e dev oted to each of these 3 parts of the program. 4.3 Some words ab out the sub-exp onen tial b ound on the lo w er tail of S ( r ) In this subsection, we turn our attention to the pro of of the first part of the program, i.e. on Theorem X.12. In fact, as in the case of T n , the following more explicit statement is pro v ed in [GPS10]. Theorem X.13 ([GPS10]) . Ther e exists an absolute c onstant θ > 0 such that for any 1 ≤ r ≤ n and any k ≥ 1 , ˆ P  | S ( r ) | = k  ≤ 2 θ log 2 2 ( k +2) ˆ P  | S ( r ) | = 1  . R emark X.6 . Note that the theorems from [BKS99] on the noise sensitivity of p ercola- tion are all particular cases ( r = 1) of this intermediate result in [GPS10]. The main idea in the pro of of this theorem is in some sense to assign a tr e e structur e to each p ossible set S ( r ) . The adv an tage of working with a tree structure is that it is eas- ier to work with inductiv e arguments. In fact, once a mapping S ( r ) 7→ “tree structure” has b een designed, the proof proceeds similarly as in the case of T ( r ) b y double induction on the depth of the tree as well as on k ≥ 1. Of course, this mapping is a delicate affair: 122 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION it has to be designed in an “efficien t” w a y so that it can comp ete against en trop y effects caused b y the exp onen tial gro wth of the num b er of tree structures. W e will not giv e the details of how to define such a mapping, but let us describ e informally how it works. More sp ecifically than a tree structure, we will in fact assign an annulus structur e to eac h set S ( r ) . Definition X.3. L et A b e a finite c ol le ction of disjoint (top olo gic al) annuli in the plane. We c al l this an ann ulus structure . F urthermor e, we wil l say that a set S ⊂ R 2 is compatible with A (or vic e versa) if it is c ontaine d in R 2 \ S A and interse cts the inner disk of e ach annulus in A . Note that it is al lowe d that one annulus is “inside” of another annulus. A 1 A 2 A 3 S f n Figure X.5: An example of an ann ulus structure A := { A 1 , A 2 , A 3 } compatible with a sp ectral sample S f n . The mapping procedure in [GPS10] assigns to each S ( r ) an annulus structure A ⊂ [0 , n ] 2 in suc h a w a y that it is compatible with S ( r ) . See Figure X.5 for an example. Again, w e will not describ e this pro cedure nor discuss the obvious b oundary issues whic h arise here, but let us state a crucial prop erty satisfied by annulus structures. Lemma X.14 ( clustering Lemma). If A is an annulus structur e c ontaine d in [0 , n ] 2 , 4. BACK TO THE SPECTRUM: AN EXPOSITION OF THE PROOF 123 then ˆ P  S ( r ) is c omp atible with A  ≤ Y A ∈A α 4 ( A ) 2 , wher e α 4 ( A ) denotes the pr ob ability of having a four-arm event in the annulus A . R emark X.7 . T o deal with b oundary issues, one would also need to incorp orate within our ann ulus structures half-annuli centered on the b oundaries as well as quarter disks cen tered at the corners of [0 , n ] 2 . Let us briefly comment on this lemma. • First of all, its pro of is an elegant combination of linear algebra and percolation. It is a short and relatively elementary argument. See Lemma 4.3 in [GPS10]. • It is very p ow erful in dealing with the p ossible non-injectivit y of the mapping S ( r ) 7→ A . Indeed, while describing the setup abov e, one might ha v e ob jected that if the mapping w ere not injective enough, then the cardinality of the “fib ers” ab o v e eac h annulus structure would hav e to b e taken in to accoun t as w ell. F ortunately , the ab o v e lemma reads as follows: for any fixed annulus structure A , X S ( r ) : S ( r ) 7→A ˆ P  S ( r )  ≤ ˆ P  S ( r ) is compatible with A  ≤ Y A ∈A α 4 ( A ) 2 . • Another essen tial feature of this lemma is that it quan tifies v ery efficiently the fact that the clustering effect wins o v er the entrop y effect in the sense of Figure X.4. The mec hanism resp onsible for this is that the probability of the four-arm ev en t squared has an exp onent (equal to 5 / 2 on T ) larger than the volume exp onent equal to 2. T o illustrate this, let us analyze the situation when k = 2 (still neglecting b oundary issues). The probability that the sp ectrum S f n in tersects tw o and only t w o r -squares at macroscopic distance Ω( n ) from each other can b e easily estimated using the lemma. Indeed, in suc h a case, S ( r ) w ould b e compatible with an ann ulus structure consisting of t w o annuli, eac h b eing appro ximately of the t yp e A ( r, n ). There are O ( n 2 r 2 ) × O ( n 2 r 2 ) suc h p ossible ann ulus structures. Using the lemma eac h of them costs (on T ) ( r n ) 5+ o (1) . An easy exercise shows that this is m uc h smaller than ˆ P  | S ( r ) | = 2  . In other w ords, if | S ( r ) | is conditioned to b e small, it tends to b e lo calized. Also, the wa y that the lemma is stated mak es it v ery con v enien t to work with higher v alues of k . The details of the pro of of Theorem X.13 can b e found in [GPS10]. The double induction there is in some sense v ery close to the one w e carried out in detail in sub- section 3.4 in the case of the tree; this is the reason why we included this latter pro of. F or those who might read the pro of in [GPS10], there is a notion of over cr owde d cluster defined there; it exactly corresp onds in the case of the tree to stopping the analysis ab o v e scale J k instead of going all the wa y to scale 1 (note that without stopping at this scale J k , the double induction in subsection 3.4 w ould hav e failed). 124 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION 4.4 Some words on the w eak indep endence prop ert y pro v ed in [GPS10] This part of the program is in some sense the main one. T o introduce it, let us start by a naiv e but tempting strategy . What the first part of the program (Theorem X.13) tells us is that for any mesoscopic scale 1 ≤ r ≤ n , if S f n is non-empt y , it is v ery unlik ely that it will intersect few squares of size r . In other words, it is very unlikely that | S ( r ) | will b e small. Let B 1 , . . . , B m denote the set of O ( n 2 /r 2 ) r -squares which tile [0 , n ] 2 . One might try the follo wing sc anning pr o c e dur e : explore the sp ectral sample S f n inside the squares B i one at a time. More precisely , b efore starting the scanning pro cedure, w e consider our sp ectral sample S f n as a random subset of [0 , n ] 2 ab out whic h w e do not know anything yet. Then, at step one, w e reveal S | B 1 . This giv es us some partial information ab out S f n . What w e still hav e to explore is a random set of [0 , n ] 2 \ B 1 whic h follows the la w of a sp ectral sample conditioned on what was seen in B 1 and we k eep going in this wa y . By Theorem X.13, many of these squares will b e non-empty . No w, it is not hard to prov e the following lemma (using similar metho ds as in Problem X.6). Lemma X.15. Ther e is a universal c onstant c > 0 such that for any r -squar e B in the bulk [ n/ 4 , 3 n/ 4] 2 , one has ˆ P  | S f n ∩ B | > c r 2 α 4 ( r )   S f n ∩ B 6 = ∅  > c . This lemma in fact holds uniformly in the p osition of B inside [0 , n ] 2 . If one could pro v e the following (m uc h) stronger result: there exists a universal constan t c > 0 suc h that uniformly on the sets S ⊂ [0 , n ] 2 \ B one has ˆ P  | S f n ∩ B | > c r 2 α 4 ( r )   S f n ∩ B 6 = ∅ and S | B c = S  > c , (X.12) then it would not be hard to make the ab o v e scanning strategy work together with Theorem X.13 in order to obtain Theorem X.3. (Note that such a result would indeed giv e a strong hint of indep endence within S f n .) Ho wev er, as w e discussed b efore, the current understanding of the indep endence within S f n is far from giving such a statemen t. Instead, the following result is prov ed in [GPS10]. W e pro vide here a sligh tly simplified v ersion. Theorem X.16 ([GPS10]) . Ther e exists a uniform c onstant c > 0 such that for any set W ⊂ [0 , n ] 2 and any r -squar e B such that B ∩ W = ∅ , one has ˆ P  | S f n ∩ B | > c r 2 α 4 ( r )   S f n ∩ B 6 = ∅ and S f n ∩ W = ∅  > c . Note that this theorem in some sense interpolates b etw een part of Lemma X.10 and Lemma X.15 which corresp ond resp ectiv ely to the sp ecial cases W = B c and W = ∅ . 4. BACK TO THE SPECTRUM: AN EXPOSITION OF THE PROOF 125 Y et it lo oks very weak compared to the exp ected (X.12) whic h is stated uniformly on the b eha vior of S f n outside of B . Assuming this w eak hin t of indep endence (Theorem X.16), it seems we are in bad shap e if we try to apply the ab ov e scanning pro cedure. Indeed, we face the following t w o obstacles: 1. The first obstacle is that one would k eep a go o d con trol only as far as one would not see any “sp ectrum”. Namely , while revealing S | B i one at a time, the first time one finds a square B i suc h that S | B i 6 = ∅ , one w ould b e forced to stop the scanning pro cedure there. In particular, if the size of the sp ectrum in this first non-trivial square do es not exceed r 2 α 4 ( r ), then w e cannot conclude anything. 2. The second obstacle is that, b esides the conditioning S ∩ W = ∅ , our estimate is also conditioned on the even t that S ∩ B 6 = ∅ . In particular, in the ab o v e “naiv e” scanning strategy where squares are rev ealed in a sequential wa y , at each step one w ould hav e to up date the probability that S ∩ B i +1 6 = ∅ based on what w as disco v ered so far. It is the purp ose of the third part of the program to adapt the ab o v e scanning strategy to these constrain ts. Before describing this third part in the next subsection, let us say a few words on how to prov e Theorem X.16. A crucial step in the pro of of this theorem is to understand the follo wing “one-p oin t function” for an y x ∈ B at distance at least r/ 3 from the b oundary: ˆ P  x ∈ S f n and S f n ∩ W = ∅  . A v ery useful observ ation is to rewrite this one-p oin t function in terms of an explicit coupling of tw o i.i.d. p ercolation configurations. It works as follows: let ( ω 1 , ω 2 ) b e a coupling of t w o i.i.d. p ercolations on [0 , n ] 2 whic h are suc h that  ω 1 = ω 2 on W c ω 1 , ω 2 are indep enden t on W One can c hec k that the one-p oint function w e are in terested in is related to this coupling in the follo wing simple wa y: ˆ P  x ∈ S f n and S f n ∩ W = ∅  = P  x is pivotal for ω 1 AND ω 2  . R emark X.8 . Y ou may chec k this iden tit y in the sp ecial cases where W = ∅ or W = { x } c . Thanks to this observ ation, the pro of of Theorem X.16 proceeds by analyzing this W -coupling. See [GPS10] for the complete details. 126 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION 4.5 Adapting the setup to the w eak hin t of indep endence As w e discussed in the previous subsection, one faces t w o main obstacles if, on the basis of the w eak indep endence given by Theorem X.16, one tries to apply the naiv e sequen tial scanning pro cedure describ ed earlier. Let us start with the first obstacle. Assume that we scan the domain [0 , n ] 2 in a se- quen tial w ay , i.e., w e c ho ose an increasing family of subsets ( W l ) l ≥ 1 = ( { w 1 , . . . , w l } ) l ≥ 1 . A t eac h step, w e rev eal what S |{ w l +1 } is, conditioned on what w as disco v ered so far (i.e., conditioned on S | W l ). F rom the weak indep endence Theorem X.16, it is clear that if w e wan t this strategy to hav e an y chance to b e successful, we ha v e to choose ( W l ) l ≥ 1 in such a w a y that ( S f n ∩ W l ) l ≥ 1 will remain empty for some time (so that w e can con tin ue to rely on our w eak indep endence result); of course this cannot remain empty forev er, so the game is to choose the increasing family ( W l ) l ≥ 1 in suc h a wa y that the first time S f n ∩ { w l } will happ en to b e non-empty , it should give a strong indication that S f n is large in the r -neighborho o d of w l . As we ha v e seen, rev ealing the entire mesoscopic b o xes B i one at a time is not a successful idea. Here is a m uc h b etter idea (whic h is not yet the right one due to the second obstacle, but w e are getting close): in eac h r -square B i , instead of rev ealing all the bits, let us reveal only a very small prop ortion δ r of them. Lemma X.15 tells us that if S ∩ B i 6 = ∅ , then each p oint x ∈ B i has probabilit y of order α 4 ( r ) to b e in S f n . Therefore if w e c ho ose δ r  ( r 2 α 4 ( r )) − 1 , then with high probabilit y , by rev ealing only a prop ortion δ r of the p oin ts in B i , we will “miss” the sp ectral sample S f n . Hence, we ha v e to c hoose δ r ≥ ( r 2 α 4 ( r )) − 1 . In fact choosing δ  ( r 2 α 4 ( r )) − 1 is exactly the righ t balance. Indeed, we know from Theorem X.13 that man y r -squares B i will b e touched b y the sp ectral sample; now, in this more sophisticated scanning pro cedure, if the first suc h square encoun tered happ ens to con tain few p oin ts (i.e.  r 2 α 4 ( r )), then with the previous scanning strategy , we would “lose”, but with the presen t one, due to our c hoice of δ r , most likely w e will keep S f n ∩ W l = ∅ so that we can con tin ue further on un til w e reach a “goo d” square (i.e. a square containing of order r 2 α 4 ( r ) p oin ts). No w, Theorems X.13 and X.16 together tell us that with high probabilit y , one will ev en tually reac h suc h a go o d square. Indeed, supp ose the m first r -squares touched b y the sp ectral sample happ ened to contain few p oints; then, most lik ely , if W l m is the set of bits rev ealed so far, by our c hoice of δ r w e will still ha v e S ∩ W l m = ∅ . This allo ws us to still rely on Theorem X.16, whic h basically tells us that there is a p ositiv e conditional probability for the next one to b e a “go o d” square (we are neglecting the second obstacle here). This sa ys that the probabilit y to visit m consecutive bad squares seems to decrease exp onen tially fast. Since m is t ypically very large (b y Theorem X.13), w e conclude that, with high probability , we will finally reach go o d squares. In the first go o d square encoun tered, b y our c hoice of δ r , there is now a positive probability to rev eal a bit present in S f n . In this case, the sequen tial scanning will hav e to stop, since w e will not b e able to use our w eak indep endence result an ymore, but this is not a big issue: indeed, assume that y ou hav e some random set S ⊂ B . If by rev ealing each bit only with probabilit y δ r , y ou end up finding a p oint in S , most lik ely y our set S is at 4. BACK TO THE SPECTRUM: AN EXPOSITION OF THE PROOF 127 least of size Ω( r 2 α 4 ( r )). This is exactly the size we are lo oking for in Theorem X.3. No w, only the second obstacle remains. It can b e rephrased as follows: assume y ou applied the ab ov e strategy in B 1 , . . . , B h (i.e. y ou revealed each p oint in B i , i ∈ { 1 , . . . , h } only with probabilit y δ r ) and that y ou did not find any sp ectrum yet. In other w ords, if W l denotes the set of p oin ts visited so far, then S f n ∩ W l = ∅ . No w if B h +1 is the next r -square to b e scanned (still in a “dilute” wa y with intensit y δ r ), w e seem to b e in go o d shap e since w e know how to control the conditioning S f n ∩ W l = ∅ . Ho w ev er, if w e w an t to rely on the uniform control given by Theorem X.16, w e also need to further condition on S f n ∩ B h +1 6 = ∅ . In other w ords, we need to control the follo wing conditional exp ectation: ˆ P  S f n ∩ B h +1 6 = ∅   S f n ∩ W l = ∅  . It is quite in v olv ed to estimate suc h quan tities. F ortunately , by c hanging our sequen tial scanning pro cedure in to a sligh tly more “abstract” pro cedure, one can a v oid dealing with suc h terms. More precisely , within eac h r -square B , w e will still reveal only a δ r prop ortion of the bits (so that the first obstacle is still taken care of ), but instead of op erating in a sequential wa y (i.e. scanning B 1 , then B 2 and so on), we will gain a lot b y considering the combination of Theorem X.13 and Theorem X.16 in a more abstract fashion. Namely , the follo wing large deviation lemma from [GPS10] captures exactly what w e need in our present situation. Lemma X.17 ([GPS10]) . L et X i , Y i ∈ { 0 , 1 } , i ∈ { 1 , . . . , m } b e r andom variables such that for e ach i Y i ≤ X i a.s. If ∀ J ⊂ [ m ] and ∀ i ∈ [ m ] \ J , we have P  Y i = 1   Y j = 0 , ∀ j ∈ J  ≥ c P  X i = 1   Y j = 0 , ∀ j ∈ J  , (X.13) then if X := P X i and Y := P Y i , one has that P  Y = 0   X > 0  ≤ c − 1 E  e − ( c/e ) X   X > 0  . Recall that B 1 , . . . , B m denotes the set of r -squares whic h tile [0 , n ] 2 . F or eac h i ∈ [ m ], let X i := 1 S ∩ B i 6 = ∅ and Y i := 1 S ∩ B i ∩W 6 = ∅ , where W is an indep enden t uniform random subset of [0 , n ] 2 of in tensit y δ r . This lemma enables us to com bine our t w o main results, Theorems X.16 and X.13, in a v ery nice w a y: By our choice of the in tensit y δ r , Theorem X.16 exactly states that the assumption (X.13) is satisfied for a certain constant c > 0. Lemma X.17 then implies that ˆ P  Y = 0   X > 0  ≤ c − 1 E  e − ( c/e ) X   X > 0  . No w, notice that X = P X i exactly corresp onds to | S ( r ) | while the even t { X > 0 } corresp onds to { S f n 6 = ∅} and the ev en t { Y = 0 } corresp onds to { S f n ∩ W = ∅} . Therefore Theorem X.13 leads us to 128 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION ˆ P  S f n ∩ W = ∅ , S f n 6 = ∅  ≤ c − 1 E  e − ( c/e ) | S ( r ) | , S f n 6 = ∅  ≤ c − 1 X k ≥ 1 ˆ P  | S ( r ) | = k  e − ( c/e ) k ≤ c − 1  X k ≥ 1 2 θ log 2 2 ( k +2) e − ( c/e ) k )  ˆ P  | S ( r ) | = 1  ≤ C ( θ ) ˆ P  | S ( r ) | = 1   n 2 r 2 α 4 ( r , n ) 2 , (X.14) where (X.10) is used in the last step. This shows that on the ev en t that S f n 6 = ∅ , it is very unlik ely that w e do not detect the sp ectral sample on the δ r -dilute set W . This is enough for us to conclude using the follo wing iden tit y: ˆ P  S f n ∩ W = ∅   S f n  = (1 − δ r ) | S f n | = (1 − 1 r 2 α 4 ( r ) ) | S f n | . Indeed, b y a v eraging this iden tit y we obtain ˆ P  S f n ∩ W = ∅ , S f n 6 = ∅  = ˆ E  ˆ P  S f n ∩ W = ∅   S f n  1 S f n 6 = ∅  = ˆ E  (1 − 1 r 2 α 4 ( r ) ) | S f n | 1 S f n 6 = ∅  ≥ Ω(1) ˆ P  0 < | S f n | < r 2 α 4 ( r )  , whic h, combined with (X.14) yields the desired upp er b ound in Theorem X.3. See Problem X.7 for the low er b ound. 5 The radial case The next chapter will fo cus on the existence of exc eptional times in the mo del of dy- namical p ercolation. A main to ol in the study of these exceptional times is the sp ectral measure ˆ Q g R where g R is the Bo olean function g R := {− 1 , 1 } O ( R 2 ) → { 0 , 1 } defined to b e the indicator function of the one-arm ev en t { 0 ← → ∂ B (0 , R ) } . Note that by definition, g R is suc h that k g R k 2 2 = α 1 ( R ). In [GPS10], the following “sharp” theorem on the lo w er tail of S g R is pro v ed. Theorem X.18 ([GPS10]) . L et g R b e the one-arm event in B (0 , R ) . Then for any 1 ≤ r ≤ R , one has ˆ Q g R  0 < | S g R | < r 2 α 4 ( r )   α 1 ( R ) 2 α 1 ( r ) . (X.15) 5. THE RADIAL CASE 129 The pro of of this theorem is in many w a ys similar to the c hordal case (Theorem X.3). An essen tial difference is that the “clustering v.s. en trop y” mechanism is very differen t in this case. Indeed in the chordal left to righ t case, when S f n is conditioned to b e very small, the pro of of Theorem X.3 shows that t ypically S f n lo calizes in some r -square whose lo cation is “uniform” in the domain [0 , n ] 2 . In the radial case, the situation is v ery different: S g R conditioned to b e very small will in fact tend to lo calize in the r -square cen tered at the origin. This means that the analysis of the mesoscopic b eha vior (i.e. the analogue of Theorem X.13) has to b e adapted to the radial case. In particular, in the definition of an annulus structure, the annuli containing the origin pla y a distinguished role. See [GPS10] for complete details. 130 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION Exercise sheet on c hapter X Exercise X.1. Prov e Prop osition X.4. Exercise X.2. Consider the fractal p ercolation pro cess T i , i ≥ 1 in troduced in this c hapter. (Recall that T 2 i ≡ T i ). Recall that in Section 3, it was imp ortan t to estimate the quan tit y P  |T i | = 1  . This is one of the purp oses of the presen t exercise. (a) Let p i := P  |T i | = 1  . By recursion, sho w that there is a constant c ∈ (0 , 1) so that, as i → ∞ p i ∼ cµ i , where µ := 4 p (1 − p + pq ) 3 and q is the probabilit y of extinction for the Galton- W atson tree correp onding to ( T i ) i ≥ 1 . (b) Using the generating function s 7→ f ( s )(= E ( s n um ber of offspring ) of this Galton- W atson tree, and by studying the b eha vior of its i -th iterates f ( i ) , prov e the same result with µ := f 0 ( q ). Chec k that it gives the same formula. (c) Recall the definition of p m,b from Section 3. Let p m, ∞ b e the probability that exactly 1 p erson at generation m survives forever. Prov e that p m, ∞ = (1 − q ) µ m for the same exp onent µ . Pro v e Lemma X.9. Finally , prov e that lim b →∞ p m,b = p m, ∞ . Exercise X.3. Extract from the pro of of Lemma X.8 the answer to the question asked in Figure X.4. Exercise X.4. Prov e that Theorem X.3 ⇒ Theorem X.2 ⇒ Theorem X.1 Exercise X.5. Consider an r -square B ⊂ [ n/ 4 , 3 n/ 4] 2 in the “bulk” of [0 , n ] 2 . (a) Pro v e using Proposition IX.3 that ˆ P  S f n 6 = ∅ and S f n ⊂ B   α 4 ( r , n ) 2 131 132 CHAPTER X. SHARP NOISE SENSITIVITY OF PERCOLA TION (b) Chec k that the clustering Lemma X.14 is consistent with this estimate. Problem X.6. The purp ose of this exercise is to prov e Lemma X.10. (a) Using Prop osition IX.3, pro v e that for an y x ∈ B at distance r / 3 from the bound- ary , P  x ∈ S f n and S f n ∩ B c = ∅   α 4 ( r ) α 4 ( r , n ) 2 . (b) Reco v er the same result using Prop osition IX.4 instead. (c) Conclude using Exercise X.5 that ˆ E  | S f n ∩ ¯ B |   S f n 6 = ∅ and S f n ⊂ B   r 2 α 4 ( r ), where ¯ B ⊂ B is the set of p oints x ∈ B at distance at least r / 3 from the b oundary . (d) Study the second-moment ˆ E  | S f n ∩ ¯ B | 2   S f n 6 = ∅ and S f n ⊂ B  . (e) Deduce Lemma X.10. Problem X.7. Most of this chapter was dev oted to the explanation of the pro of of Theorem X.3. Note that we in fact only discussed how to pro v e the upp er b ound. This is b ecause the low er b ound is muc h easier to pro v e and this is the purp ose of this problem. (a) Deduce from Lemma X.10 and Exercise X.5(a) that the low er b ound on ˆ P  0 < | S f n | < r 2 α 4 ( r )  giv en in Theorem X.3 is correct. I.e., show that there exists a constant c > 0 such that ˆ P  0 < | S f n | < r 2 α 4 ( r )  > c n 2 r 2 α 4 ( r , n ) 2 . (b) (Hard) In the same fashion, prov e the low er b ound part of Theorem X.18. Chapter XI Applications to dynamical p ercolation In this section, we present a very natural mo del where p ercolation undergo es a time- ev olution: this is the mo del of dynamical p ercolation describ ed b elow. The study of the “dynamical” b ehavior of p ercolation as opp osed to its “static” b ehavior turns out to b e very rich: interesting phenomena arise especially at the phase transition p oin t. W e will see that in some sense, dynamical planar p ercolation at criticality is a v ery unstable or c haotic pro cess. In order to understand this instabilit y , sensitivit y of p ercolation (and therefore its F ourier analysis) will pla y a key role. In fact, the original motiv ation for the pap er [BKS99] on noise sensitivity was to solve a particular problem in the sub ject of dynamical p ercolation. [Ste09] provides a recen t survey on the sub ject of dynamical p ercolation. W e mention t hat one can read all but the last section of the presen t c hapter without ha ving read Chapter X. 1 The mo del of dynamical p ercolation This mo del was introduced b y H¨ aggstr¨ om, Peres and Steif [HPS97] inspired by a ques- tion that P aul Malliavin asked at a lecture at the Mittag-Leffler Institute in 1995. This mo del w as in v en ted indep endently by Itai Benjamini. In the general v ersion of this mo del as it w as introduced, giv en an arbitrary graph G and a parameter p , the edges of G switc h bac k and forth according to indep enden t 2-state con tin uous time Marko v chains where closed switc hes to open at rate p and op en switc hes to closed at rate 1 − p . Clearly , the pro duct measure with density p , denoted b y π p in this chapter, is the unique stationary distribution for this Mark o v pro cess. The general question studied in dynamical p ercolation is whether, when we start with the stationary distribution π p , there exist atypical times at which the p erco- 133 134 CHAPTER XI. APPLICA TIONS TO DYNAMICAL PER COLA TION lation structure lo oks markedly different than that at a fixed time. In almost all cases, the term “markedly different” refers to the existence or nonexistence of an infinite con- nected comp onent. Dynamical p ercolation on site p ercolation mo dels, whic h includes our most imp ortan t case of the hexagonal lattice, is defined analogously . W e v ery briefly summarize a few early results in the area. It was sho wn in [HPS97] that b elo w criticalit y , there are no times at which there is an infinite cluster and ab ov e criticalit y , there is an infinite cluster at all times. See the exercises. In [HPS97], exam- ples of graphs which do not percolate at criticality but for whic h there exist exceptional times where p ercolation o ccurs were giv en. (Also giv en were examples of graphs which do p ercolate at criticality but for which there exist exceptional times where p ercolation do es not o ccur.) A fairly refined analysis of the case of so-called spheric al ly symmetric trees w as giv en. See the exercises for some of these. Giv en the ab o v e results, it is natural to ask what happens on the standard graphs that w e w ork with. Recall that for Z 2 , w e hav e seen that there is no percolation at criticalit y . It turns out that it is also known (see b elow) that for d ≥ 19, there is no p ercolation at criticalit y for Z d . It is a ma jor op en question to prov e that this is also the case for intermediate dimensions; the consensus is that this should be the case. 2 What’s going on in high dimensions: Z d , d ≥ 19 ? F or the high dimensional case, Z d , d ≥ 19, it w as sho wn in [HPS97] that there are no exceptional times of p ercolation at criticality . Theorem XI.1 ([HPS97]) . F or the inte ger lattic e Z d with d ≥ 19 , dynamic al critic al p er c olation has no exc eptional times of p er c olation. The key reason for this is a highly non trivial result due to work of Hara and Slade ([HS94]), using earlier work of Barsky and Aizenman ([BA91]), that says that if θ ( p ) is the probabilit y that the origin p ercolates when the parameter is p , then for p ≥ p c θ ( p ) = O ( p − p c ) . (XI.1) (This implies in particular that there is no p ercolation at criticalit y .) In fact, this is the only thing which is used in the pro of and hence the result holds whenev er the p ercolation function satisfies this “finite deriv ativ e condition” at the critical p oint. Outline of Pro of. By countable additivity , it suffices to show that there are no times at which the origin p ercolates during [0 , 1]. W e use a first momen t argument. W e break the time interv al [0 , 1] into m in terv als each of length 1 /m . If w e fix one of these in terv als, the set of edges which are op en at some time during this in terv al is i.i.d. with densit y ab out p c + 1 /m . Hence the probability that the origin p ercolates with resp ect to these set of edges is by (XI.1) at most O (1 /m ). It follows that if N m is the num ber of in terv als where this o ccurs, then E [ N m ] is at most O (1). It is not hard to c hec k that N ≤ lim inf m N m , where N is the cardinality of the set of times during [0 , 1] at which 3. D = 2 AND BKS 135 the origin p ercolates. F atou’s Lemma now yields that E ( N ) < ∞ and hence there are at most finitely man y exceptional times during [0 , 1] at which the origin p ercolates. T o go from here to having no exceptional times can either b e done b y using some rather abstract Mark o v pro cess theory or b y a more hands on approach as w as done in [HPS97] and whic h w e refer to for details. R emark XI.1 . It is kno wn that (XI.1) holds for an y homogeneous tree (see [Gri99] for the binary tree case) and hence there are no exceptional times of p ercolation in this case also. R emark XI.2 . It is w as pro v ed b y Kesten and Zhang [KZ87], that (XI.1) fails for Z 2 and hence the pro of metho d ab o v e to sho w that there are no exceptional times fails. This infinite deriv ativ e in this case migh t suggest that there are in fact exceptional times for critical dynamical p ercolation on Z 2 , an imp ortan t question left op en in [HPS97]. 3 d = 2 and BKS One of the questions p osed in [HPS97] was whether there are exceptional times of p ercolation for Z 2 . It w as this question which was one of the main motiv ations for the pap er [BKS99]. While they did not pro v e the existence of exceptional times of p ercolation, they did obtain the following very interesting result which has a very similar fla v or. Theorem XI.2 ([BKS99]) . Consider an R × R b ox on which we run critic al dynamic al p er c olation. L et S R b e the numb er of times during [0 , 1] at which the c onfigur ation changes fr om having a p er c olation cr ossing to not having one. Then S R → ∞ in pr ob ability as R → ∞ . Noise sensitivit y of p ercolation as well as the ab o v e theorem tells us that certain large scale connectivit y prop erties decorrelate v ery quic kly . This suggests that in some v ague sense ω p c t “c hanges” v ery quic kly as time goes on and hence there migh t be some chance that an infinite cluster app ears since we are given many “chances”. In the next section, w e b egin our study of exceptional times for Z 2 and the hexagonal lattice. 4 The second momen t metho d and the sp ectrum In this section, w e reduce the question of exceptional times to a “second moment metho d” computation which in turn reduces to questions concerning the sp ectral b e- ha vior for specific Boolean functions inv olving percolation. Since p = 1 / 2, our dynamics can b e equiv alen tly defined by having each edge or hexagon b e rerandomized at rate 1. 136 CHAPTER XI. APPLICA TIONS TO DYNAMICAL PER COLA TION The k ey random v ariable which one needs to lo ok at is X = X R := Z 1 0 1 0 ω t ← → R dt where 0 ω t ← → R is of course the even t that at time t there is an op en path from the origin to distance R aw a y . Note that the ab ov e in tegral is simply the Leb esgue measure of the set of times in [0 , 1] at which this o ccurs. W e wan t to apply the second moment metho d here. W e isolate the easy part of the argumen t so that the reader who is not familiar with this method understands it in a more general con text. Ho w ev er, the reader should k eep in mind that the difficult part is alw a ys to pro v e the needed b ound on the second moments whic h in this case is (XI.2). Prop osition XI.3. If ther e exists a c onstant C such that for al l R E ( X 2 R ) ≤ C E ( X R ) 2 , (XI.2) then a.s. ther e ar e exc eptional times of p er c olation. Pro of. F or an y nonnegativ e random v ariable Y , the Cauch y-Sc hw arz inequalit y applied to Y I { Y > 0 } yields P ( Y > 0) ≥ E ( Y ) 2 / E ( Y 2 ) . Hence b y (XI.2), we ha v e that for all R , P ( X R > 0) ≥ 1 /C and hence b y coun table additivit y (as we ha v e a decreasing sequence of even ts) P ( ∩ R { X R > 0 } ) ≥ 1 /C . Had the set of times that a fixed edge is on b een a closed set, then the ab ov e w ould ha v e yielded by compactness that there is an exceptional time of p ercolation with probabilit y at least 1 /C . Ho w ev er, this is not a closed set. On the other hand, this p oin t is very easily fixed b y mo difying the pro cess so that the times eac h edge is on is a closed set and observing that a.s. no new times of p ercolation are introduced by this mo dification. The details are left to the reader. Once we ha v e an exceptional time with p ositiv e probability , ergo dicit y immediately implies that this o ccurs a.s. The first moment of X R is, due to F ubini’s Theorem, simply the probability of our one-arm ev en t, namely α 1 ( R ). The second moment of X R is easily seen to b e E ( X 2 ) = E ( Z 1 0 Z 1 0 1 0 ω s ← → R 1 0 ω t ← → R ds dt ) = Z 1 0 Z 1 0 P (0 ω s ← → R, 0 ω t ← → R ) ds dt (XI.3) whic h is, b y time in v ariance, at most 2 Z 1 0 P (0 ω s ← → R, 0 ω 0 ← → R ) ds. (XI.4) 5. PROOF OF EXISTENCE OF EX CEPTIONAL TIMES ON T 137 The key observ ation no w, which brings us bac k to noise sensitivit y , is that the in tegrand P (0 ω s ← → R, 0 ω 0 ← → R ) is precisely E [ f R ( ω ) f R ( ω  )] where f R is the indicator of the even t that there is an op en path from the origin to distance R aw ay and  = 1 − e − s since lo oking at our pro cess at t w o different times is exactly lo oking at a configuration and a noisy version. What w e hav e seen in this subsection is that pro ving the existence of exceptional times comes down to pro ving a second momen t estimate and furthermore that the in tegrand in this second moment estimate concerns noise s ensitivit y , something for whic h w e hav e already developed a fair num ber of tools to handle. 5 Pro of of existence of exceptional times for the hexagonal lattice via randomized algorithms In [SS10b], exceptional times w ere shown to exist for the hexagonal lattice; this was the first transitive graph for which such a result was obtained. How ever, the metho ds in this pap er did not allow the authors to pro v e that Z 2 had exceptional times. Theorem XI.4 ([SS10b]) . F or dynamic al p er c olation on the hexagonal lattic e T at the critic al p oint p c = 1 / 2 , ther e exist almost sur ely exc eptional times t ∈ [0 , ∞ ) such that ω t has an infinite cluster. Pro of. As we noted in the previous section, tw o differen t times of our mo del can b e view ed as “noising” where the probability that a hexagon is rerandomized within t units of time is 1 − e − t . Hence, b y (IV.2), we ha v e that P  0 ω 0 ← → R, 0 ω t ← → R  = E  f R  2 + X ∅6 = S ⊆ B (0 ,R ) ˆ f R ( S ) 2 exp( − t | S | ) (XI.5) where B (0 , R ) are the set of hexagons in v olv ed in the ev en t f R . W e see in this expression that, for small times t , the frequencies con tributing in the correlation b et w een { 0 ω 0 ← → R } and { 0 ω t ← → R } are of “small” size | S | . 1 /t . Therefore, in order to detect the existence of exceptional times, one needs to achiev e go o d control on the lower tail of the F ourier sp ectrum of f R . The approac h of this section is to find an algorithm minimizing the revealmen t as m uc h as p ossible and to apply Theorem VI I I.1. How ev er there is a difficult y here, since our algorithm migh t ha v e to lo ok near the origin, in which case it is difficult to k eep the rev ealmen t small. There are other reasons for a p otential problem. If R is v ery large and t very small, then if one conditions on the even t { 0 ω 0 ← → R } , since few sites are up dated, the op en path in ω 0 from 0 to distance R will still b e preserv ed in ω t at least up to some distance L ( t ) (further aw ay , large scale connections start to decorrelate). In some sense the geometry asso ciated to the even t { 0 ω ← → R } is “frozen” on a certain scale b etw een time 0 and time t . Therefore, it is natural to divide our correlation 138 CHAPTER XI. APPLICA TIONS TO DYNAMICAL PER COLA TION analysis into tw o scales: the ball of radius r = r ( t ) and the annulus from r ( t ) to R . Ob viously the “frozen radius” r = r ( t ) increases as t → 0. W e therefore pro ceed as follo ws instead. F or any r , we hav e P  0 ω 0 ← → R, 0 ω t ← → R  ≤ P  0 ω 0 ← → r  P  r ω 0 ← → R, r ω t ← → R  ≤ α 1 ( r ) E  f r,R ( ω 0 ) f r,R ( ω t )  , (XI.6) where f r,R is the indicator function of the ev en t, denoted b y r ω ← → R , that there is an op en path from distance r aw ay to distance R a w a y . No w, as ab o v e, we ha v e E  f r,R ( ω 0 ) f r,R ( ω t )  ≤ E  f r,R  2 + ∞ X k =1 exp( − tk ) X | S | = k ˆ f r,R ( S ) 2 . (XI.7) The Bo olean function f r,R someho w av oids the singularity at the origin, and it is p ossible to find algorithms for this function with small rev ealmen ts. In an y case, letting δ = δ r,R b e the rev ealmen t of f r,R , it follo ws from Theorem VI II.1 and the fact that P k k exp( − tk ) ≤ O (1) /t 2 that E  f r,R ( ω 0 ) f r,R ( ω t )  ≤ α 1 ( r , R ) 2 + O (1) δ α 1 ( r , R ) /t 2 . (XI.8) The follo wing prop osition giv es a b ound on δ . W e will sketc h why it is true after- w ards. Prop osition XI.5 ([SS10b]) . L et 2 ≤ r < R . Then δ r,R ≤ O (1) α 1 ( r , R ) α 2 ( r ) . (XI.9) Putting together (XI.6), (XI.8), Prop osition XI.5 and using quasi-m ultiplicativit y of α 1 yields P  0 ω 0 ← → R, 0 ω t ← → R  ≤ O (1) α 1 ( R ) 2 α 1 ( r )  1 + α 2 ( r ) t 2  . This is true for all r and t . If we c ho ose r = r ( t ) = (1 /t ) 8 and ignore o (1) terms in the critical exp onen ts (which can easily b e handled rigorously), w e obtain, using the explicit v alues for the one and t w o-arm critical exp onents, that P  0 ω 0 ← → R, 0 ω t ← → R  ≤ O (1) t − 5 / 6 α 1 ( R ) 2 . (XI.10) No w, since R 1 0 t − 5 / 6 dt < ∞ , b y in tegrating the ab o v e correlation b ound ov er the unit in terv al, one obtains that E  X 2 R  ≤ C E  X R  2 for some constan t C as desired. Outline of pro of of Prop osition XI.5. W e use an algorithm that mimics the one w e used for p ercolation crossings except the presen t setup is “radial”. As in the c hordal case, w e randomize the starting p oint of our exploration pro cess b y choosing a site uniformly on the ‘circle’ of radius R . Then, 5. PROOF OF EXISTENCE OF EX CEPTIONAL TIMES ON T 139 w e explore the picture with an exploration path γ directed tow ards the origin; this means that as in the case of crossings, when the interface encounters an op en (resp. closed) site, it turns say to the left (resp. right), the only difference b eing that when the exploration path closes a lo op around the origin, it contin ues its exploration inside the connected comp onent of the origin. (It is known that this discrete curve conv erges to w ards r adial SLE 6 on T , when the mesh go es to zero.) It turns out that the so- defined exploration path gives all the information we need. Indeed, if the exploration path closes a clo ckwise lo op around the origin, this means that there is a closed circuit around the origin making f r,R equal to zero. On the other hand, if the exploration path do es not close any clo ckwise lo op until it reac hes radius r , it means that f r,R = 1. Hence, we run the exploration path un til either it closes a clockwise lo op or it reaches radius r . This is our algorithm. Neglecting b oundary issues (points near radius r or R ), if x is a p oin t at distance u from 0, with 2 r < u < R / 2, in order for x to be examined b y the algorithm, it is needed that there is an op en path from 2 u to R and the t w o-arm ev en t holds in the ball centered at u with radius u/ 2. Hence for | x | = u , P  x ∈ J  is at most O (1) α 2 ( u ) α 1 ( u, R ). Due to the explicit v alues of the one and t w o-arm exp onen ts, this expression is decreasing in u . Hence, ignoring the b oundary , the rev ealmen t is at most O (1) α 2 ( r ) α 1 ( r , R ). See [SS10b] for more details. W e now assume that the reader is familiar with the notion of Hausdorff dimen- sion. W e let E ⊆ [0 , ∞ ] denote the (random) set of these exceptional times at which p ercolation o ccurs. It is an immediate consequence of F ubini’s Theorem that E has Leb esgue measure zero and hence we should lo ok at its Hausdorff dimension if we wan t to measure its “size”. The first result is the following. Theorem XI.6 ([SS10b]) . The Hausdorff dimension of E is an almost sur e c onstant in [1 / 6 , 31 / 36] . 140 CHAPTER XI. APPLICA TIONS TO DYNAMICAL PER COLA TION It w as conjectured there that the dimension of the set of exceptional times is a.s. 31 / 36. Outline of Pro of. The fact that the dimension is an almost sure constant follo ws from easy 0-1 Laws. The low er b ounds are obtained b y placing a random measure on E with finite so-called α –energies for any α < 1 / 6 and using a result called F rostman’s Theorem. (This is a standard tec hnique once one has go o d control of the correlation structure.) Basically , the 1 / 6 comes from the fact that for an y α < 1 / 6, one can m ultiply the integrand in R 1 0 t − 5 / 6 dt by (1 /t ) α and still b e in tegrable. It is the amount of “ro om to spare” y ou ha v e. If one could obtain b etter estimates on the correlations, one could thereb y improv e the low er b ounds on the dimension. The upp er b ound is obtained via a first moment argument similar to the pro of of Theorem XI.1 but no w using (I I.1). Before moving on to our final metho d of dealing with the sp ectrum, let us consider what w e might ha v e lost in the ab o v e argument. Using the ab o v e argument, w e op- timized things b y taking r ( t ) = (1 /t ) 8 . Ho w ev er, at time t compared to time 0, w e ha v e noise whic h is ab out t . Since we now know the exact noise sensitivit y exp onen t, in order to obtain decorrelation, the noise level should b e at least ab out the negative 3 / 4th p o w er of the radius of the region w e are lo oking at. So, even ts in our annu- lus should decorrelate if r ( t ) >> (1 /t ) 4 / 3 . This suggests there migh t b e p otential for impro v emen t. Note we used an inner radius whic h is 6 times larger than p oten tially necessary (8 = 6 × 4 / 3). This 6 is the same 6 b y whic h the result in Theorem VI I I.4 differed by the true exp onent (3 / 4 = 6 × 1 / 8) and the same 6 explaining the gap in Theorem XI.6 (1 − 1 / 6) = 6 × (1 − 31 / 36). This last difference is also seen b y comparing the exp onen ts in (XI.10) and the last term in (XI.11) b elow. 6 Pro of of existence of exceptional times via the geometric approac h of the sp ectrum Recall that our third approach for proving the noise sensitivity of p ercolation crossings w as based on a geometrical analysis of the spectrum, viewing the sp ectrum as a random set. This approac h yielded the exact noise sensitivit y exponent for p ercolation crossings for the hexagonal lattice. This approac h can also b e used here as w e will no w explain. Tw o big adv antages of this approach are that it succeeded in pro ving the existence of exceptional times for p ercolation crossings on Z 2 , something whic h [SS10b] w as not able to do, as well as obtaining the exact Hausdorff dimension for the set of exceptional times, namely the upp er b ound of 31 / 36 in the previous result. Theorem XI.7 ([GPS10]) . F or the triangular lattic e, the Hausdorff dimension of E is almost sur ely 31 / 36 . Pro of. As explained in the previous section, it suffices to lo w er the 5 / 6 in (XI.10) to 5 / 36. (Note that (XI.10) was really only obtained for n um b ers strictly larger than 5 / 6, 6. EXCEPTIONAL TIMES VIA THE GEOMETRIC APPRO A CH 141 with the O (1) dep ending on this num ber; the same will be true for the 5 / 36.) Let s ( r ) b e the inv erse of the map r → r 2 α 4 ( r ) ∼ r 3 / 4 . So more or less, s ( r ) := r 4 / 3 . Using Theorem X.18, we obtain the following: E  f R ( ω 0 ) f R ( ω t )  = X S exp( − t | S | ) ˆ f R ( S ) 2 = ∞ X k =1 X S : | S |∈ [( k − 1) /t,k/t ) exp( − t | S | ) ˆ f R ( S ) 2 ≤ ∞ X k =1 exp( − k ) ˆ Q  | S f R | < k /t  ≤ O (1) ∞ X k =1 exp( − k ) α 1 ( R ) 2 α 1 ( s ( k /t )) ≤ O (1) α 1 ( R ) 2 ∞ X k =1 exp( − k )( k t ) 4 / 3 × 5 / 48 ≤ O (1) α 1 ( R ) 2 ( 1 t ) 5 / 36 . (XI.11) This completes the pro of. (Of course, there are o (1) terms in these exp onen ts which w e are ignoring.) W e hav e done a lot of the w ork for proving that there are exceptional times also on Z 2 . Theorem XI.8 ([GPS10]) . F or dynamic al p er c olation on Z 2 at the critic al p oint p c = 1 / 2 , ther e exist almost sur ely exc eptional times t ∈ [0 , ∞ ) such that ω t has an infinite cluster. Pro of. s ( r ) is defined as it w as b efore but no w w e cannot sa y that s ( r ) is ab out r 4 / 3 . Ho w ev er, we can say that for some fixed δ > 0, we hav e that for all r , s ( r ) ≥ r δ (XI.12) F rom the previous proof, w e still hav e E  f R ( ω 0 ) f R ( ω t )  α 1 ( R ) 2 ≤ O (1) ∞ X k =1 exp( − k ) 1 α 1 ( s ( k /t )) . (XI.13) Exactly as in the pro of of Theorem XI.4, w e need to show that the right hand side is in tegrable near 0 in order to carry out the second moment argument. Quasi-m ultiplicativit y can b e used to show that α 1 ( s (1 /t )) ≤ k O (1) α 1 ( s ( k /t )) . (XI.14) 142 CHAPTER XI. APPLICA TIONS TO DYNAMICAL PER COLA TION (Note that if things b ehav ed exactly as pow er laws, this would b e clear.) Therefore the ab o v e sum is at most O (1) ∞ X k =1 exp( − k ) k O (1) α 1 ( s (1 /t )) ≤ O (1) 1 α 1 ( s (1 /t )) (XI.15) V. Beffara has shown that there exists  0 > 0 suc h that for all r , α 1 ( r ) α 4 ( r ) ≥ r  0 − 2 . (XI.16) Note that Theorem VI.4 and (VI.7) tell us that the left hand side is larger than Ω(1) r − 2 . The ab o v e tells us that we get an (important) extra pow er of r in (VI.7). It follo ws that 1 α 1 ( s (1 /t )) ≤ α 4 ( s (1 /t )) s (1 /t ) 2 −  0 = (1 /t ) s (1 /t ) −  0 . (XI.17) (XI.12) tells us that the last factor is at most t η for some η > 0 and hence the relev ant integral con v erges as desired. The rest of the argument is the same. One can also consider exceptional times for other even ts, such as for example times at whic h there is an infinite cluster in the upp er half-plane or times at which there are t w o infinite clusters in the whole plane, and consider the corresp onding Hausdorff dimension. A n um b er of results of this t ype, which are not sharp, are given in [SS10b] while v arious sharp results are given in [GPS10]. Exercise sheet of Chapter XI Exercise XI.1. Prov e that on an y graph b elow criticality , there are no times at which there is an infinite cluster while ab ov e criticalit y , there is an infinite cluster at all times. Exercise XI.2. Consider critical dynamical p ercolation on a general graph satisfying θ ( p c ) = 0. Show that a.s. { t : ω t p ercolates } has Leb esgue measure 0. Exercise XI.3. (Somewhat hard). A spheric al ly symmetric tree is one where all ver- tices at a giv en level ha v e the same n um b er of c hildren, although this n um b er ma y dep end on the given lev el. Let T n b e the num ber of vertices at the n th level. Show that there is p ercolation at p if X n 1 p − n T n < ∞ Hin t: Let X n b e the n um b er of vertices in the n th lev el which are connected to the ro ot. Apply the second moment metho d to the sequence of X n ’s. The con v ergence of the sum is also necessary for p ercolation but this is harder and y ou are not asked to show this. This theorem is due to Russell Lyons. Exercise XI.4. Show that if T n is n 2 2 n up to m ultiplicativ e constants, then the critical v alue of the graph is 1 / 2 and we p ercolate at the critical v alue. 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