Component sizes of the random graph outside the scaling window

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1 Alea 3, Copoet sizes of the rado graph outside the scalig widow Asaf Nachias ad Yuval Peres Departet of Matheatics, UC Berkeley, Berkeley, CA 94720, USA. E-ail address: Microsoft Research, Oe Microsoft way,, Redod, WA , USA. E-ail address: Abstract. We provide siple proofs describig the behavior of the largest copoet of the Erdős-Réyi rado graph G, p outside of the scalig widow, p = 1+ɛ where ɛ 0 but ɛ 1/3. 1. Itroductio Cosider the rado graph G, p obtaied fro the coplete graph o vertices by retaiig each edge with probability p ad deletig each edge with probability 1 p. We deote by C j the j-th largest copoet. Let ɛ be a o-egative sequece such that ɛ 0 ad ɛ 1/3. The followig theores describe the behavior of the largest copoet whe p = 1+ɛ is outside the scalig-widow. The theores, up to soe logarithic errors, were proved first by Bollobás 1984 usig euerative ethods. The logarithic errors were reoved later by Luczak Theore 1.1. [Subcritical phase] If p = 1 ɛ, the for ay δ > 0 ad iteger l > 0 we have C l P 2ɛ 2 logɛ 3 1 δ 0, as. Theore 1.2. [Supercritical phase] If p = 1+ɛ, the for ay δ > 0 we have C 1 P 2ɛ 1 δ 0, Received by the editors Jauary ; accepted July Matheatics Subject Classificatio. Priary: 05C80 Secodary: 60C05, 60G42. Key words ad phrases. rado graphs, percolatio, artigales. Microsoft Research ad U.C. Berkeley. Research of both authors supported i part by NSF grats #DMS ad #DMS

2 134 Asaf Nachias ad Yuval Peres ad for ay iteger l > 1 we have C l P 2ɛ 2 logɛ 3 1 δ 0, as. The proofs of these theores i Bollobás 1984 ad Luczak 1990 are quite ivolved ad use the detailed asyptotics fro Wright 1977, Bollobás 1984 ad Beder et al for the uber of graphs o k vertices with k + l edges. The proofs we preset here are siple ad require o hard theores. The ai advatage, however, of these proofs is their robustess. I a copaio paper see Nachias ad Peres 2007 we use siilar ethods to aalyze critical percolatio o rado regular graphs. I this case, the euerative ethods eployed i Bollobás 1984 ad Luczak 1990 are ot available. The phase trasitio i the Erdős-Réyi rado graphs G, p occurs whe p = c. Naely, if c > 1, the with high probability w.h.p C 1 is liear i, ad if c < 1, the w.h.p. C 1 is logarithic i. Whe c 1 the situatio is ore delicate. Luczak et al prove that for p = 1+λ 1/3, the law of 2/3 C 1 coverges to a positive o-costat distributio which was idetified by Aldous 1997 as the logest excursio legth of Browia otio with soe variable drift. See Nachias ad Peres 2005 for a recet accout of the case p = 1+λ 1/3 with siple proofs. Thus, C 1 is ot cocetrated ad is roughly of size 2/3 if p = 1+λ 1/3. However, if ɛ a sequece such that 1/3 ɛ ad p = 1+ɛ, the as stated i Theores 1.1 ad 1.2, the size C 1 of the largest copoet i G, p is cocetrated. 2. The exploratio process We recall a exploratio process, due to Marti-Löf 1986 ad Karp 1990, i which vertices will be either active, explored or eutral. After the copletio of step t {0, 1,..., } we will have precisely t explored vertices ad the uber of the active ad eutral vertices is deoted by A t ad N t respectively. Fix a orderig of the vertices {v 1,..., v }. I step t = 0 of the process, we declare vertex v 1 active ad all other vertices eutral. Thus A 0 = 1 ad N 0 = 1. I step t {1,..., }, if A t 1 > 0, let w t be the first active vertex; if A t 1 = 0, let w t be the first eutral vertex. Deote by η t the uber of eutral eighbors of w t i G, p, ad chage the status of these vertices to active. The, set w t itself explored. Deote by F t the σ-algebra geerated by {η 1,..., η t }. Observe that give F t 1 the rado variable η t is distributed as BiN t 1 1 {At 1=0}, p ad we have the recursios ad N t = N t 1 η t 1 {At 1=0}, t, 2.1 A t = { At 1 + η t 1, A t 1 > 0 η t, A t 1 = 0, t. As every vertex is either eutral, active or explored, 2.2

3 Copoet sizes of the rado graph outside the scalig widow 135 N t = t A t, t. 2.3 At each tie j i which A j = 0, we have fiished explorig a coected copoet. Hece the rado variable Z t defied by t 1 Z t = 1 {Aj=0}, couts the uber of copoets copletely explored by the process before tie t. Defie the process {Y t } by Y 0 = 1 ad Y t = Y t 1 + η t 1. By 2.2 we have that Y t = A t Z t, i.e. Y t couts the uber of active vertices at step t ius the uber of copoets copletely explored before step t. At each step we arked as explored precisely oe vertex. Hece, the copoet of v 1 has size i{t 1 : A t = 0}. Moreover, let t 1 < t 2... be the ties at which A tj = 0; the t 1, t 2 t 1, t 3 t 2,... are the sizes of the copoets. Observe that Z t = Z tj + 1 for all t {t j + 1,..., t j+1 }. Thus Y tj+1 = Y tj 1 ad if t {t j + 1,..., t j+1 1}, the A t > 0, ad thus Y tj+1 < Y t. By iductio we coclude that A t = 0 if ad oly if Y t < Y s for all s < t, i.e. A t = 0 if ad oly if {Y t } has hit a ew record iiu at tie t. By iductio we also observe that Y tj = j 1 ad that for t {t j + 1,... t j+1 } we have Z t = j. Also, by our previous discussio for t {t j + 1,... t j+1 } we have i s t 1 Y s = Y tj = j 1, hece by iductio we deduce that Z t = i s t 1 Y s + 1. Cosequetly, A t = Y t i s t 1 Y s Lea 2.1. For all p 2 t > 0, there exists a costat c > 0 such that for ay iteger P N t 5t e ct. Proof. Let {α i } t i=1 be a sequece of i.i.d. rado variables distributed as Bi, p. It is clear that we ca couple η i ad α i so η i α i for all i, ad thus by 2.1 N t 1 t t α i. 2.5 The su t i=1 α i is distributed as Bit, p ad p 2 so by Large Deviatios see Alo ad Specer 2000 sectio A.14 we get that for soe fixed c > 0 i=1 i=1 t P α i 3t e ct, which together with 2.5 cocludes the proof. 3. The subcritical phase Before begiig the proof of Theore 1.1 we require soe facts about processes with i.i.d. icreets. Fix soe sall ɛ > 0 ad let p = 1 ɛ for soe iteger > 1.

4 136 Asaf Nachias ad Yuval Peres Let {β j } be a sequece of rado variables distributed as Bi, p. Let {W t } t 0 be a process defied by Let τ be the hittig tie of 0, W 0 = 1, W t = W t 1 + β t 1. τ = i t {W t = 0}. By Wald s lea we have that E τ = ɛ 1. Further iforatio o the tail distributio of τ is give by the followig lea. Lea 3.1. There exists costat C 1, C 2, c 1, c 2 > 0 such that for all T ɛ 2 we have Pτ T C 1 ɛ 2 T 3/2 e ɛ2 c 1 ɛ 3 T 2, ad Furtherore, Pτ T c 2 ɛ 2 T 3/2 e ɛ2 +C 2 ɛ 3 T 2. E τ 2 = Oɛ 3. We will use the followig propositio due to Spitzer Propositio 1. Let a 0,..., a k 1 Z satisfy k 1 i=0 a i = 1. The there is precisely oe j {0,..., k 1} such that for all r {0,..., k 2} r a j+i od k 0. i=0 Proof of Lea 3.1. By Propositio 1, Pτ = t = 1 t PW t = 0. Sice t β j is distributed as a Bit, p rado variable we have t PW t = 0 = p t 1 1 p t t 1. t 1 Replacig t 1 with t i the above forula oly chages it by a ultiplicative costat which is always betwee 1/2 ad 2. A straightforward coputatio usig Stirlig s approxiatio gives { PW t = 0 = Θ t 1/2 1 ɛ t Deote q = 1 ɛ Pτ T = t T t 1 1 1, the 1 1 ɛ Pτ = t = t T This su ca be bouded above by 1 t PW t = 0 = Θ qt T 3/2 q t = T 3/2 1 q, t T 1 1 ɛ t 1 }. 3.1 t 3/2 q t. t T ad below by 2T t=t t 3/2 q t 2T 3/2 qt 1 q T 1 q.

5 Copoet sizes of the rado graph outside the scalig widow 137 Observe that as we have that q teds to 1 ɛe ɛ. By expadig e ɛ we fid that q = 1 ɛ1 + ɛ + ɛ2 2 + Θɛ3 = 1 ɛ2 2 + Θɛ3. Usig this ad the previous bouds o Pτ T we get the first two assertios of the Lea. The third assertio follows fro the followig coputatio. By 3.1 we have that for soe costat C > 0 E τ 2 = t 1 t 2 Pτ = t = t 1 tpw t = 0 C t 1 tq t. Thus, by direct coputatio or by Feller 1971, sectio XIII.5, Theore 5 1 3/2 E τ 2 O = Oɛ 3. 1 q Proof of Theore 1.1. We begi with a upper boud. Recall that copoet sizes are {t j+1 t j : j 0} where t j are record iia of the process {Y t }. For a vertex v deote by Cv the coected copoet of G, p which cotais v. We first boud P Cv 1 T 1 where T 1 = 21 + δɛ 2 logɛ 3. Recall that Cv 1 = i t {Y t = 0}. Couple {Y t } with a process {W t } as i Lea 3.1, which has icreets distributed as Bi, p 1 such that Y t W t for all t. Defie τ as i Lea 3.1. Sice p = 1 ɛ ad T 1 ɛ 2, Lea 3.1 gives that Pτ T 1 C 1 ɛɛ 3 1+δ1 c1ɛ log 3/2 ɛ 3, for soe fixed C > 0. Our couplig iplies that P Cv 1 T 1 Pτ T 1. Deote by X the uber of vertices v such that Cv T 1. If C 1 T 1, the X T 1. Also, for ay two vertices v ad u, by syetry we have that Cv ad Cu are idetically distributed. We coclude that P C 1 T 1 PX T 1 E X = P Cv 1 T 1 T 1 T 1 C 1ɛɛ 3 1+δ1 c1ɛ log 3/2 ɛ δɛ 2 logɛ 3 ɛ 3 δ1 c1ɛ+c1ɛ 0. We ow tur to prove a lower boud. Write ad defie the stoppig tie T 2 = 21 δɛ 2 logɛ 3, γ = i{t : N t δɛ 8 }. Recall that {t j } are ties i which A tj = 0 ad also that Y tj = j 1 is a record } be a process with icreets dis- 0 = j 1. Sice η t γ is stochastically rado variable we ca couple such that iiu for {Y t }. For each iteger j let {W j t tributed as Bi δɛ j 8, p ad iitially W bouded below by a Bi δɛ 8 Y tj+t γ W j t γ t j 0.

6 138 Asaf Nachias ad Yuval Peres Defie the stoppig ties {τ j } by Take τ j = i{t : W j t = j}. N = ɛ 1 ɛ 3 1 δ 8. We will prove that w.h.p. t N < γ ad that there exists k 1 < k 2 <... < k l < N such that τ ki T 2. Note that by our couplig, these two evets iply that C l T 2. Lea 2.1 shows that for soe c > 0 we have P γ δɛ e cɛ By boudig the icreets of {Y t } above by variables distributed as Bi, p 1 we lear by Wald s Lea see Durrett 1996 that E [t j+1 t j ] ɛ 1 for ay j 0, hece E t N ɛ 2 ɛ 3 1 δ 8. We coclude that Pt N δɛ 40 40ɛ 2 ɛ 3 1 δ δɛ 8 = 40 δ ɛ3 δ 8, 3.3 which goes to 0 as ɛ 1/3 teds to. Next, we take = δɛ 8 i Lea 3.1 ad ote that p =. Hece, Lea 3.1 gives that for ay j 1 1+ δ 8 ɛ 1 ɛ1 δɛ 8 Pτ j T 2 c 2 ɛɛ 3 1+ δ δ1+c 2ɛ log 3/2 ɛ 3 ɛɛ 3 1 δ 4. Let X be the uber of j N such that τ j T. The we have E X Nɛɛ 3 1 δ 4 Cɛ 3 δ 8, hece by Large Deviatios see Alo ad Specer 2000, sectio A.14, for ay fixed iteger l > 0 we have P X < l e cɛ3 δ 8, 3.4 for soe c = cl > 0. By our couplig we have that } { } { C l < T 2 X < l { t N > γ }. This together with 3.2, 3.3 ad 3.4 gives ɛ 3 δ 8 P C l < T 2 O. δ 4. The supercritical phase I this sectio we deote ξ t = η t 1. We first prove soe Leas. Lea 4.1. If p = 1+ɛ, the for all t 3ɛ ad E A t = Oɛt + t, 4.1

7 Copoet sizes of the rado graph outside the scalig widow 139 E Z t = Oɛt + t. 4.2 Proof. Write T = 3ɛ. We will use 2.4. First observe that sice η t ca always be bouded above by a Bi, p rado variable, we ca boud E [ξ t F t 1 ] ɛ for all t. Hece, the process {ɛj Y j } t j=0 is a subartigale for ay t. Deote by γ the stoppig tie γ = i{t : N t 15ɛ}. By Doob s axial L 2 iequality we have E [ ax j t γ ɛj Y j 2 ] 4E [ɛt γ Y t γ 2 ]. 4.3 The process {Y t } is stochastically bouded above by the process {X t } which has i.i.d. icreets distributed as Bi, p 1 rado variables. By defiitio, coditioed o the evet j < γ, the rado variable η j ca be stochastically bouded below by a Bi 15ɛ, p rado variable. Thus, the process {Y t γ } is stochastically bouded below by the process { X t γ }, where { X t } has i.i.d. icreets distributed as Bi 15ɛ, p 1 rado variables. Hece we ca couple such that Yt γ1 2 {Yt γ 0} Xt γ 2, Yt γ1 2 {Yt γ<0} X t γ 2. It is a iediate coputatio to verify that E Xt γ 2 = Oɛ2 t 2 + t ad that E X t γ 2 = Oɛ2 t 2 + t ad thus E Yt γ 2 = Oɛ2 t 2 + t. We use this ad the Cauchy- Schwarz iequality to boud the right had side of 4.3, E [ɛt γ Y t γ 2 ] = Oɛ 2 t 2 + t. Lea 2.1 iplies that for large eough, P N T 15ɛ e 3cɛ 1 2, 4.4 ad as {N t } is a decreasig sequece we deduce that Pγ T 2. Hece for ay t T E [ɛt Y t 2 ] E [ɛt γ Y t γ 2 1 {t<γ} ] + O 2 Pt γ = Oɛ 2 t 2 + t. We deduce by 4.3 ad Jese iequality that for ay t T E [i j t Y j ɛj] = Oɛt + t, hece E [i j t Y j ] = Oɛt + t ad so by 2.4 we obtai 4.1. Iequality 4.2 follows iediately fro the relatio Z t = A t Y t. Lea 4.2. If p = 1+ɛ, the for all t 3ɛ ad E N t = 1 p t + Oɛ 2, 4.5 E ξ t = ɛ t + Oɛ2. 4.6

8 140 Asaf Nachias ad Yuval Peres Proof. Observe that by 2.1 we have that E [N t F t 1 ] = 1 pn t 1 1 p1 {At 1=0}. By iteratig this relatio we get that E N t = 1 p t + OE Z t which by Lea 4.1 yields 4.5 observe that for t = 3ɛ we have ɛt t by our assuptio o ɛ. Sice E[ξ t F t 1 ] = pn t 1 p1 {At 1=0} 1, by takig expectatios ad usig 4.5 we get E ξ t = 1 + ɛ1 1 + ɛ t 1 + Oɛ 2 = 1 + ɛ1 1 + ɛt/ 1 + Oɛ 2 = ɛ t + Oɛ2, where we used the fact that 1 x t = 1 tx + Ot 2 x 2. Proof of Theore 1.2. Write T = 3ɛ ad ξ j = E [ξ j F j 1 ]. The process M t = Y t t ξj, is a artigale. By Doob s axial L 2 iequality we have that E ax t T M 2 t 4E M 2 T. Sice M t has orthogoal icreets with bouded secod oet, we deduce that E MT 2 = OT. Hece, by Jese s iequality, [ t ] E Y t O T = O ɛ. 4.7 ax t T As ξ j = pn j 1 p1 {Aj 1=0} 1 by 2.3 we have ξ j E ξ j E ξ j = pe A j {Aj 1=0} E A j 1 E 1 {Aj 1=0}. By the triagle iequality ad Lea 4.1 we coclude that for all j T j t E ξ j E ξ j p Oɛj + j, ad hece for ay t T [ ] E ξj E ξ j p Oɛt 2 + t 3/2 Oɛ 3. By the triagle iequality we get [ E ax t T t ] ξj E ξ j Oɛ Usig the triagle iequality, 4.7, 4.8 ad Markov s iequality give that for ay a > 0 P ax t T Y t t Lea 4.2 iplies that for ay b > 0 E ξ j aɛ 2 a 1 Oɛ + Oɛ 3 1/

9 Copoet sizes of the rado graph outside the scalig widow 141 bɛ bɛ E ξ j = ɛ t + Oɛ2 = b b2 2 ɛ2 + Oɛ By 4.9 ad 4.10 we deduce that for δ > 0 sall eough, with probability tedig to 1, the process Y t is strictly positive at all ties i [δɛ, 2 δɛ] ad hece P C 1 21 δɛ 1 O δ 1 ɛ + ɛ 3 1/2. We also deduce by 4.9 ad 4.10 that at tie t = 2 + δɛ we have Y t δ2 3 ɛ2 ad at all ties t δɛ we have that Y t > δ2 3 ɛ2 with probability tedig to 1. Sice copoet sizes are excursio legths of Y t above its past iia, we coclude that w.h.p. by tie 21 + δɛ we have explored copletely at least oe copoet of size at least 21 δɛ. Coditio o the tie t 0 of the first record iiu after tie 21 δɛ. The uber of eutral vertices reaiig at that tie is t 0. The subgraph of G, p iduced o these reaiig vertices is distributed as G t 0, p. Sice t 0 21 δɛ ad p = 1+ɛ, Theore 1.1 iplies that w.h.p. G t 0, p has o copoets of size at least ɛ. Thus we have proved that w.h.p. i G, p there exists a uique copoet of size betwee 21 δɛ ad 21 + δɛ. Coditio o this evet ad o the size of this uique copoet ad cosider the graph G iduced by the copleet of this copoet. This graph has vertices where ad sice p = 1+ɛ we have that p 2ɛ 2δɛ, 1 ɛ 2δɛ + Oɛ 2. The graph G is distributed as G, p coditioed o the evet A that it does ot cotai a copoet of size betwee 21 δɛ ad 21 + δɛ. By Theore 1.1 we have that PA = 1 o1. Thus for ay collectio of graphs B A we have that P,pB = 1 + o1p,p B where P,p is the distributio of G, p ad P,p is the easure P,p coditioed o A. Thus, we coclude by Theore 1.1 that for ay iteger l > 1 ad δ > 0 C l P 2ɛ 2 logɛ 3 1 δ 0, cocludig the proof of the theore. Reark. With a little ore effort it is possible to show for the supercritical case, that i the exploratio process for ay fixed l > 1, the l-th largest copoet is explored after the largest copoet is explored. Ackowledgets. The first author would like to thak Microsoft Research, i which parts of this research were coducted, for their kid hospitality. The work was doe partially while the authors were visitig the Istitute for Matheatical Scieces, Natioal Uiversity of Sigapore i The visit was supported by the Istitute.

10 142 Asaf Nachias ad Yuval Peres Refereces D. Aldous. Browia excursios, critical rado graphs ad the ultiplicative coalescet. A. Probab. 25, N. Alo ad J. H. Specer. The probabilistic ethod. Wiley, New York, 2d editio E. A. Beder, E. R. Cafield ad B. D. McKay. The asyptotic uber of labeled coected graphs with a give uber of vertices ad edges. Rado Structures Algoriths 1, B. Bollobás. The evolutio of rado graphs. Tras. Aer. Math. Soc. 286, R. Durrett. Probability: Theory ad Exaples. Duxbury Press, Belot, Califoria, 2d editio W. Feller. A itroductio to probability theory ad its applicatios, volue II. Joh Wiley & Sos, Ic., New York-Lodo-Sydey, 2d editio R. M. Karp. The trasitive closure of a rado digraph. Rado Structures Algoriths 1, T. Luczak. Copoet behavior ear the critical poit of the rado graph process. Rado Structures Algoriths 1, T. Luczak, B. Pittel ad J. C. Wiera. The structure of a rado graph at the poit of the phase trasitio. Tras. Aer. Math. Soc. 341, A. Marti-Löf. Syetric saplig procedures, geeral epideic processes ad their threshold liit theores. J. Appl. Probab. 23, A. Nachias ad Y. Peres. The critical rado graph, with artigales. To appear i Israel J. of Math A. Nachias ad Y. Peres. Critical percolatio o rado regular graphs Preprit. Available at F. Spitzer. A cobiatorial lea ad its applicatio to probability theory. Aer. Math. Soc. 82, E. M. Wright. The uber of coected sparsely edged graphs. Joural of Graph Theory 1,