A new proof of the sharpness of the phase transition for Bernoulli percolation on Z d

Transcription

1 A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation on Z. More precisely, we show that for p < p c, the probability that the origin is connecte by an open path to istance n ecays exponentially fast in n. for p > p c, the probability that the origin belongs to an infinite cluster satisfies the mean-fiel lower boun θ(p) p pc p( p. c) In [DT5], we give a more general proof which covers long-range Bernoulli percolation (an the Ising moel) on arbitrary transitive graphs. This article presents the argument of [DT5] in the simpler framework of nearest-neighbour Bernoulli percolation on Z. Statement of the result Motivation. Bernoulli percolation was introuce by Broabent an Hammersley [BH57] as a moel for liqui poure in a porous meium. Since then, Bernoulli percolation has foun many applications in statistical physics an beyon, an it has been one of the most stuie moels of ranom graph. In this moel, each ege of the lattice Z is open with probability p, an close with probability p, thus giving us a ranom graph ω p given by the vertices of Z an the open eges. Of special interest for mathematicians an physicists are the connectivity properties of ω p. For 2, one may show that there exists a critical parameter p c = p c () (0, ) separating a supercritical phase p > p c where ω p almost surely contains an infinite connecte component from a subcritical phase p < p c where the connecte components of ω p are almost surely finite. The efinition of the subcritical phase implies that for p < p c, the probability of 0 being connecte to istance n by eges in ω p ecays to 0. This

2 result was refine in the following way: [Men86] an [AB87] prove that the probability is in fact smaller than exp( cn) where c = c(p) > 0. This result, sometimes referre to as the sharpness of the phase transition, is a milestone in the area (many of the elicate properties of the subcritical phase are base on this property). In this paper, we provie an alternative (short) proof of this result. Notation. Fix an integer 2. We consier the -imensional hypercubic lattice (Z, E ). Let Λ n = { n,..., n}, an let Λ n = Λ n Λ n be its vertex-bounary. Throughout this paper, S always stans for a finite set of vertices containing the origin. Given such a set, we enote its ege-bounary by S, efine by all the eges {x, y} with x S an y S. Consier the Bernoulli bon percolation measure P p on {0, } E for which each ege of E is eclare open with probability p an close otherwise, inepenently for ifferent eges. Two vertices x an y are connecte in S V if there exists a path of vertices (v k ) 0 k K in S such that v 0 = x, v K = y, an {v k, v k+ } is open for every 0 k < K. We enote this event by x S y. If S = Z, we rop it from the notation. We set 0 Λ n if 0 is connecte to a vertex in Λ n, an 0 if 0 Λ n hols for all n. Phase transition. usually efine by The critical parameter for Bernoulli percolation is p c = sup{p [0, ] s.t. P p [0 ] = 0}. A new iea of this paper is to use a ifferent efinition of the critical parameter. This new efinition relies on the following quantity. For p [0, ] an 0 S Z, efine ϕ p (S) = p {x,y} S x]. This can be interprete as the expecte number of open eges at the bounary of S, that are connecte to 0 in S. Base on this new quantity, we introuce: p c = sup {p [0, ] s.t. there exists a finite set 0 S Z with ϕ p (S) < }. We are now in a position to state our main result. Theorem.. For any, p c = p c. Furthermore,. For p < p c, there exists c = c(p) > 0 such that for every n, P p [0 Λ n ] e cn. 2

3 2. For p > p c, P p [0 ] p p c p( p c ). Remarks.. On Z, we easily fin that p c = p c =, an Item 2 is then irrelevant. 2. We refer to [DT5] for a etaile bibliography, an for a version of the proof vali in greater generality. The aim of this paper is to provie a proof in the simplest possible framework. 3. Theorem. was prove by Aizenman an Barsky [AB87] in the more general framework of long-range percolation. In their proof, they consier an aitional parameter h corresponing to an external fiel, an they erive the results from ifferential inequalities satisfie by the thermoynamical quantities of the moel. A ifferent proof, base on the geometric stuy of the pivotal eges, was obtaine at the same time by Menshikov [Men86]. These two proofs are also presente in [Gri99]. 4. In the efinition of p c, the set of parameters p such that there exists a finite set 0 S Z with ϕ p (S) < is an open subset of [0, ]. Thus, p c o not belong to this set, as illustrate below. S, ϕ p (S) < S, ϕ p (S) 0 p c p Therefore, we obtain that the expecte size of the cluster of the origin satisfies for every p p c, P p [0 x] x Z p ϕ p (Λ n ) = +. n 0 This result was originally prove in [AN84]. 5. Since ϕ p ({0}) = 2p, we obtain p c /2. 6. Item 2 is calle the mean-fiel lower boun for the infinite cluster ensity. This is ue to the fact that θ(p) p p c hols as p p c for Bernoulli percolation on a regular tree. This mean-fiel behavior is expecte to hol for Bernoulli percolation on Z when 6 (it is prove for 9 [HS90]). 3

4 7. On the square lattice, the inequality p c /2 was first obtaine by Harris in [Har60] (see also the short proof of Zhang presente in [Gri99]). The other inequality p c /2 was first prove by Kesten in [Kes80] using a elicate geometric construction involving crossing events. Since then, many other proofs invoking exponential ecay in the subcritical phase (see [Gri99]) or sharp threshol arguments (see e.g. [BR06]) have been foun. Here, Theorem. provies a short proof of exponential ecay an therefore a short alternative to these proofs. For completeness, let us sketch how exponential ecay implies that p c /2: Item implies that for p < p c the crossing probability for a n by n square tens to 0 as n goes to infinity. But self-uality implies that this oes not happen when p = /2, thus implying that p c /2. 2 Proof of the theorem It is sufficient to show Items an 2 with p c replace by p c (since it immeiately implies the equality p c = p c ). 2. Proof of Item The proof of Item (with p c in place of p c ) can be erive from the BKinequality [vbk85]. We present here an exploration argument, similar to the one in [Ham57], which oes not rely on the BK-inequality. Let p < p c. By efinition, one can fix a finite set S containing the origin, such that ϕ p (S) <. Let L > 0 such that S Λ L. Let k an assume that the event 0 Λ kl hols. Let C = {z S s.t. 0 S z}. Since S Λ kl =, there exists an ege {x, y} S such that the following events occur: 0 is connecte to x in S, {x, y} is open, y is connecte to Λ kl in C c. Using first the union boun, an then a ecomposition with respect to possible values of C, we fin P p [0 Λ kl ] {x,y} S C S = p {x,y} S C S P p [{0 S x, C = C} {{x, y} is open} {y Z C Λ kl }] x, C = C]P p [y Z C Λ kl ]. 4

5 In the secon line, we use the fact that the three events epen on ifferent sets of eges: the first event {0 S x, C = C} epens on eges between a vertex of C an one of S (which may be in C too), the secon on the state of {x, y} only an the thir on the state of eges not sharing any enpoint with C (this exclues the ege {x, y} or the eges involve in the first event). As a consequence, these three events are inepenent. Since y Λ L, one can boun P p [y Z C Λ kl ] by P p [0 Λ (k )L ] in the last expression. Hence, we fin which by inuction gives P p [0 Λ kl ] ϕ p (S)P p [0 Λ (k )L ] P p [0 Λ kl ] ϕ p (S) k. This proves the esire exponential ecay. 2.2 Proof of Item 2 Let us start by the following lemma proviing a ifferential inequality vali for every p. Lemma 2.. Let p [0, ] an n, p P p[0 Λ n ] p( p) inf ϕ p (S) ( P p [0 Λ n ]). (2.) Let us first see how it implies Item 2 of Theorem.. Let n an set f(p) = P p [0 Λ n ]. For p ( p c, ), ϕ p (S) for every S 0 so that the ifferential inequality (2.) becomes f (p) f(p) p( p). Integrating this inequality between p c an p > p c gives f( p c ) f(p) p p p c p c. Using the trivial boun f( p c ) 0, we fin P p [0 Λ n ] = f(p) p c( p) p( p c ) = p p c p( p c ). By letting n ten to infinity, we obtain the esire boun on P p [0 ]. 5

6 Proof of Lemma 2.. Recall that {x, y} is pivotal for the configuration ω an the event {0 Λ n } if ω {x,y} {0 Λ n } an ω {x,y} {0 Λ n }. (The configuration ω {x,y}, resp. ω {x,y}, coincies with ω except that the ege {x, y} is close, resp. open.) By Russo s formula (see [Gri99, Section 2.4]), we have p P p[0 Λ n ] = e Λ n P p [e is pivotal] Define the following ranom subset of Λ n : = p P p [e is pivotal, 0 / Λ n ]. e Λ n S = {x Λ n s.t. x / Λ n }. The bounary of S correspons to the outmost blocking surface (which can be obtaine by exploring from the outsie the set of vertices connecte to the bounary). When 0 is not connecte to Λ n, the set S is always a subset of Λ n containing the origin. By summing over the possible values for S, we obtain p P p[0 Λ n ] = p P p [e is pivotal, S = S]. e Λ n Observe that on the event S = S, the pivotal eges are the eges {x, y} S such that 0 is connecte to x in S. This implies that p P p[0 Λ n ] = p {x,y} S x, S = S]. The event {S = S} is measurable with respect to the state of eges having one enpoint in Z S, while {0 S x} epens by efinition on eges with both enpoints in S. As a consequence, they are inepenent. We obtain as esire. p P p[0 Λ n ] = = p {x,y} S p( p) p( p) inf x]p p [S = S] ϕ p (S)P p [S = S] ϕ p (S) P p [0 / Λ n ], 6

7 Acknowlegments This work was supporte by a grant from the Swiss NSF an the NCCR SwissMap also fune by the Swiss NSF. References [AB87] [AN84] Michael Aizenman an Davi J. Barsky. Sharpness of the phase transition in percolation moels. Comm. Math. Phys., 08(3): , 987. M. Aizenman an C. M. Newman. Tree graph inequalities an critical behavior in percolation moels. Journal of Statistical Physics, 36(- 2):07 43, 984. [BH57] S. R. Broabent an J. M. Hammersley. Percolation processes. I. Crystals an mazes. Proc. Cambrige Philos. Soc., 53:629 64, 957. [BR06] [DT5] [Gri99] [Ham57] [Har60] Béla Bollobás an Oliver Rioran. A short proof of the Harris-Kesten theorem. Bull. Lonon Math. Soc., 38(3): , H. Duminil-Copin an V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation an the Ising moel. arxiv: , 205. Geoffrey Grimmett. Percolation, volume 32 of Grunlehren er Mathematischen Wissenschaften [Funamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, secon eition, 999. J. M. Hammersley. Percolation processes: Lower bouns for the critical probability. Ann. Math. Statist., 28: , 957. T. E. Harris. A lower boun for the critical probability in a certain percolation process. Proc. Cambrige Philos. Soc., 56:3 20, 960. [HS90] Takashi Hara an Goron Slae. Mean-fiel critical behaviour for percolation in high imensions. Comm. Math. Phys., 28(2):333 39, 990. [Kes80] Harry Kesten. The critical probability of bon percolation on the square lattice equals 2. Comm. Math. Phys., 74():4 59, 980. [Men86] M. V. Menshikov. Coincience of critical points in percolation problems. Dokl. Aka. Nauk SSSR, 288(6):308 3, 986. [vbk85] J. van en Berg an H. Kesten. Inequalities with applications to percolation an reliability. J. Appl. Probab., 22(3): , 985. Département e Mathématiques Université e Genève Genève, Switzerlan hugo.uminil@unige.ch, vincent.tassion@unige.ch 7