Abstract. We define generic ensembles of infinite trees. These are limits as N → ∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension dh = 2. Our main result is that their spectral dimension is ds = 4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is

ABSTRACT. We consider the nearest-neighbor simple random walk on Z d, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy> 0 exceeds the threshold for bond percolation on Z d. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step return probability P 2n ω (0,0). We prove that P 2n ω (0,0) is bounded by a random constant times n −d/2 in d = 2,3, while it is o(n −2) in d ≥ 5 and O(n −2 log n) in d = 4. By producing examples with anomalous heat-kernel decay approaching 1/n 2 we prove that the o(n −2) bound in d ≥ 5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d = 4. 1.

Abstract. Let C1 denote the largest connected component of the critical Erdős-Rényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.

We show that the total variation mixing time of the simple random walk on the giant component of supercritical G(n,p) and G(n,m) is Θ(log 2 n). This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are “decorated expanders ” — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. 1

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its r-point function for any r ≥ 2 and of its volume both at a given height and below a given height. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay towards the critical value for ordinary percolation, but only slowly, and this slow decay causes an anomalous scaling behaviour. MSC 2000. 60K35, 82B43 Key words and phrases. Invasion percolation cluster, incipient infinite cluster, r-point function, cluster size, simple random walk, Poisson point process.

1. Simply generated trees and Galton–Watson trees We suppose that we are given a fixed weight sequence w = (wk)k�0 of non-negative real numbers. We then define the weight of a finite rooted and ordered (a.k.a. plane) tree T by w(T): = ∏ wd +(v), (1.1) v∈T taking the product over all nodes v in T, where d + (v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let Tn be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w0> 0 and that there exists some k � 2 with wk> 0. We consider only n such that there exists some tree with n vertices and positive weight.) One particularly important case is when ∑ ∞ k=0 wk = 1, so the weight sequence (wk) is a probability distribution on Z�0. (We then say that (wk) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = wk. It is easily seen that the simply generated random tree Tn equals the conditioned Galton–Watson tree with offspring distribution ξ, i.e., the random Galton–Watson tree defined by ξ conditioned on having exactly n vertices. One of the reasons for the interest in these trees is that many kinds of random trees occuring in various applications (random ordered trees, unordered trees, binary trees,...) can be seen as simply generated random trees and conditioned Galton–Watson trees, see e.g. Aldous [3, 4], Devroye [9] and Drmota [10]. It is easily seen that if a, b> 0 and we change wk to ˜wk: = ab k wk, (1.2)

Abstract. We show that for all p> pc(Z d) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. This implies for instance that simple random walk on the largest cluster of a finite box [−n,n] d with high probability has L ∞-mixing time Θ(n 2), and that the heat kernel (return probability) on the infinite cluster a.s. decays like pn(o, o) = O(n −d/2). Versions of these results have been proven by Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau (2006). We also give a short proof of a theorem of Angel, Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedge in Z 3 is a.s. transient whenever the wedge itself is transient. 1. Introduction and

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E l e c t r o n

We use the concept of unimodular random graph to show that the branching simple random walk on Z d indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if d � 4.