On the Probability of the Existence of Fixed-Size Components in Random Geometric Graphs

On the Probabilit y of the Existence of Fixed-Size Comp onen ts in Random Geometric Graphs ∗ J. D ´ ıaz 1 D. Mitsc he 2 X. P ´ erez-Gim ´ enez 1 1 Llenguatges i Sistemes Inform` atics, UPC, 0 8034 Bar celona 2 Institut f ¨ ur Theoretische Informatik, ETH Z ¨ uric h, 8092 Z ¨ urich { diaz ,xpere z } @lsi.upc.edu, dmitsc he@in f.ethz.ch Abstract In this work w e give precise asympto tic expressio ns on the pro bability of the existence o f fix e d- size co mpo nent s at the thres hold of connectivity for rando m geometric gra phs. 1 In tro duction and basic result s on Ra ndom Geometric Graphs. Recen tly , quite a bit of w ork h as b een d on e on R andom Ge om etric gr a phs , du e to th e imp ortance of these grap h s as th eoretical models for ad h o c net w orks (for applications w e refer to [5]). Most of the theoretical r esu lts on random geometric graph s can b e found in the b o ok by M. D. P enrose [7]. In this sectio n w e succinctly recal l the results needed to motiv ate and pro v e our main theorem. Giv en a set of n v ertices and a non-negativ e real r = r ( n ), eac h v ertex is placed at some random p osition in the unit torus [0 , 1) 2 selected indep end en tly and un iformly at random (u.a.r.). W e denote b y X i = ( x i , y i ) the random p osition of v ertex i for i ∈ { 1 , . . . , n } , and let X = X ( n ) = { X 1 , . . . , X n } . Note that with p robabilit y 1, no t wo v ertices c ho ose the same p osition and th us we restrict the attent ion to the case that |X | = n . W e define G ( X ; r ) as the random graph h a vin g X as the vertex set, and with an edge connecting eac h pair of v ertices X i and X j in X at d istance d ( X i , X j ) ≤ r , where d ( · , · ) denotes t he Euclidean distance in the torus. Unless otherwise sta ted, all o ur stated r esults are asymptotic as n → ∞ . W e use the follo wing standard n otatio n for the asymptotic b eha viour of sequences of n on- negativ e num b ers a n and b n : a = O ( b ), if there exist constan ts C and n 0 suc h that a n ≤ C b n for n ≥ n 0 . F ur thermore, a = Ω( b ) i f b = O ( a ), a = Θ( b ) if a = O ( b ) and ∗ P artially supp orted by the Spanish CYCIT: TIN2007-66523 (FORMALISM). The first and third author are partial ly supp orted by 7th F ramew ork under con tract ICT-2007.82 (FRONTS). The first author was also supp orted by L a distinci´ o p er a la pr omo ci´ o de la r e c er c a de la Gener ali tat de Catalunya, 2002 . 1 a = Ω( b ) and fin ally a = o ( b ) if a n /b n → 0 a s n → ∞ . As usual, th e ab b reviation a.a.s. stands for asymptotic al ly almost sur ely , i.e. w ith probabilit y 1 − o (1 ). All log arithms in this pap er are natural logarithms. Let K 1 b e the random v ariable count ing the num b er of isolate d vertices in G ( X ; r ). By multiplying the probabilit y that one ve rtex is isolate d by the num b er of v ertices w e obtain E ( K 1 ) = n (1 − π r 2 ) n − 1 = ne − π r 2 n − O ( r 4 n ) . (1) Define µ := n e − π r 2 n . Observe from the previous expression that µ is closely r elated to E ( K 1 ). In fact, µ = o (1) iff E ( K 1 = o (1) ), and if µ = Ω(1) then E ( K 1 ) ∼ µ . Moreo ve r, the asymptotic b ehavio ur of µ charact erizes the connectivit y of G ( X ; r ). The follo wing prop osition is w ell kno wn: a result similar to it em ( 1) can b e found in Corollary 3.1 of [4 ] and it can also b e found in Section 1.4, p.10 of [7], Item (2) is Theorem 13.11 of [7], and Item (3) can as w ell b e found in Section 1.4, p.10 of [7]. F or th e sake of co mpleteness, w e giv e a simple pro of of Proposition 1 in Section 4. Prop osition 1. In terms of µ the c onne ctivity c an b e char acterize d as fol lows: 1. If µ → 0 , th en a.a.s. G ( X ; r ) is c onne cte d. 2. If µ = Θ(1) , then a.a.s. G ( X ; r ) c onsists of one giant c omp onent of size > n/ 2 and a Poisson numb er (with p ar ameter µ ) of isolate d vertic es. 3. If µ → ∞ , then a.a .s. G ( X ; r ) is disc onne cte d. F rom the defin ition of µ w e ha v e th at µ = Θ(1) iff r = q log n ± O (1) π n . Therefore w e conclude that the prop erty of connectivit y of G ( X ; r ) exhibits a sharp thr eshold at r = q log n π n . Note that the pr evious classification of the connectivit y of G ( X ; r ), indicates that if µ = Θ(1), the comp onen ts of size 1 are p redominan t and those comp onent s hav e the main con trib u tion to the connectivit y of G ( X ; r ). In fact if µ = Θ(1), the probabilit y that G ( X ; r ) has some comp onent of size greater than 1 other t han th e giant comp onent is o (1 ). On the other hand, M.D. Penrose [7] studied th e n um b er of comp onents in G ( X ; r ) that are isomorp h ic to a giv en fixed graph; equiv alen tly , he stud ied the probabilit y of finding c omp onents of a given size in G ( X ; r ). Ho w ev er the range of radii r co v ered b y P enrose do es not exceed the thermo dynamic al thr eshold Θ( p 1 /n ) where a gian t comp onent app ears at G ( X ; r ), wh ic h is b elow the connectivit y threshold treated in th e pr esen t pap er. In fact, a p ercolation argumen t in [7] only sho ws that with probabilit y 1 − o (1) no comp onents other t han isol ated vertices and the gian t one exist at the connectivit y threshold, wh ithout gi ving accurate b ounds on this probabilit y (see Section 1.4 of [7] and Prop osition 13.12 an d Prop osition 13.13 of [7]). Throughout the pap er w e shall consider G ( X ; r ) with r = q log n ± O (1) π n . W e pro ve that for suc h a c h oice of r , giv en a fixed ℓ > 1, the pr obabilit y of ha ving co mp onents of size exactly ℓ is Θ  1 log i − 1 n  . Moreo v er , in the pro cess of the pro of we c haracterize the differen t t yp es of comp onents th at could exist for suc h a v alue of r . 2 Figure 1: Non-em b eddable comp onents on the unit torus . T o the left tw o non- em b eddable and non-solitary comp onen ts, to the right a solitary non-em b eddable and an em b eddable comp onen t. 2 Basic definitions and statemen ts of results Giv en a comp onent Γ of G ( X ; r ), Γ is emb e ddable if it can b e mapp ed in to the square [ r , 1 − r ] 2 b y a translation in the toru s. Embedd able comp onent s d o not wrap around the t orus. Comp onents whic h are n ot embed d able m ust h a ve a large size (at least Ω(1 /r )). Sometimes several non-em b eddable comp onen ts can coexist together (see Figure 1). Ho wev er, th ere are some non-em b eddable comp onen ts whic h are so spread around the torus, that they do not allo w any ro om for other non-em b eddable ones. Call these comp onent s solitary . Clearly , w e can ha ve at most one solitary comp onen t. W e cannot disprov e the existence of a solitary component, since with probabilit y 1 − o (1) there exists a gian t comp onent of this nature (see Corollary 2.1 of [4], implicitly it is also in Th eorem 13.11 of [7]). F or comp onen ts whic h are not solitary , w e give asymptotic b ound s on th e probability of their existence according to their size. Giv en a fixed in teger ℓ ≥ 1 , let K ℓ b e the n umb er of comp onen ts in G ( X ; r ) of size exactly ℓ . F or large enough n , we can assume these to b e em b eddable, sin ce r = o (1). Moreo ve r, for an y fixed ǫ > 0, let K ′ ǫ,ℓ b e the num b er of comp onen ts of size exactly ℓ , whic h ha ve all their v ertices at distance at most ǫr from t heir leftmost one. Finally , e K ℓ denotes th e n um b er of comp onen ts of size at least ℓ and wh ic h are not solitary . In Figure 2 an example of a comp onen t Γ of size exactly ℓ = 9 is giv en , whic h has all its vertices at distance at mo st ǫr from the leftmost one u . Notice that K ′ ǫ,ℓ ≤ K ℓ ≤ e K ℓ . Ho w ev er, in the follo w ing we sho w that asymp tot- ically all th e weigh t in the probabilit y that e K ℓ > 0 comes from comp onent s w h ic h also con tribute to K ′ ǫ,ℓ for ǫ arb itrarily small. Th is means that the more common comp onent s of size at least ℓ are cliques of siz e exactly ℓ w ith all their vertices close together. W e n o w ha ve all definitions to state our m ain theo rem, which is pro ved in Section 3 . Theorem 2. L et ℓ ≥ 2 b e a fixe d i nte ger. L et 0 < ǫ < 1 / 2 b e fixe d. A ssume that 3 Γ ǫr Figure 2: A comp onen t Γ b elonging to K ′ ǫ, 9 µ = Θ(1) . Then Pr h e K ℓ > 0 i ∼ Pr [ K ℓ > 0] ∼ Pr  K ′ ǫ,ℓ > 0  = Θ  1 log ℓ − 1 n  . Giv en a r an d om set X of n points in [0 , 1) 2 , let ( G ( X ; r )) r ∈ R + b e th e con tin uous random graph p ro cess describing the ev olution of G ( X ; r ) for r b etw een 0 and + ∞ ( X remains fixed for the whole pro cess). Observe th at the graph pr o cess starts at r = 0 with all n v ertices b eing isolated, then ed ges are progressive ly add ed, and finally at r ≥ √ 2 / 2 w e ha ve the complete graph on n vertic es. In this con text, consider the random v ariables r c = r c ( n ) = inf { r ∈ R + : G ( X ; r ) is connected } and r i = r i ( n ) = inf { r ∈ R + : G ( X ; r ) has no isolat ed v ertex } . As a corollary of Theorem 2 w e obtain an alternativ e pro of of the f ollo wing w ell kno w n r esult (see Theorem 1 of [6]): in tu itiv ely sp eaking, we show that a.a.s. ( G ( X ; r )) r ∈ R + b ecomes connected exactly at the same moment when the last isolated v ertex disapp ears. Note that this is stron ger than the results state d in the int ro duc- tion, wh ic h ju st sa y that the p rop erties of connectivit y and havi ng n o isolated v ertex ha v e a sh arp threshold with the same asymptotic characte rization (see P rop osition 1). Corollary 3. W i th pr ob ability 1 − o (1) , we h ave r c = r i . The p ro of of Corollary 3 is giv en in S ection 4. 3 Pro of of Th eorem 2 W e state and p ro v e three lemmata from wh ic h Theorem 2 will follo w easily . Lemma 4. L e t ℓ ≥ 2 b e a fixe d inte ger, and 0 < ǫ < 1 / 2 b e also fixe d. Assume that µ = Θ(1) . Then, E  K ′ ǫ,ℓ  = Θ(1 / log ℓ − 1 n ) . Pr o of. First observe that with p robabilit y 1, for eac h comp onen t Γ wh ic h con tributes to K ′ ǫ,ℓ , Γ has a unique leftmost vertex X i and the v ertex X j in Γ at greatest distance from X i is also unique. Hence, w e can restrict our atten tion to th is case. 4 X j ρ X i r r Γ S Figure 3: The set S for the comp onent Γ of Figure 2 Fix an arbitrary set o f indices J ⊂ { 1 , . . . , n } of size | J | = ℓ , with tw o d istinguished elemen ts i and j . Denote by Y = S k ∈ J X k the set of random p oints in X with ind ices in J . L et E b e the follo wing eve n t: All vertice s in Y are at distance at most ǫr from X i and to the righ t of X i ; v ertex X j is the one in Y with g reatest distance f rom X i ; and the v ertices of Y form a comp onent Γ of G ( X ; r ). If Pr ( E ) is m u ltiplied by the n um b er of p ossible choice s of i , j and the remaining ℓ − 2 eleme n ts of J , w e get E K ′ ǫ,ℓ = n ( n − 1)  n − 2 ℓ − 2  Pr ( E ) . (2) In order to b ound the p robabilit y of E w e need some definitions. Let ρ = d ( X i , X j ) and let S b e the set of all p oin ts in the torus [0 , 1) 2 whic h are at distance at most r from some vertex in Y (see Figure 3). Notice that ρ and S dep end on the set of random points Y . W e first need b ounds of Area ( S ) in terms of ρ . Observe that S is conta ined in the circle o f radius r + ρ and cen ter X i , and th us Area ( S ) ≤ π ( r + ρ ) 2 . (3) Let i L = i , i R , i T and i B b e resp ectiv ely the indices of the leftmo st, righ tmost, topmost and b ottommost v ertices in Y (some of these ind ices p ossibly equal). Assume w.l.o.g. that the v ertical length of Y (i.e. th e vertical distance b et ween X i T and X i B ) is at least ρ/ √ 2. Otherwise, the horizon tal length of Y has this prop erty and w e can rotate the descriptions in th e argument. Th e upp er halfcircle with cent er X i T and the lo w er halfcircle with cen ter X i B are disj oint and are con tained in S . If X i R is at greater ve rtical d istance from X i T than from X i B , then consid er the rectangle of heigh t ρ/ (2 √ 2) and width r − ρ/ (2 √ 2) with o ne corner on X i R and ab ov e a nd to th e righ t of X i R . Otherwise, consid er the same rectangle b elo w and to the r igh t of X i R . This rectangle is also conta ined in S and its inte rior do es n ot intersect the pr eviously describ ed halfcircles. Analog ously , w e can find another r ectangle of h eigh t ρ/ (2 √ 2) and width r − ρ/ (2 √ 2) to the left of X i L and either ab ov e or b elo w X i L with the same prop erties. Hence, Area ( S ) ≥ π r 2 + 2  ρ 2 √ 2   r − ρ 2 √ 2  . (4) 5 F rom ( 3), (4) and the fact that ρ < r / 2, we can write π r 2  1 + 1 6 ρ r  < Area ( S ) < π r 2  1 + 5 2 ρ r  < 9 π 4 r 2 . (5) No w consid er the p robabilit y P that the n − ℓ v ertices n ot in Y lie outside S . Clearly P = (1 − Area ( S )) n − ℓ . Moreo v er , by (5) and using the fact that e − x − x 2 ≤ 1 − x ≤ e − x for all x ∈ [0 , 1 / 2], w e obta in e − (1+5 ρ/ (2 r )) πr 2 n − (9 π r 2 / 4) 2 n < P < e − (1+ ρ/ (6 r )) π r 2 n (1 − 9 πr 2 / 4) ℓ , and after plugging in the definition o f µ (recall th at µ = ne − r 2 π n ) we hav e  µ n  1+5 ρ/ (2 r ) e − (9 πr 2 / 4) 2 n < P <  µ n  1+ ρ/ (6 r ) 1 (1 − 9 πr 2 / 4) ℓ . (6) Ev en t E can also b e describ ed as follo ws : Th ere is some non-negativ e real ρ ≤ ǫr suc h that X j is placed at distance ρ from X i and to the r igh t of X i ; all the r emaining v ertices in Y are in side the halfcircle o f center X i and r adius ρ ; an d the n − ℓ v ertices not in Y lie outside S . Hence, Pr ( E ) can be boun ded from ab o v e (b elo w) b y in tegrating with resp ect to ρ the probabilit y den s it y fu nction of d ( X i , X j ) times the probabilit y that the remaining ℓ − 2 selected v ertices lie inside the right halfcircle of center X i and r adius ρ times the u pp er (lo wer) b ound on P w e ob tained in (6): Θ(1) I (5 / 2) ≤ Pr ( E ) ≤ Θ(1) I (1 / 6) , (7) where I ( β ) = Z ǫr 0 π ρ  π 2 ρ 2  ℓ − 2 1 n 1+ β ρ/r dρ = 2 n  π 2 r 2  ℓ − 1 Z ǫ 0 x 2 ℓ − 3 n − β x dx (8) Since ℓ is fixed, for β = 5 / 2 o r β = 1 / 6, I ( β ) = Θ log ℓ − 1 n n ℓ ! Z ǫ 0 x 2 ℓ − 3 n − β x dx = Θ log ℓ − 1 n n ℓ ! (2 ℓ − 3)! ( β log n ) 2 ℓ − 2 = Θ  1 n ℓ log ℓ − 1 n  . (9) The statement follo w s from (2), (7) and ( 9). Lemma 5. L et ℓ ≥ 2 b e a fixe d inte ger. L et ǫ > 0 b e also fixe d. Assume tha t µ = Θ(1) . Then Pr h e K ℓ − K ′ ǫ,ℓ > 0 i = O (1 / log ℓ n ) . 6 Pr o of. W e assume thr ou gh ou t th is pr o of that ǫ ≤ 10 − 18 , and prov e the claim for this case. The case ǫ > 10 − 18 follo ws from the fact that ( e K ℓ − K ′ ǫ,ℓ ) ≤ ( e K ℓ − K ′ 10 − 18 ,ℓ ). Consider all the p ossible comp onents in G ( X ; r ) whic h are not solitary . Remo v e from these comp onen ts the ones of s ize at most ℓ and diameter at most ǫr , and den ote b y M the num b er of remaining comp onents. By construction e K ℓ − K ′ ǫ,ℓ ≤ M , and therefore it is sufficien t to p ro ve that Pr ( M > 0) = O (1 / log ℓ n ). The comp onen ts coun ted b y M are classified int o several types according to th eir s ize and diameter. W e deal with eac h t yp e separat ely . Part 1 . Consider all the p ossible co mp onents in G ( X ; r ) whic h ha v e diameter at m ost ǫr and size b et ween ℓ + 1 and log n/ 37. Call them comp onents of t yp e 1, and let M 1 denote th eir n um b er. F or eac h k , ℓ + 1 ≤ k ≤ log n/ 37, let E k b e the exp ected num b er of comp onen ts of t yp e 1 and size k . W e observe that these co mp onents ha ve all of t heir ve rtices at distance at most ǫr from th e leftmost one. Th erefore, we c an apply the same argumen t w e used for boun ding E K ′ ǫ,ℓ in the pro of of Lemma 4 . Note that (2), (7) and (8) are also v alid for sizes not fixed but dep ending on n . Thus, we obtain E k ≤ O (1) n ( n − 1)  n − 2 k − 2  I (1 / 6 ) , where I (1 / 6) is defined in (8 ). W e use the fact that  n − 2 k − 2  ≤ ( ne k − 2 ) k − 2 and get E k = O (1) log n  e 2 log n k − 2  k − 2 Z ǫ 0 x 2 k − 3 n − x/ 6 dx. (10) The expression x 2 k − 3 n − x/ 6 can b e maximized for x ∈ R + b y elemen tary tec hn iques, and we dedu ce that x 2 k − 3 n − x/ 6 ≤  2 k − 3 ( e/ 6) log n  2 k − 3 . W e can b ound the integ ral in (10) a nd ge t E k = O (1) log n  e 2 log n k − 2  k − 2 ǫ  2 k − 3 ( e/ 6) log n  2 k − 3 = O (1)  36 2 e (2 k − 3) 2 ( k − 2) log n  k − 2 k . Note that for k ≤ log n/ 37 the expr ession k  36 2 e (2 k − 3) 2 ( k − 2) log n  k − 2 is decreasing with k . Hence w e can write E k = O  1 log ℓ +1 n  , ∀ k : ℓ + 3 ≤ k ≤ 1 37 log n. Moreo ve r the b ounds E ℓ +1 = O (1 / log ℓ n ) and E ℓ +2 = O (1 / log ℓ +1 n ) are obtained from Lemm a 4, and hence E M 1 = 1 37 log n X k = ℓ +1 E k = O  1 log ℓ n  + O  1 log ℓ +1 n  + log n 37 O  1 log ℓ +1 n  = O  1 log ℓ n  , 7 y 2 y Figure 4: Th e tessellat ion for coun ting comp onen ts of t yp e 2 w ith t w o particular b o xes mark ed. and then Pr ( M 1 > 0) ≤ E M 1 = O (1 / log ℓ n ) . Part 2 . Consider all the p ossible co mp onents in G ( X ; r ) whic h ha v e diameter at m ost ǫr and size g reater than log n/ 37. Call them comp onents of t yp e 2, and let M 2 denote their n umb er. W e tessellate the torus with square cells of side y = ⌊ ( ǫr ) − 1 ⌋ − 1 ( y ≥ ǫr b ut also y ∼ ǫr ). W e d efi ne a b ox to b e a s quare of side 2 y c onsisting of the u nion of 4 cells of the tessellatio n. Consider the set of all p ossible b oxe s. Note that an y comp onen t of t yp e 2 m ust b e fu lly con tained i n s ome b o x (see Figure 4). Let us fi x a box b . Let W b e the num b er of v ertices which are cont ained inside b . Notice that W has a b inomial d istr ibution with mean E W = (2 y ) 2 n ∼ (2 ǫ ) 2 log n/π . By setting δ = log n 37 E W − 1 and applyin g the Chernoff inequ ality to W (see e.g. [3], Theorem 12.7), w e hav e Pr ( W > 1 37 log n ) = Pr ( W > (1 + δ ) E W ) ≤  e δ (1 + δ ) 1+ δ  E W = n − (log(1+ δ ) − δ 1+ δ ) 37 . Note th at δ ∼ π 148 ǫ 2 − 1 > e 79 , therefore Pr ( W > 1 37 log n ) < n − 2 . 1 . T aking a union b ound o ver the set of all Θ( r − 1 ) = Θ( n/ log n ) b o xes, the probability that there is some b o x with more than 1 37 log n v ertices is O (1 / ( n 1 . 1 log n )). Since eac h comp onent of type 2 is co n tained in some b o x, we hav e Pr ( M 2 > 0) = O (1 / ( n 1 . 1 log n )) . Part 3 . Consider all the p ossible comp onen ts in G ( X ; r ) which are em b edd able and ha v e diameter at lea st ǫr . Call them c omp onents of t yp e 3, and let M 3 denote their n um b er. W e tessellate the torus into square cells of side αr , for some α = α ( ǫ ) > 0 fixed but sufficien tly small. Let Γ b e a comp onen t of t yp e 3. Let S = S Γ b e the set of all p oin ts in the torus [0 , 1) 2 whic h are at distance at m ost r from some vertex in Γ. Remo ve from S the ve rtices of Γ and the edges (represented by straigh t line segments) 8 S S ∗ Figure 5: The tessel lation for count ing co mp onents of typ e 3. and denote by S ′ the o uter co nnected to p ologic comp onen t of the r emaining se t. By construction, S ′ m ust cont ain no v ertex in X (see Fi gure 5 , left picture). No w let i L , i R , i T and i B b e resp ectiv ely the indices of the leftmost, righ tmost, topmost and b ottommost v ertices in Γ (some of these indices possib ly equal). As in the pr evious setting, assume that the v ertical length of Γ (i.e. the v ertical distance b et w een X i T and X i B ) is at least ǫr/ √ 2. Otherwise, the horizonta l le ngth of Γ has this prop erty and w e can rotate the descriptions in the argument. The upp er h alfcircle with cen ter X i T and th e low er halfcircle with cente r X i B are d isjoin t and are con tained in S ′ . If X i R is at greater v ertical d istance from X i T than from X i B , then consid er the rectangle of height ǫr/ (2 √ 2) and width r − ǫr / (2 √ 2) with one corner on X i R and ab o v e and to the r igh t of X i R . Otherwise, consider the same rectangle b elo w and to the right of X i R . This rectangle is also con tained in S ′ and its in terior do es not in tersect th e previously describ ed halfcircles. Analogously , w e can fi nd another rectangle of height ǫr / (2 √ 2) and wid th r − ǫr / (2 √ 2) to th e left of X i L and either ab o ve or b elo w X i L , with the same p r op erties. Hence, taking into accoun t that ǫ ≤ 10 − 18 , we hav e Area ( S ′ ) ≥ π r 2 + 2  ǫr 2 √ 2   r − ǫr 2 √ 2  >  1 + ǫ 5  π r 2 . (11) Let S ∗ b e the union of all the cells in th e tessellation whic h are fu lly contai ned in S ′ . W e lo ose a bit of area compared t o S ′ . Ho wev er, if α was chosen sm all enough, we can guaran tee that S ∗ is top ologically connected and has area Area ( S ∗ ) ≥ (1 + ǫ/ 6) π r 2 . This α can b e c h osen to b e the same f or all comp onents of t yp e 3 (see Figure 5, r igh t picture). Hence, w e sho we d th at the ev ent ( M 3 > 0) implies that some conn ected un ion of cells S ∗ of area Area ( S ∗ ) ≥ (1 + ǫ/ 6) π r 2 con tains n o ve rtices. By removing some cells from S ∗ , we can a ssume that (1 + ǫ/ 6) π r 2 ≤ Area ( S ∗ ) < (1 + ǫ/ 6) π r 2 + α 2 r 2 . Let S ∗ b e a n y union of ce lls with these prop erties. Note th at there are Θ (1 /r 2 ) = Θ( n/ log n ) man y possible c hoices for S ∗ . The pr obabilit y that S ∗ con tains no vertices is (1 − Area ( S ∗ )) n ≤ e − (1+ ǫ/ 6) πr 2 n =  µ n  1+ ǫ/ 6 . Therefore, we can tak e the union b ound ov er all the Θ ( n/ log n ) p ossible S ∗ , and obtain an upp er b ound of the probabilit y that there is some comp onent of the t yp e 3: Pr ( M 3 > 0) ≤ Θ  n log n   µ n  1+ ǫ/ 6 = Θ  1 n ǫ/ 6 log n  . 9 Part 4 . C on s ider all the p ossible comp onents in G ( X ; r ) which are n ot em b eddable and n ot s olitary either. Call them comp onent s of t yp e 4, and let M 4 denote th eir n um b er. W e tessellate the torus [0 , 1) 2 in to Θ( n/ log n ) small square cells of side length αr , where α > 0 is a sufficient ly small p ositiv e constant. Let Γ b e a comp onent of t yp e 4 . Let S = S Γ b e the set of all p oin ts in the torus [0 , 1) 2 whic h are at distance at m ost r from some v ertex in Γ. Remo v e from S the v ertices of Γ and the edges (represen ted b y straigh t segmen ts) and denote b y S ′ the remaining set . By constru ction, S ′ m ust co n tain n o vertex in X . Supp ose there is a horizon tal or a v ertical band of width 2 r in [0 , 1) 2 whic h does not in tersect th e compon ent Γ (assume w.l.o.g. that it is the topmost h orizon tal band consisting of all p oints with the y -co ordinate in [1 − 2 r , 1)). Let us divide the torus in to v ertical band s of width 2 r . All of them m ust con tain at least one verte x of Γ , since otherw ise Γ w ould b e embedd able. Select an y 9 consecutiv e v ertical bands and pic k one v ertex of Γ with maximal y -co ordinate in eac h one. F or eac h one of these 9 v ertices, w e select the left upp er quartercircle cen tered at the vertex if the vertex is closer to the r igh t side of the b and o r the right upp er quartercircle otherwise. These nine quartercircles we c hose are disjoin t and m ust co n tain no v ertices b y constru ction. Moreo ve r, they b elong to the same connected co mp onent of the set S ′ , whic h we denote b y S ′′ , and w hic h has an area of Area ( S ′′ ) ≥ (9 / 4) π r 2 . Let S ∗ b e the union of all the cells in the tessell ation of the torus wh ic h are completely con tained in S ′′ . W e lose a bit of area compared to S ′′ . Ho w ev er, as usu al, by c ho osing α small enough w e can guaran tee that S ∗ is co nnected and it h as an area of Area ( S ∗ ) ≥ (11 / 5) πr 2 . Note that this α can b e chosen to be the s ame for all co mp onents Γ o f this kind. Supp ose otherwise that all horizon tal and v ertical b an d s of width 2 r in [0 , 1) 2 con tain at least on e v ertex of Γ. S ince Γ is n ot solitary it must b e p ossible that it co exists with some other n on-em b eddable comp onent Γ ′ . Then all v ertical band s or all horizonta l bands of width 2 r m u st also conta in some v ertex of Γ ′ (assume w.l.o.g. the v ertical bands do). Let us divide the to rus into v ertical bands of width 2 r . W e can find a simple path Π with ve rtices in Γ ′ whic h passes th rough 11 consecutiv e band s. F or eac h one of the 9 in ternal b ands, pic k the u pp ermost vertex of Γ in the band b elo w Π (in the torus sense). As b efore eac h one of these vertice s contributes with a disjoin t quartercircle whic h must b e empt y of vertice s, and b y the same argumen t w e obtain a connected union of cells of the tessellation, whic h w e denote by S ∗ , with Area ( S ∗ ) ≥ (11 / 5) πr 2 and conta ining no ve rtices. Hence, we show ed that the ev en t ( M 4 > 0) implies that some connected union of cells S ∗ with Area ( S ∗ ) ≥ (11 / 5) πr 2 con tains no v ertices. By rep eating th e same argumen t we used for comp onent s of t yp e 3 but replacing (1 + ǫ/ 6) π r 2 b y (11 / 5) π r 2 , w e get Pr ( M 4 > 0) = Θ  1 n 6 / 5 log n  . F or a r andom v ariable X and any k ≥ 1, w e denote by E [ X ] k the k th factorial 10 momen t of X , i.e. E [ X ] k = E [ X ( X − 1) . . . ( X − k + 1) ]. Lemma 6. L et ℓ ≥ 2 b e a fixe d inte ger. L et 0 < ǫ < 1 / 2 b e fixe d. Assume that µ = Θ(1) . Then E  K ′ ǫ,ℓ  2 = O (1 / log 2 ℓ − 2 n ) . Pr o of. As in the pro of of Lemma 4, w e assume that eac h comp onen t Γ wh ic h con- tributes to K ′ ǫ,ℓ has a unique leftmost v ertex X i , and the v ertex X j in Γ at greate st distance f r om X i is also unique. I n fact, this happ ens with probabilit y 1. Cho ose an y tw o disjoint subsets of { 1 , . . . , n } of size ℓ eac h, namely J 1 and J 2 , with four distinguished e lemen ts i 1 , j 1 ∈ J 1 and i 2 , j 2 ∈ J 2 . F or k ∈ { 1 , 2 } , denote by Y k = S l ∈ J k X l the set of r andom points in X with indices in J k . Let E b e the ev en t that the follo w ing c onditions h old for b oth k = 1 and k = 2: All vertic es in Y k are at distance at most ǫr from X i k and to the right of X i k ; ve rtex X j k is the one in Y k with greatest distance f r om X i k ; and the v ertices of Y k form a comp onen t Γ of G ( X ; r ). If Pr ( E ) is m u ltiplied by the n umb er of p ossib le choic es of i k , j k and the remaining v ertices of J k , we get E [ K ′ ǫ,ℓ ] 2 = O ( n 2 ℓ ) Pr ( E ) . (12) In order to b ound the prob ab ility of E we n eed some definitions. F or eac h k ∈ { 1 , 2 } , let ρ k = d ( X i k , X j k ) and let S k b e the se t of a ll the p oin ts in the torus [0 , 1) 2 whic h are at distance at m ost r from some v ertex in Y k . Obvio usly ρ k and S k dep end on the set of rand om p oints Y k . Also defin e S = S 1 ∪ S 2 . Let F b e th e even t that d ( X i 1 , X i 2 ) > 3 r . T his holds with pr ob ab ility 1 − O ( r 2 ). In order to b ound Pr ( E | F ), w e apply a similar approac h to the on e in the pr o of o f Lemma 4. In fact, observe th at if F holds th en S 1 ∩ S 2 = ∅ . Therefore in vi ew of (5) w e can write π r 2 (2 + ( ρ 1 + ρ 2 ) / (6 r )) < Area ( S ) < 18 π 4 r 2 , (13) and u sing the same tec hn iques that ga ve us (6) w e get (1 − Area ( S )) n − 2 ℓ <  µ n  2+( ρ 1 + ρ 2 ) / (6 r ) 1 (1 − 18 πr 2 / 4) 2 ℓ . (14) Observe th at E can also b e describ ed as follo ws: F or eac h k ∈ { 1 , 2 } there is some non-negativ e real ρ k ≤ ǫr suc h t hat X j k is placed at distance ρ k from X i k and to the righ t of X i k ; all the remainin g ve rtices in Y k are in side the h alfcircle of cen ter X i k and radius ρ k ; and the n − ℓ v ertices not in Y k lie outside S k . In fact, rather than this last condition, we only require for our b oun d that all v ertices in X \ ( Y 1 ∪ Y 2 ) are placed outside S , w hic h has probabilit y (1 − Area ( S )) n − 2 ℓ . Then, from (14) and follo wing an analogous argumen t to the one t hat le ads to (7 ), we obtain the b ound Pr ( E | F ) ≤ Θ(1) Z ǫr 0 Z ǫr 0 π ρ 1  π 2 ρ 2 1  ℓ − 2 π ρ 2  π 2 ρ 2 2  ℓ − 2 1 n 2+( ρ 1 + ρ 2 ) / (6 r ) dρ 1 dρ 2 = Θ(1) I (1 / 6) 2 , 11 where I (1 / 6) is defined in (8 ). Th us from (9) w e c onclude Pr ( E ∧ F ) ≤ Θ(1) P ( F ) I (1 / 6) 2 = O  1 n 2 ℓ log 2 ℓ − 2 n  . (15) Otherwise, supp ose that F do es n ot hold ( i.e. d ( X i 1 , X i 2 ) ≤ 3 r ). Observe that E implies that d ( X i 1 , X i 2 ) > r , since X i 1 and X i 2 m ust b elong to differen t comp onen ts. Hence the circles with cente rs on X i 1 and X i 2 and radius r ha ve an intersectio n of area less than ( π / 2) r 2 . These tw o circles are cont ained in S and then we can write Area ( S ) ≥ (3 / 2) π r 2 . Note that E imp lies that all v ertices in X \ ( Y 1 ∪ Y 2 ) are p laced outside S and that for eac h k ∈ { 1 , 2 } all the v ertices in Y k \ { X i k } are at distance at most ǫ r and to the r igh t of X i k . This give s us the foll o w ing rough b ound Pr ( E | F ) ≤  π 2 ( ǫr ) 2  2 ℓ − 2  1 − 3 π 2 r 2  n − 2 ℓ = O (1)  log n n  2 ℓ − 2  µ n  3 / 2 . Multiplying t his b y Pr ( F ) = O ( r 2 ) = O (log n/n ) w e o btain Pr ( E ∧ F ) = O log 2 ℓ − 1 n n 2 ℓ +1 / 2 ! , (16) whic h is negligible compared to (15). The state men t follo ws fr om (12), (15) and (16). Our main th eorem n o w follo ws easily: F rom Corollary 1.12 in [2], w e ha ve E K ′ ǫ,ℓ − 1 2 E [ K ′ ǫ,ℓ ] 2 ≤ Pr ( K ′ ǫ,ℓ > 0) ≤ E K ′ ǫ,ℓ , and th er efore b y Lemmata 4 and 6 w e obtain Pr ( K ′ ǫ,ℓ > 0) = Θ(1 / log ℓ − 1 n ) . Com bining this and L emm a 5, yie lds the statemen t. 4 Pro of of Corollary 3 Before pro ving Corollary 3, w e g iv e a pro of of Prop osition 1, since w e will mak e use of th e arguments used in the pr o of of this prop osition. Pr o of of Pr op osition 1. Recall that µ = ne − π r 2 n and r = q log n − log µ π n . Ob serv e that r ∈ [0 , + ∞ ) is mon otonically d ecreasing with resp ect to µ ∈ (0 , n ]. Hence, the probabilit y t hat G ( X ; r ) is connected is also decreasing with resp ect to µ . Supp ose first that µ = Θ(1). F rom (1) and since O ( r 4 n ) = o (1) w e ha v e that E K 1 ∼ µ . W e shall compute the factorial momen ts of K 1 and sho w that E [ K 1 ] k ∼ µ k for eac h fixed k . As in Lemma 4, for k ≥ 2, w e fi x an arbitrary set of ind ices 12 J ⊂ { 1 , . . . , n } of size | J | = k . Denote by Y = S k ∈ J X k the set of random p oin ts in X with indices in J . Let E b e the ev en t that all ve rtices in Y are isolated, and denote b y S the s et of p oin ts in [0 , 1) 2 that are at distance at most r from some v ertex in Y . W e h a ve E [ K 1 ] k ∼ n k Pr ( E ). Note that in order for the ev ent E to happ en, w e m ust ha v e S ∩ ( X \ Y ) = ∅ . T o compute Pr ( E ), w e distinguish t w o case s: Case 1: Supp ose that ∀ i 6 = j ∈ J , d ( X i , X j ) > 4 r . In this case, Area ( S ) = k r 2 π , and th us the probabilit y of E is (1 − k r 2 π ) n − k ∼ e − k r 2 π n . Case 2: Otherwise there exists ∃ i 6 = j ∈ J such that d ( X i , X j ) ≤ 4 r . Define J ′ = { j ∈ J | ∃ i ∈ J, i < j, d ( X i , X j ) ≤ 4 r } and let ℓ = | J ′ | . Note that 1 ≤ ℓ ≤ k − 1. Let j ′ b e the sm allest elemen t of J ′ and let i ′ < j ′ b e the (smallest) elemen t of J with d ( X i ′ , X j ′ ) ≤ 4 r . Denote by C i ′ the circle of radius r centered at X i ′ , and consider the h alfcircle of radiu s r cen tered at X j ′ delimited b y the line going through X j ′ , p erp endicular to the l ine connecting X i ′ with X j ′ , and w h ic h do es not intersect C i ′ (note th at d ( X i ′ , X j ′ ) > r , so this halfcircle exists). This circle and halfcircle con trib ute to A rea ( S ) b y 3 2 r 2 π , and th us in total Area ( S ) ≥ ( k − ℓ + 1 2 ) r 2 π . Moreo v er, the pr ob ab ility that any j ∈ J to belongs to J ′ is at most Θ( r 2 ). Hence, if we denote b y J ℓ the ev ent that such a set J ′ with | J ′ | = ℓ e xists, we ha v e for an y 1 ≤ ℓ ≤ k − 1, Pr ( E | J ℓ ) Pr ( J ℓ ) ≤ (1 − ( k − ℓ + 1 / 2)) r 2 π n Θ( r 2 ) ℓ = o ( e − k r 2 π n ) . Then, the main contribution to Pr ( E ) comes f rom Case 1, and therefore E K 1 ∼ n k e − k r 2 π n = µ k , so the random v ariable K 1 is asymptotically P oisson with p arameter µ . By Theorem 2, a.a.s. G ( X ; r ) consists only of isolated v ertices and a solitary comp onent , a nd the second statement in the result i s pro ven. The fi rst and third statemen ts follo w directly from the fact that, for an y µ = Θ(1), Pr ( G ( X ; r ) is connected) ∼ e − µ , com bined with the decreasing monotonicit y of this probabilit y with r esp ect to µ . Pr o of of Cor ol lary 3. F or an y ǫ > 0, one can find a large enough constan t κ = κ ( ǫ ) suc h that e − e κ < ǫ/ 2 and 1 − e − e − κ < ǫ/ 2. Let r ℓ = q log n − κ π n and r u = q log n + κ π n . By Prop osition 1, K 1 is asymptotically P oisson in G ( X ; r ℓ ) an d G ( X ; r u ), with p arameter µ = e κ and µ = e − κ resp ectiv ely . Therefore, in G ( X ; r ℓ ) we h a ve Pr ( K 1 = 0) ∼ e − e κ < ǫ/ 2 , and in G ( X ; r u ) w e ha v e Pr ( K 1 > 0) ∼ 1 − e − e − κ < ǫ/ 2. Moreo v er, by Theorem 2, a.a.s. b oth G ( X ; r ℓ ) and G ( X ; r u ) consist only of isolate d vertices and a gian t solitary comp onen t. Hence, with p robabilit y at le ast 1 − ǫ , the r an d om pro cess ( G ( X ; r )) r ∈ R + has the follo win g evo lution: for r ≤ r ℓ , the graph sta ys disconnected; at r = r ℓ , there are only a few isolated vertices and a gian t comp onen t; for r b et ween r ℓ and r u , all isolated v ertices merge together or with other comp onen ts; fin ally for r ≥ r u , the graph is connected. F or this particular evo lution of the pr o cess, r c = r i unless for an r with r ℓ < r < r u some isolated v ertices merge toget her and create a small comp onent before b eing absorb ed b y the gian t one. Then , it is su fficien t for our purp oses to sho w that a.a.s. an y t wo isolated vertices in G ( X ; r ℓ ) are at a distance bigger than r u . Define Z to b e th e random v ariable that count s the p airs of v ertices i and j whic h are b oth isolated in G ( X ; r ℓ ) and suc h that d ( X i , X j ) ≤ r u . By the same 13 argumen t as in the proof of Pr op osition 1, setting S to be the set of p oin ts in [0 , 1) 2 at distance at most r ℓ from either X i or X j , w e obtain Area ( S ) ≥ 3 2 r 2 ℓ π . Moreo ve r, since r ℓ < d ( X i , X j ) ≤ r u , X j m ust lie in an ann ulus of a rea Θ(1 /n ) around X i , whic h o ccurs with probability Θ(1 /n ). T aking a union b ound o ver all pairs of v ertices i and j , Pr ( Z > 0 ) ≤ n ( n − 1)  1 − 3 2 r 2 ℓ π  n − 2 Θ(1 /n ) = Θ  n − 1 / 2  . Therefore, when gradually in cr easing r from r ℓ to r u , a.a.s. no pair of isolated v er- tices in G ( X ; r ℓ ) gets connected b efore joinin g the solitary comp onent, and thus no comp onent of size 2 or larger (except for the solitary comp onen t) app ears in this part of the pro cess. Hence, with p robabilit y at least 1 − ǫ , we ha ve that r c = r i , and the statemen t follo ws, since ǫ can b e chosen to be arb itrarily sm all. Ac knowledgmen t. 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