ELECTRONICCOMMUNICATIONSinPROBABILITY RANDOM WALK ATTRACTED BY PERCOLATION CLUSTERS

Starting with a percolation model in Z d in the subcritical regime, we consider a random walk described as follows: the probability of transition from x to y is proportional to some function f of the size of the cluster of y . This function is supposed to be increasing, so that the random walk is attracted by bigger clusters. For f ( t ) = e ﬂt we prove that there is a phase transition in ﬂ , i.e., the random walk is subdiﬁusive for large ﬂ and is diﬁusive for small ﬂ .


Introduction and results
First, we describe the usual site percolation model in Z d . This model is defined as follows. For fixed p ∈ (0, 1), consider i.i.d. random variables ω(x), x ∈ Z d , where ω(x) = 1 with probability p and ω(x) = 0 with probability 1 − p. Note that, if ω(x) = 0, then C(x) = ∅. It is a well-known fact (see e.g. [9]) that there exists p cr (depending on d; obviously, p cr = 1 in dimension 1) such that if p < p cr , then a.s. there is no infinite open cluster, and if p > p cr , then a.s. there exists an infinite open cluster (also, with positive probability |C(0)| = ∞). Throughout this paper we assume that p < p cr , i.e, the model is in the (strictly) subcritical regime. Fix a parameter β > 0. The percolation configuration is regarded as random environment. Fixed the environment, we start a discrete time random walk on Z d with transition probabilities P ω xy = e β|C(y)| z∼x e β|C(z)| , if x ∼ y. Since β is positive, one can note that the random walk is in some sense "attracted" by bigger clusters, and the strength of this attraction grows with β. Denote by ξ(t) the position of this random walk at time t. Let P be the probability measure with respect to ω and P x ω the (so-called quenched) probability for the random walk starting from x in the fixed environment ω. Denote also P x = P x ω ⊗ P (P x is usually called the annealed probability); throughout the paper · stands for the L ∞ norm. Our main result is that there is a phase transition in β, i.e., the random walk exhibits different behaviors for large and small β: it is diffusive for small values of β and subdiffusive for large values of β. Theorem 1.1 Suppose that the random walk ξ(t) starts from the origin. There exist β 0 and One can prove also that the same result holds for the bond percolation model in the subcritical regime. The method of the proof remains the same; the reason why we have chosen the site percolation is that for bond percolation there are some technical difficulties (easily manageable, though; they relate to the fact that, in the bond percolation model, two neighboring sites can belong to different large clusters) in the proof of the part (ii) of Theorem 1.1.
Recently much work has been done on the (simple or not) random walk on the unique infinite cluster for the supercritical (bond or site) percolation in Z d (see e.g. [2,4,10,15]; see also [11] for some results for the random walk on the incipient infinite cluster in dimension 2). Another related subject is the class of models (see e.g. [5,8]) that can be described as follows. Into each edge of Z d we place a random variable that represents the transition rate between the sites. The new features of the model of the present paper are, first, the fact that the random environment is not independent, and secondly, the absence of the uniform ellipticity. Speaking of uniform ellipticity, we should mention that in the paper [7] there was considered a simple symmetric one-dimensional random walk with random rates, where the time spent at site i before taking a step has an exponential distribution with mean τ i , and τ i 's are i.i.d. positive random variables with distribution function F having a polinomial tail. One may find that there are similarities of the d-dimensional analog of the model of [7] with our model, because clusters of size n will have "density" e −Cn , and the mean time spent there is roughly e βn , so, thinking of clusters as "sites", we indeed obtain a polinomial tail of mean time spent at a given site. However, the facts that the random environment is no longer independent and that here the random walk is not a time-change of the simple random walk make the model of the present paper considerably more difficult to analyze.
2 Proof of Theorem 1.1 We begin by introducing some notations and recalling a well-known fact from the percolation theory. Namely, we will use the following result (see [12,9]): if p < p cr , then there exists Now, to prove Theorem 1.1, an important idea is to consider ξ(t) in finite region. Take Λ n = (−n/2, n/2] d and let the process ξ (n) (t) be the random walk ξ(t) restricted on Λ n .
Proof of part (i). It can be easily seen that ξ(t) is reversible with the reversible measure and thus the finite Markov chain ξ (n) (t) is also reversible, with the invariant (and reversible) measure is the normalizing constant, so that x∈Λn π (n) (x) = 1.
Consider also a random walkξ (n) (t) that is a continuization of ξ (n) (t). That is, where N t is a Poisson process with rate 1, independent of anything else (in other words,ξ (n) is a continuous time Markov chain with the transition rates equal to the transition probabilities of ξ (n) ). Let T i be the time interval between the jumps (i − 1) and i of N t , S n = n i=1 T i , andT A (respectively, T A ) be hitting time of set A by random walkξ (n) (t) (respectively, ξ (n) (t)). It can be easily seen (cf., for example, s., many other results concerningξ (n) can be easily translated into the corresponding results for ξ (n) .
Remark 2.1 Using this technique, it is elementary to obtain that Theorem 1.1 holds for ξ(t) iff it holds forξ(t), whereξ(t) is the continuization of ξ(t) defined in the same way.
So, now we consider the finite continuous time Markov chainξ (n) (t). Denote by λ the spectral gap ofξ (n) (t).
so, first we successively change the d-th coordinate of x to obtain the d-th coordinate of y, then we do the same with (d − 1)-th coordinate, and so on. With this construction it is clear that the length of γ(x, y) is at most dn. For an edge (for the edges of other directions the computations are quite analogous). We have then as y∈Ru π (n) (y) ≤ 1. Now our goal is to prove that with large probability, for all such u, x∈Iu π (n) (x) is of order n/Z. DenoteĨ Note that I u ⊂Ĩ u , so we will concentrate on obtaining an upper bound for x∈Ĩu π (n) (x). It is important to observe that the variables π (n) (x) are not independent. For the sake of simplicity, suppose that √ n is an integer, the general case can be treated analogously. DivideĨ u into √ n equal (connected) parts of size √ n, denote √ n for some c 2 > 0, so For y = n α (α > 0 will be chosen later), using (2.1), we have According to Corollary 1.5 from [13], if X 1 , . . . , X k are independent random variables, , then for any set y 1 , . . . , y k of positive numbers and any t, t ∈ (0, 1], where y ≥ max{y 1 , . . . , y k } and DenoteF i (u) = 1−F i (u). We apply Corollary 1.5 from [13] to random variables 2de 2β|C(xij )| 1 B , i = − √ n 2 + 1, . . . , √ n 2 , with x = k √ n, y i ≡ y = n α , and t = 1. First term of the right-hand side of (2.7) was estimated in (2.6). For the second term, we have so, to guarantee that x y → 0 as n → ∞, it is sufficient to take k large enough and β/c 3 < α < 1/2. We proved that for β sufficiently small as e β(|C(z1)|+|C(z2)|) ≥ 1, with probability at least 1−c 7 n − c 10 β +1 . Since β can be made arbitrarily small, Borel-Cantelli lemma implies that for almost all environments for n large enough it holds that A ≤ c 13 n 2 and thus λ ≥ c 14 n −2 . Lemma 2.1 is proved. Now, using Lemma 2.1.4 from [14] with f (x) = 1 { x ≥n/4} , where, as before, · is the L ∞ norm, we prove (1.1). By Lemma 2.1.4 from [14] we have that In what follows we show that π (n) (f ) is of constant order. We have Since |C(x)| ≥ 0, it is easy to obtain that for all ω it holds we have π (n) (f ) ≥ const. Then, using that Var (n) π (f ) ≤ 1, taking t = c 16 n 2 for c 16 large enough yields that the random walkξ (n) (t), and thus ξ (n) (t), will be at distance of order n from the origin (as both random walks start from 0) after a time of order n 2 with probability bounded away from 0. Now, for any fixed ε > 0, divide the time interval (0, t] into t ε intervals of length t 1−ε . Borel-Cantelli lemma implies then that for t large enough there will be at least one time interval such that at the end of this interval ξ (n) will be at distance at least t It is a well-known fact that a reversible Markov chain with a "well-behaved" reversible measure cannot go much farther than t 1/2 by time t, see [3,6,11,16]. By Theorem 1 from [6] we have, for any ε > 0 Proof of part (ii). For x ∈ Z d let T 0 (x) = 0, T 0 (x) = 0, and define is the moment of ith entry to the cluster of x, and T i (x) is the time spent there (i.e., after T i (x) and before going out of C(x)). It is important to note that the cluster C(x) is surrounded by sites with ω(·) = 0. Comparing T i (x) with geometric random variable with parameter (2d−1+e β|C(x)| ) −1 , one can easily see that if |C(x)| ≥ δ log n, then ET i (x) ≥ c 17 n βδ . Moreover, it is elementary to obtain that for ε > 0 and for any δ > 0 we can choose β large enough so that with probability bounded away from 0 we have for x such that |C(x)| ≥ δ log n. Now, we use a dynamic construction of the percolation environment usually called the generation method (see [12]). That is, we proceed in the following way: we assign generation index 0 to the origin, and put ω Then, choose ω(x) for x ∈ H • 1 and start the random walk ξ(t) from the origin. Let Note that C(ξ(τ 1 )) ∩ C(0) = ∅, and construct, using the above method H 2 = C(ξ(τ 1 )) ∪ ∂C(ξ(τ 1 )) and Then, define τ 2 = min{t : ξ(t) / ∈ G 2 }, and so on. For all i, we have where p is the percolation parameter. This is so due to the fact that, to have C(ξ(τ i )) ≥ δ log n, it is sufficient to choose a path of length δ log n emanating from ξ(τ i ) which does not intersect G i (it is possible by the construction of τ i , since ξ(τ i ) cannot be completely surrounded by points of G i ), and such path will be open with probability p δ log n . Thus, for any ε > 0 (one can take the same ε from (2.18)), we can choose δ small enough (take δ such that δ log p −1 < ε) so that where F i is the σ-algebra generated by {ω(x), x ∈ G i } and {ξ(m), m ≤ τ i }. Fix θ > 0 in such a way that 1 − θ > ε. Note that, as p < p cr , using (2.1) and Borel-Cantelli lemma, for n large enough min{k : ξ(τ k ) / ∈ Λ n } (the number of times that we repeat the basic step in the above construction) will be of order at least n 1−θ for all n large enough, P 0 -a.s. (recall that Λ n = (−n/2, n/2] d ; with overwhelming probability all the clusters inside Λ n will be of sizes at most n θ ). On each step, by (2.19), with probability at least n −ε the random walk enters the cluster of size at least δ log n. By (2.18), it stays in that cluster (if β is large enough) for at least n 2+ε time units with large probability. If 1 − θ > ε, with overwhelming probability on some step (of the above construction) the random walk will delay (in the corresponding cluster) for more than n 2+ε time units before going out of Λ n . In other words, we will have max s≤n 2+ε ξ(s) ≤ dn 2 , which implies (1.2). This concludes the proof of Theorem 1.1.