Strong convergence of partial match queries in random quadtrees

Strong con v ergence of partial matc h queries in random quadtrees Nicolas Curien ´ Ec ole Normale Sup ´ erieur e Abstract W e pro ve that the rescaled costs of partial match queries in a random tw o-dimensional quadtree con v erge almost sur ely tow ards a random limi t which is identified as the terminal v alue of a martingale. Our approach shares man y similarities with the theory of self-similar fragmen tations. 1 In tro duction The quadtree structure is a storage system designed for retri eving m ultidimensional data. It has first b een introduced b y Fink el & Bentley [7] and was studied thoroughly in computer science. The goal of this work is t o study fine properties of t he so-called partial matc h queries in random (uniform) t wo-dimensional quadt rees. Let us briefly recall the mo del. Consider a Poisson point pro cess Π on R + × [0 , 1] 2 with in tensity d t ⊗ d x d y . Let (( τ i , x i , y i ) , i ≥ 1) b e the atoms of Π rank ed in the increasing order of their τ -component. W e define a pro cess (Quad( t )) t ≥ 0 with v alues in finite co verings of [0 , 1] 2 b y closed rectangles with disjoint in teriors as follo ws. W e initially start with the unit square Quad(0) := [0 , 1] 2 . At eac h time an atom ( τ i , x i , y i ) of the Poisson pro cess Π falls in a rectangle of Quad( τ − i ) it splits this rectangle into four subrectangles according to the horizontal and vertical co ordinates of x i and y i . Observ e that a.s., for ev ery i ≥ 1, there exists a unique rectangle of Quad( τ i ) such that ( x i +1 , y i +1 ) is in its in terior, hence the pro cess (Quad( t )) t ≥ 0 is a.s. well defined. In this w ork, we c hose to fo cus on the contin uous time version of the random quadtree but all the results can b e transferred to the random quadtree with a fixed num b er of p oin ts by standard dep oissonization techniques, see e.g. [3] or [5, Lemma 1]. Figure 1: The first 7 splittings of a quadtree. W e shall b e int erested in the so-called partial matc h query (see [9, p 523]). Equiv alently , for x ∈ [0 , 1], we fo cus on the num b er N t ( x ) of rectangles in the quadtree at time t whose horizon tal co ordinate intersects x , that is, N t ( x ) := #  R ∈ Quad( t ) : R ∩ [( x, 0) , ( x, 1)] 6 = ∅  . 1 1 INTR ODUCTION 2 The first study of the partial match has b een carried out b y Fla jolet, Gonnet, Puec h and Robson in [8]. They prov ed that if U is uniformly distributed ov er [0 , 1] and indep enden t of Π then E [ N t ( U )] is asymptotically equiv alent to κ · t β as t tends to infinit y , where β := √ 17 − 3 2 , and κ > 0 is some explicit constant. The asymptotic of the exp ected v alue of N t ( x ) for a fixed p oin t x ∈ [0 , 1] has recently b een obtained in [5], it reads t − β E [ N t ( x )] − − − → t →∞ K 0 · h ( x ) , (1) where K 0 := Γ(2 β + 2)Γ( β + 2) 2Γ 3 ( β + 1)Γ 2 ( β / 2 + 1) and h : u ∈ [0 , 1] 7− →  u (1 − u )  β / 2 . In a very recen t breakthrough [4], Broutin, Neininger and Sulzbac h used the “contrac- tion method” to obtain a conv ergence in distribution as t → ∞ of the rescaled pro cesses { t − β N t ( x ) : 0 ≤ x ≤ 1 } tow ards a random con tinuous pro cess { ˜ M ∞ ( x ) : 0 ≤ x ≤ 1 } charac- terized by a recursive decomp osition. The main result of the present wor k is to show that this conv ergence actually holds in a stronger sense: Theorem 1. F or every x ∈ [0 , 1] we have the fol lowing almost sure c onver genc e t − β N t ( x ) a.s. − − − → t →∞ K 0 · ˜ M ∞ ( x ) . 1 I NT R O D UC T I O N 2 T h e fi r s t s t u d y of t h e p ar t i al m at c h h as b e e n c ar r i e d ou t b y F l a j ol e t , G on n e t , P u e c h an d R ob s on i n [ ? ] . T h e y p r o v e d t h at i f U i s u n i f or m l y d i s t r i b u t e d o v e r [ 0 , 1] an d i n d e p e n d e n t of Π then E [ N t ( U ) ] i s as y m p t ot i c al l y e q u i v al e n t t o κ · t β as t te nd s to in fin it y , whe re β := √ 17 − 3 2 , an d κ > 0 i s s om e e x p l i c i t c on s t an t . T h e as y m p t ot i c of t h e e x p e c t e d v al u e of N t ( x ) f or a fi x e d p oi n t x ∈ [0 , 1] h as r e c e n t l y b e e n ob t ai n e d i n [ ? ] , i t r e ad s t − β E [ N t ( x )] −− −→ t →∞ K 0 · h ( x ) , ( 1) wh er e K 0 := Γ (2 β + 2) Γ ( β + 2) 2 Γ 3 ( β + 1) Γ 2 ( β / 2 + 1) an d h : u ∈ [0 , 1] �− →  u (1 − u )  β / 2 . I n a v e r y r e c e n t b r e ak t h r ou gh [ ? ] , B r ou t i n , Ne i n i n ge r an d S u l z b ac h u s e d t h e “c on t r ac - t i on m e t h o d ” t o ob t ai n a c on v e r ge n c e i n d i s t r i b u t i on as t →∞ of t h e r e s c al e d p r o c e s s e s { t − β N t ( x ): 0 ≤ x ≤ 1 } t o w ar d s a r an d om c on t i n u ou s p r o c e s s { ˜ M ∞ ( x ): 0 ≤ x ≤ 1 } c h ar ac - t e r i z e d b y a r e c u r s i v e d e c om p os i t i on . T h e m ai n r e s u l t of t h e p r e s e n t w or k i s t o s h o w t h at t h i s c on v e r ge n c e ac t u al l y h ol d s i n a s t r on ge r s e n s e : T heo rem 1 . Fo r e v e r y x ∈ [0 , 1] we ha v e the f ol l owi ng al m os t s u r e c onv e r g e nc e t − β N t ( x ) a.s. −− − → t →∞ K 0 · ˜ M ∞ ( x ) . ! 0.5 0 0.5 1 ! 0.5 0 0.5 1 ! 0.5 0 0.5 1 0 0.5 1 0 0.5 1 F i gu r e 2: An i l l u s t r at i on of t h e s t r on g c on v e r ge n c e of t h e p ar t i al m at c h q u e r i e s . T h e c u r v e s ab o v e t h e q u ad t r e e r e p r e s e n t t h e r e n or m al i z e d p r o c e s s e s t − β ( N t ( x )) x ∈ [0 , 1] f or t = 20 , 50 , 100 , 500 an d 3000. T h e r an d om v ar i ab l e ˜ M ∞ ( x ) i s ob t ai n e d as t h e l i m i t i n g v al u e of a c on t i n u ou s - t i m e m ar - t i n gal e { M t ( x ): t ≥ 0 } de fin ed b y ( ?? ) w h i c h i s a v ar i at i on on t h e m ar t i n gal e i n t r o d u c e d i n [ ? , S e c t i on 3. 2] . T h e m ar t i n gal e M t ( x ) m u s t b e c on s i d e r e d as an an al ogou s of t h e w e l l - k n o w n M al t h u s i an m ar t i n gal e i n f r agm e n t at i on t h e or y , s e e [ ? ] . I n d e e d , t h e c on v e r ge n c e of t − β N t ( x ) t o w ar d s t h e l i m i t i n g v al u e of M t ( x ) i s s i m i l ar t o t h e p r o of of [ ? , T h e or e m 5] an d r e q u i r e s s om e of t h e e s t i m at e s of [ ? ] . S e e S e c t i on ?? f or c om m e n t s . He r e i s a d i r e c t c or ol l ar y of T h e or e m ?? c om b i n e d w i t h t h e r e s u l t s of [ ? ]: Figure 2: An illustration of the strong con v ergence of the partial match queries. The curv es ab o v e the quadtree represent the renormalized processes t − β ( N t ( x )) x ∈ [0 , 1] for t = 20 , 50 , 100 , 500 and 3000. The random v ariable ˜ M ∞ ( x ) is obtained as the limiting v alue of a contin uous-time mar- tingale { M t ( x ) : t ≥ 0 } defined by (2) which is a v ariation on the martingale introduced in [4, Section 3.2]. The martingale M t ( x ) m ust be considered as an analogous of the w ell-kno wn Malth usian martingale in fragmentat ion theory , see [1]. Indeed, the conv ergence of t − β N t ( x ) to wards the limiting v alue of M t ( x ) is similar to the pro of of [2, Theorem 5] and requires some of the estimates of [4]. See Section 5 for comments. Here is a direct corollary of Theorem 1 combined with the results of [4]: 2 THE MAR TINGALES 3 Corollary 2. We have the fol lowing c onver genc e in pr ob ability  t − β N t ( x )  x ∈ [0 , 1] ( P ) − − − → t →∞ K 0 ·  ˜ M ∞ ( x )  x ∈ [0 , 1] , for the uniform metric k . k ∞ . The note is organized as follows: W e first introduce the martingales whose limit v alue furnishes the process { ˜ M ∞ ( x ) : 0 ≤ x ≤ 1 } and recall some of its prop erties. The third section is devoted to an estimate on the smallest and the largest rectangle in the quadtree at time t > 0 which is used in the pro of of the main result. In the last section we give some commen ts related to fragmentation theory . Ac knowledg ment: I am grateful to Nicolas Broutin, Ralph Neininger and Henning Sulzbac h for keeping me informed about their recent work on quadtrees. Sp ecial thanks go to Adrien Joseph and Henning Sulzbach for precious comments on a first ver sion of this w ork. 2 The martingales In this section we introduce the martingale which the pro of of Theorem 1 is based on and compare it with the one introduced in [4, Section 3.2]. W e start by setting some notation. Recall the definitions of β and of the map h given in the Introduction. The genealogy of the rectangles app earing in the quadtree pro cess (Quad( t )) t ≥ 0 can b e enco ded on the full infinite 4-ary tree T 4 := [ n ≥ 0 { 1 , 2 , 3 , 4 } n . The first square [0 , 1] 2 corresponds to the w ord ∅ ∈ T 4 and when a rectangle enco ded by a word u ∈ T 4 is split, w e enco de the four resulting subrectangles by u 1 , u 2 , u 3 and u 4 in coun ter clo ckwise order starting with the north-east rectangle. This genealogy induces a notion of ancestor, offspring... on the rectangles of ∪ Quad( t ). The generation of a rectangle R that app ears in the quadt ree pro cess is the length of its encoding w ord in T 4 and is denoted b y Gen( R ) (the length of ∅ is 0 by conv enti on). The time of app earance of R is the first t > 0 such that R ∈ Quad( t ) and is denoted by Time( R ). F or t > 0 and x ∈ [0 , 1] w e denote by Q t ( x ) := { Q i t ( x ) } i ≥ 1 the set of all rectangles b elonging to Quad( t ) whose first co ordinate in tersects x . The left-most and right-most horizon tal co ordinates of Q i t ( x ) are denoted b y ` i t and r i t and w e write x i t := x − ` i t r i t − ` i t , for the “p osition” of x inside the rectangle Q i t ( x ). By standard properties of t he P oisson point pro cess Π, conditionally on the sigma-field F t generated by (Quad( u )) 0 ≤ u ≤ t the num b er of rectangles in Q t + s ( x ) whose ancestor at time t is Q i t ( x ) has the same distribution as N 0 s · λ ( Q i t ( x )) ( x i t ) , where λ stands for the tw o-dimensional Leb esgue measure on [0 , 1] 2 and N 0 . ( . ) is an indep en- den t copy of the pro cess N . ( . ). Prop osition 3. F or every x ∈ [0 , 1] , the pr o c ess t 7− → M t ( x ) := X i ≥ 1 λ  Q i t ( x )  β h  x i t  , (2) 3 A GEOMETRIC ESTIMA TE 4 is a c ontinuous-time non-ne gative martingale whose limiting value is denote d by ˜ M ∞ ( x ) . Before going into the proof of Prop osition 3 let us emphasize the difference b etw een this martingale and the one considered in [4]. Fix a generation n ≥ 0 and denote by { ˜ Q i n ( x ) } 1 ≤ i ≤ 2 n the rectangles at generation n that are ab o ve the p oint x ∈ [0 , 1] and write ˜ x i n for the p osition of x inside ˜ Q i n ( x ). Then from [4, Section 3.2], ˜ M n ( x ) := 2 n X i =1 λ  ˜ Q i n ( x )  β h ( ˜ x i n ) , is a discrete-time non-negative martingale whose limiting v alue is ˜ M ∞ ( x ). The pro of of this fact is based on the following lemma that sho ws that the exp ectation is kept after one splitting. Lemma 4. We have E " 2 X i =1 λ ( ˜ Q i 1 ( x )) β h ( ˜ x i 1 ) # = h ( x ) . This lemma was prov ed in [4] but is also rigorously equiv alent to the fact h solv es the in tegral equation that was already considered in [5, Section 5]. The difference b et ween the martingales ˜ M n ( x ) and M t ( x ) is that in the latter case we consider the splittings chronologically as they o ccur whereas in the first case w e consider them generation after generation. It should b e clear that the order in which the splittings are considered do es not change the martingale prop ert y and we could use Lemma 4 to show directly that M t ( x ) is a martingale for ev ery x ∈ [0 , 1]. How ev er it will be useful for our purpose to link M t ( x ) to its discrete time analog ˜ M n ( x ). Pr o of of Pr op osition 3. Fix x ∈ [0 , 1]. It easily follo ws from [4, Section 3.2] (see also [3]) that ˜ M n ( x ) con verges in L p for an y p > 1 tow ards ˜ M ∞ ( x ) and th us E [ ˜ M ∞ ( x ) | F t ] = lim E [ ˜ M n ( x ) | F t ] almost surely as n → ∞ . By the Marko v prop ert y applied at time t > 0 and using the martingale structure of ˜ M n ( x ) w e deduce that E [ ˜ M n ( x ) | F t ] = X i ≥ 1 λ ( Q i t ( x )) β h ( x i t ) 1 Gen( Q i t ( x )) 1 tow ards ˜ M ∞ ( x ). This completes the pro of of the prop osition. The pro cess x ∈ [0 , 1] 7→ ˜ M ∞ ( x ) was used in [4] to construct a fixed point to a recur- siv e equation in distribution. In particular it is prov ed that x 7→ ˜ M ∞ ( x ) is almost surely con tinuous. 3 A geometric estimate In this section w e establish a rough contr ol on the area of the largest and the smallest rectangle of Quad( t ). The reader can skip this part on first reading. F or t > 0, let I t := inf λ ( R ) and S t := sup λ ( R ) where the infimum and suprem um run ov er all the rectangles R ∈ Quad( t ). W e will roughly pro ve that t − 4+ o (1) ≤ I t and S t ≤ t − 1+ o (1) . The formal statemen t is the following: 3 A GEOMETRIC ESTIMA TE 5 Lemma 5. F or every ε > 0 we have − 4 − ε < lim inf t →∞ log  I t ) log( t ) a.s. and lim sup t →∞ log  P ( S t > t − 1+ ε )  log( t ) = −∞ . Pr o of. Lower bound. Let ( x i ) 1 ≤ i ≤ n and ( y i ) 1 ≤ i ≤ n b e the co ordinates of the p oin ts of Π that o ccur b efore time t . By standard prop erties of Poisson point processes, conditionally on n , ( x i ) and ( y i ) are indep enden t sequences of n i.i.d. uniform v ariables o ver [0 , 1]. A simple geometric argument (see Fig. 3 b elo w) shows that I t ≥ min i 6 = j 1 ≤ i,j ≤ n | x i − x j | · min i 6 = j 1 ≤ i,j ≤ n | y i − y j | . Figure 3: Illustration of the low er b ound. Let ε > 0. By classical results on the uniform sieve of the interv al [0 , 1], if x 1 , x 2 , ... are i.i.d. uniform p oin ts o ver [0 , 1] then min {| x i − x j | : 1 ≤ i, j ≤ n, i 6 = j } is asymptotically larger than n − 2 − ε a.s. . Indeed we hav e P x n ∈ n − 1 [ i =1 [ x i − n − 2 − ε , x i + n − 2 − ε ] ! ≤ 2 n − 1 − ε , and an application of Borel Can telli’s lemma prov es the claim. Since ev entuall y t/ 2 ≤ n ≤ 2 t and t 7→ I t is decreasing we almost surely hav e I t ≥ t − 4 − 2 ε ev entually . Upper bound. W e use a common tec hnique in fragmen tation theory: the tagged particle. Assume that indep enden tly of the quadtree pro cess (Quad( t )) t ≥ 0 w e are giv en an i ndep enden t v ariable ( U, V ) uniformly distributed ov er [0 , 1] 2 . The rectangle R • t ∈ Quad( t ) cont aining ( U, V ) is called the “tagged rectangle” at time t . The distribution of ( R • t ) t ≥ 0 is equiv alently described as follo ws. W e start with R • 0 := [0 , 1] 2 and define the pro cess R • t iterativ ely: when a splitting occurs at time τ inside the tagged rectangle R • τ − , then R • τ is one of the four subrectangles of R • τ − c hosen prop ortionally to its tw o-dimensional Leb esgue measure. It is clear from the ab ov e construction that the tagged rectangle at generation n has a t wo-dimensional Leb esgue measure which is distributed according to 2 n Y i =1 U i , where U i are indep enden t ident ically distributed v ariables with densit y 2 m 1 0 0. F or every ε > 0 we hav e P ( λ ( R • t ) > t − 1+2 ε ) ≤ P (Gen( R • t ) ≤ t ε , λ ( R • t ) > t − 1+2 ε ) + P (Gen( R • t ) > t ε , λ ( R • t ) > t − 1+2 ε ) . 4 PR OOF OF THE MAIN RESUL TS 6 By our preceding remark, for large t > 0, the second term of the right-hand side is b ounded ab o ve b y 2 −b t ε c . F or the first term, remark that if λ ( R • t ) > t − 1+2 ε then for every 0 ≤ s ≤ t , the intensit y at whic h a particle falls inside R • s is larger than t − 1+2 ε , th us by standard properties of exponential v ariables w e ha ve P (Gen( R • t ) ≤ t ε , λ ( R • t ) > t − 1+2 ε ) ≤ P ( P ( t 2 ε ) ≤ t ε ) where P ( t 2 ε ) is a Poisson distribution of mean t 2 ε . Let us make this more precise. F or n ≥ 0, denote ˜ R • n the tagged rectangle at generation n . The rectangle ˜ R • n th us lives for an exp onen tial time of parameter λ ( ˜ R • n ) b efore it splits. W e deduce that if E 0 , E 1 , ... is an i.i.d. sequence of exponential v ariables of parameter one whic h is also independent of λ ( ˜ R • 0 ) , λ ( ˜ R • 1 ) , ... then P (Gen( R • t ) ≤ t ε , λ ( R • t ) > t − 1+2 ε ) = P   Gen( R • t ) X i =0 λ ( ˜ R • i ) − 1 · E i > t , λ ( ˜ R • t ) > t − 1+2 ε , Gen( R • t ) ≤ t ε   ≤ P   t 1 − 2 ε Gen( R • t ) X i =0 E i > t , Gen( R • t ) ≤ t ε   ≤ P   b t ε c X i =0 E i > t 2 ε   = P  P ( t 2 ε ) ≤ t ε  . The last probability b eing b ounded ab o ve by c − 1 exp( − ct ε ) for some c > 0. Gathering-up the pieces, there exists c 0 > 0 such that we hav e P ( λ ( R • t ) > t − 1+2 ε ) ≤ c 0− 1 exp( − c 0 t ε ) for ev ery t > 0. W e then use the tagged fragmen t to b ound S t from ab ov e. Notice that at any time t > 0 the tagged fragment R • t has a probabilit y S t of b eing the largest fragment, thus P ( λ ( R • t ) > t − 1+2 ε ) ≥ P ( S t > t − 1+2 ε ) t − 1+2 ε , whic h together with the previous b ound easily completes the pro of of the lemma. 4 Pro of of the main results 4.1 Pro of of Theorem 1 Pr o of. Let us first describe the main idea of the proof, whic h is similar to [2, Theorem 5] and roughly sp eaking reduces to apply a law of large num b er after conditioning at a large time t > 0. Fix x ∈ [0 , 1] and let T be muc h larger than t . Conditionally on F t the v ariable N T ( x ) is the sum of N t ( x ) indep enden t con tributions corresp onding to the offsprings of the rectangles ab ov e x at time t . By standard prop erties of the quadtree construction, the num b er of descendan ts of the rectangle Q i t ( x ) inside Q i T ( x ) is distributed as N 0 λ ( Q i t ( x ))( T − t ) ( x i t ) where N 0 . ( . ) is an independent copy of the pro cess N . ( . ). Thus if for x ∈ [0 , 1] and t ≥ 0 we set f ( t, x ) := E [ N t ( x )], w e hav e E  N T ( x ) | F t  = X i ≥ 1 f  λ ( Q i t ( x ))( T − t ) , x i t  . W e now let T → ∞ in the last display . Since for eac h rectangle Q i t ( x ) of Q t ( x ) w e ha ve ( T − t ) λ ( Q i t ( x )) → ∞ then T − β f ( λ ( Q i t ( x ))( T − t ) , x i t ) tends to K 0 λ ( Q i t ) β h ( x i t ) as T → ∞ . 4 PR OOF OF THE MAIN RESUL TS 7 Henceforth we hav e       T − β E  N T ( x ) | F t  − K 0 · X i ≥ 1 λ ( Q i t ( x )) β h ( x i t )       =   T − β E  N T ( x ) | F t  − K 0 · M t ( x )   − − − − → T →∞ 0 . (3) The strategy of the proof is now clear: conditionally on F t , b y the law of large n umbers, T − β N T ( x ) will b e very close to its (conditional) mean which is close to K 0 · M t ( x ) which con verges tow ards K 0 · ˜ M ∞ . This will imply the theorem. T o make this statemen t precise, and in particular get an almost sure conv ergence (a con vergence in probability w ould b e muc h easier to prov e), we shall need the estimates on the exp ectation and the v ariance of the pro cess N t ( x ) prov ed b y Broutin, Neininger and Sulzbac h. It follows from Prop osition 11 and Theorem 5 in [4] that there exist t wo constan ts C > 0 and δ > 0 suc h that for every t > 0 we hav e sup 0 ≤ x ≤ 1   t − β E [ N t ( x )] − K 0 · h ( x )   ≤ C t − δ (4) sup 0 ≤ x ≤ 1  V ar( N t ( x ))  ≤ C ( t 2 β + t ) , (5) the term t app earing in the last line since the v ariance of N t ( x ) is of order t near t = 0. W e first make (3) quan titative in T . Fix α ≥ 5 such that δ ( α − 5) > 1 and for t > 0, set T := t α . By the choice of t versus T and the lo wer bound of Lemma 5 we get that almost surely , there exists a random time τ suc h that for t ≥ τ w e hav e ( T − t ) inf { λ ( Q i t ( x )) : i ≥ 1 } ≥ t α − 5 . Henceforth using the b ound (4), we hav e for t ≥ τ   ( T − t ) − β E  N T ( x ) | F t  − K 0 · M t ( x )   ≤ X i ≥ 1    ( T − t ) − β f  λ ( Q i t ( x ))( T − t ) , x i t ) − K 0 · λ ( Q i t ( x )) β h ( x i t )    ≤ C N t ( x ) t − δ ( α − 5) . Since N t ( x ) is clearly less that the num b er of p oin ts fallen so far, the definition of α implies that N t ( x ) t − δ ( α − 5) go es to 0 almost su rely . Since M t ( x ) is almost surely bounded, we pro v ed that with our c hoice of α w e ha ve | T − β E [ N T ( x ) | F t ] − K 0 · M t ( x ) | → 0 almost surely as t → ∞ and using Prop osition 3 if follows that   T − β E [ N T ( x ) | F t ] − K 0 · ˜ M ∞ ( x )   a.s. − − − → t →∞ 0 . (6) Recall that conditionally on F t the contributions of each rectangle Q i t ( x ) to N T ( x ) are independent, thus we hav e T − 2 β E h  N T ( x ) − E [ N T ( x ) | F t ]  2 | F t i = T − 2 β X i ≥ 1 V ar  N λ ( Q i t ( x ))( T − t ) ( x i t )  ≤ C  X i ≥ 1 λ ( Q i t ( x )) 2 β + T 1 − 2 β X i ≥ 1 λ ( Q i t ( x ))  ≤ C  S 2 β − 1 t + T 1 − 2 β  X R ∈ Quad( t ) λ ( R ) = C ( S 2 β − 1 t + T 1 − 2 β ) , where w e used the b ound (5) to reach the second line and the fact that 2 β > 1 to go from the third to the last line. Let ε > 0. Applying the standard Marko v inequality conditionally on F t w e obtain P  T − β   N T ( x ) − E [ N T ( x ) | F t ]   ≥ ε   F t  ≤ C ε − 2  S 2 β − 1 t + T 1 − 2 β  . (7) 5 FRA GMENT A TION PR OCESS WITH P ARAMETER 8 W e no w tak e T k := (1 + η ) k and t k := (1 + η ) k/α for k = 1 , 2 , 3 ... and η > 0. Since E [ S 2 β − 1 t k ] is less than t (2 β − 1)( ε − 1) k + P ( S t k > t − 1+ ε k ), using Lemma 5 and (7) w e see that P ( T − β k | N T k ( x ) − E [ N T k ( x ) | F t k ] | ≥ ε ) is summable in k . Applying Borel-Cantelli’s lemma and using (6) we deduce that for every η > 0 we ha ve the following almost sure conv ergence (1 + η ) − kβ N (1+ η ) k ( x ) a.s. − − − − → k →∞ K 0 · ˜ M ∞ ( x ) . T o extend this result to the whole pro cess we use the fact that t 7→ N t ( x ) is increasing in t whic h implies (1 + η ) − ( k +1) β N (1+ η ) k ( x ) ≤ s − β N s ( x ) ≤ (1 + η ) − kβ N (1+ η ) k +1 ( x ) for every (1 + η ) k ≤ s ≤ (1 + η ) k +1 and k ≥ 1. Since this holds for any η > 0 w e easily deduce that t − β N t ( x ) almost surely con verges to wards K 0 · ˜ M ∞ ( x ). This comple tes the proof of Theorem 1. 4.2 Pro of of Corollary 2 Pr o of. (Sketc h) Theorem 1 implies the con vergence of the finite dimensional marginals of t − β N t ( . ) to wards those of K 0 · ˜ M ∞ ( . ) in probabilit y: F or any 0 ≤ x 1 , ..., x k ≤ 1 we hav e t − β  N t ( x i )  1 ≤ i ≤ k ( P ) − − − → t →∞ K 0  ˜ M ∞ ( x i )  1 ≤ i ≤ k . (8) F urthermore Theorem 1 of [4] provides the tigh tness of the processes t − β ( N t ( . )) for the uniform metric: for every ε > 0 there exists η > 0 such that for t > 0 large enough we hav e P ( ω t − β N t ( . ) ( η ) ≤ ε ) ≥ 1 − ε, (9) where ω g ( η ) = sup {| g ( x ) − g ( y ) | , | x − y | ≤ η } is the mo dulus of contin uit y of the function g . Recalling that x 7→ ˜ M ∞ ( x ) is almost surely contin uous, w e can combine (8) and (9) to get that t − β N t ( . ) conv erges in probability for the L ∞ metric tow ards ˜ M ∞ ( . ). W e leav e the details to the reader. Op en Question. It is b eliev able that the conv ergence of Corollary 2 actually holds almost surely , that is  t − β N t ( x )  x ∈ [0 , 1] a.s. − − − → t →∞ K 0 ·  ˜ M ∞ ( x )  x ∈ [0 , 1] , for the uniform metric k . k ∞ . 5 F ragmenta tion pro cess with parameter In this section we comment at an informal level on the strategy adopted in this work and on p ossible extensions of our techniques. F ragmen tation theory . Let us briefly recall some basics ab out fragmentation theory . W e stic k to a v ery simple case for sake of clarity . F or more details, we refer to [1]. T o define a self-similar fragmentation process 1 F w e need one input: a probabilit y measure ν on { ( s 1 , s 2 ) : s 1 ≥ s 2 > 0 and s 1 + s 2 ≤ 1 } . The process F with v alues in the set S ↓ = { ( s 1 , s 2 , . . . ) : s 1 ≥ s 2 ≥ · · · ≥ 0 and P i s i ≤ 1 } is informally characterized as follows: 1 binary , without erosion and with dislo cation measure of mass one 5 FRA GMENT A TION PR OCESS WITH P ARAMETER 9 if at time t w e hav e F ( t ) = ( s 1 ( t ) , s 2 ( t ) , . . . ), then for every i ≥ 1, the i -th “particle” of mass s i ( t ) lives an exponential time with parameter s i ( t ) b efore splitting into t wo particles of masses r 1 s i ( t ) and r 2 s i ( t ), where ( r 1 , r 2 ) has b een sampled from ν indep endently of the past and of the other particles. In other words, eac h particle undergo es a self-similar fragmentation with time rescaled by its mass. It is classical that under mild assumption there exists a unique β ∈ (0 , 1] (called the Malthusian exp onen t) such that Z d ν ( s 1 , s 2 ) s β 1 + s β 2 = 1 , (10) and that M t := P s i ( t ) β is a contin uous-time non-negative martingale which plays a centr al role in the asymptotic b ehavior of the fragmentation pro cess, see [1, 2]. P arametrized fragmentation. In the problem of the partial matc h query , one can think of the rectangles ab ov e the point x at time t > 0 as a fr agmentation pr o c ess wher e the p articles have an additional p ar ameter , in our case the p osition x i t ∈ [0 , 1]. This leads us to extend the notion of dislo cation measure and to define a fragmentation pro cess with parameter: a parametrized (binary) dislo cation probability is a collection ν = ( ν x ) x ∈ [0 , 1] suc h that for every x ∈ [0 , 1], ν x is a probabilit y measure on { ( s 1 , x 2 , s 2 , x 2 ) ∈ [0 , 1] 4 : s 1 ≥ s 1 and s 1 + s 2 ≤ 1 } . A parametrized fragmentation pro cess F with dislo cation measure ν is then a pro cess with v alues in { ( s 1 , x 1 , s 2 , x 2 , ... . . . ) ∈ [0 , 1] N : s 1 ≥ s 2 ≥ · · · ≥ 0 and P i s i ≤ 1 } whose evolution is informally described as follows: W e start with a particle of mass 1 given with a p osition x ∈ [0 , 1]. If F ( t ) = ( s 1 ( t ) , x 1 ( t ) , s 2 ( t ) , x 2 ( t ) , . . . ) then for ev ery i ≥ 1, the i -th “particle” of mass s i ( t ) with p osition x i ( t ) liv es an exp onential time with parameter s i ( t ) b efore splitting in to t w o particles of masses r 1 s i ( t ) and r 2 s i ( t ) with respective p ositions y 1 and y 2 , where ( r 1 , y 1 , r 2 , y 2 ) has b een sampled from ν x i ( t ) independently of the past and of the other parti- cles. In this setting, (10) is replaced by the follo wing assumption: ( H ) There exists β ∈ [0 , 1] and h : x ∈ [0 , 1] 7→ h ( x ) ∈ R + suc h that for every x ∈ [0 , 1] we hav e Z d ν x ( s 1 , x 1 , s 2 , x 2 )  s β 1 h ( x 1 ) + s β 2 h ( x 2 )  = h ( x ) . ( 11) Then under this assumption the process M t ( x ) := P s i ( t ) β h ( x i ( t )) is a contin uous-time non-negativ e martingale playing the role of the Malthusian martingale. It is b eliev able that substan tial parts of self-similar fragmentations theory can b e adapted to this parametrized case. Examples. P ar tial Ma tch queries in Quadtree. Within this formalism the process of th e masses of the rectangles of Q t ( x ) is a parametrized fragmen tation process starting with a single particle of mass 1 and parameter x . Its parametrized dislocation measure ν quad is giv en by Z d ν quad x ( s 1 , x 1 , s 2 , x 2 ) f ( s 1 , x 1 , s 2 , x 2 ) = Z Z [0 , 1] 2 d u d v 1 xu f  (1 − u ) v , x − u 1 − u , (1 − u )(1 − v ) , x − u 1 − u  ! , for every x ∈ [0 , 1] and every Borel function f : [0 , 1] 4 → R + . In particular d ν quad ( s 1 , x 1 , s 2 , x 2 )- almost surely we hav e x 1 = x 2 and Hyp othesis ( H ) is fulfilled with β = √ 17 − 3 2 and h ( x ) = ( x (1 − x )) β / 2 . REFERENCES 10 T o con clude, besides the application of t he metho d to the problem of partial matc h q ueries in higher-dimensional random quadtrees or in random k -d trees, w e presen t another setup tak en from [6] where the con cept of “parametrized fragmentat ion” could be applied (although the results there only rely on the standard fragmentation theory). Random chords. W e recall the random chord construction of [6]. W e consider a sequence U 1 , V 1 , U 2 , V 2 , . . . of indep enden t random v ariables, which are uniformly distributed o ver the unit circle S 1 . W e then construct inductiv ely a sequence L 1 , L 2 , . . . of random closed subsets of the (cl osed) unit disk D . T o begin with, L 1 just consists of t he chord with endp oin ts U 1 , and V 1 , which we denote b y [ U 1 V 1 ]. Then at step n + 1, we consider tw o cases. Either the chord [ U n +1 V n +1 ] intersects L n , and we put L n +1 := L n . Or the chord [ U n +1 V n +1 ] do es not intersect L n , and w e put L n +1 := L n ∪ [ U n +1 V n +1 ]. Thus, for every integer n ≥ 1, L n is a disjoint union of random chords. If x, y ∈ S 1 then one defines the fragmen ts separating x from y as the connected comp onen ts of D \ L n in tersecting [ x, y ]. Figure 4: The fragments separating tw o p oin ts. After con tracting the chords of L n , each fragment F separating x from y can be seen as a particle with t wo distinguished p oints (see Fig. 4) whose mass corresp onds to the one- dimensional Lebesgue measure of F ∩ S 1 , see [6]. The “position” or parameter of eac h particle is then the relative p ositions of its tw o distinguished p oin ts in [0 , 1]. It was shown in [6] that if U, V are independent and uniformly distributed o ver S 1 then the fragmen ts separating U from V form (in a prop er cont inuous time parametrization) a fragmentation pro cess. In the case when x, y ∈ S 1 are fixed, the pro cess of fragments separating x from y (in a prop er con tinuous tim e parameterization) can be seen as a parametrized fragmentation pro cess with assumption ( H ) fulfilled with β = √ 17 − 3 2 and h ( x ) = ( x (1 − x )) β , whic h is equiv alen t to equation (14) of [6]. References [1] J. Bertoin. R andom F r agmentations and Co agulation Pr o c esses . Num b er 102 in Cam- bridge Studies in Adv anced Mathematics. Cam bridge Universit y Press, 2006. [2] J. Bertoin and A. Gnedin. Asymptotic laws for nonconserv ative self-similar fragmenta- tions. Ele ctr on. J. Pr ob ab. , 9(19):575–593, 2004. [3] N. Broutin, R. Neininger, and H. Sulzbac h. Asymptotic analysis of Partial Match retriev al In pr ep ar ation , 2011. REFERENCES 11 [4] N. Broutin, R. Neininger, and H. Sulzbach. Partial match queries in random quadtrees. arXiv:1107.2231 , 2011. [5] N. Curien and A. Joseph. Partial match queries in random quadtrees : A probabilistic approac h. A dv. in Appl. Pr ob ab. , 43:178–194, 2011. [6] N. Curien and J.-F. Le Gall. Random recursive triangulations of the disk via fragmen- tation theory . Ann. Pr ob ab. to app ear. [7] R. A. Fink el and J. L. Bentley . Quad trees a data structure for retriev al on comp osite k eys. A cta Informatic a , 4(1):1–9, mars 1974. [8] P . Fla jolet, G. Gonnet, C. Puech, and J. M. Robson. Analytic v ariations on quadtrees. Al gorithmic a , 10(6):473–500, 1993. [9] P . Fla jolet and R. Sedgewick. A nalytic c ombinatorics . Cam bridge Universit y Press, Cam bridge, 2009. D ´ epartemen t de Math´ ematiques et Applications ´ Ecole Normale Sup´ erieure, 45 rue d’Ulm 75230 P aris cedex 05, F rance nicolas.curien@ens.fr