Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap |λ 1 − λ 2 |) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that for general graphs, if the relaxation time τ 2 satisfies τ 2 = O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.

Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle /N are given.

We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius k around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit. . This reprint differs from the original in pagination and typographic detail. 1 2 BERGER, BORGS, CHAYES AND SABERI the paper [11] also introduces a notion of graph limits for sparse graphs with bounded degrees in terms of graph homomorphisms; using expansion methods from mathematical physics, Borgs et al.[10] establishes some general results on this type of limit for sparse graphs. Another recent work [8] concerns limits for graphs which are neither dense nor sparse in the above senses; they have average degrees which tend to infinity. Earlier, a notion of a weak local limit of a sequence of graphs with bounded degrees was given by Benjamini and Schramm [5] (this notion was in fact already implicit in [3]). Interestingly, it is not hard to show that the Benjamini-Schramm limit coincides with the limit defined via graph homomorphisms in the case of sparse graphs of bounded degree; see [16] for yet another equivalent notion of convergent sequences of graphs with bounded degrees.As observed by Lyons [21], the notion of graph convergence introduced by Benjamini and Schramm is meaningful even when the degrees are unbounded, provided the average degree stays bounded. Since the average degree of the Barabási-Albert graph is bounded by construction, it is therefore natural to ask whether this graph sequence converges in the sense of Benjamini and Schramm.In this paper, we establish the existence of the Benjamini-Schramm limit for the Barabási-Albert graph by giving an explicit construction of the limit process, and use it to derive various properties of the limit. Our results cover the case of both uniform and preferential attachments graphs. 1 Moreover, our methods establish the finite-volume corrections which give the approach to the limit.Our proof uses a representation, which we first introduced in [6], to analyze processes that model the spread of viral infections on preferential attachment graphs. Our representation expresses the preferential attachment model process as a combination of several Pólya urn processes. The classic Pólya urn model was of course proposed and analyzed in the beautiful work of Pólya and Eggenberger in the early twentieth century [15]; see [14] for a basic reference. Despite the fact that our Pólya urn representation is a priori only valid for a limited class of preferential attachment graphs, we give an approximating coupling which proves that the limit constructed here is the limit of a much wider class of preferential attachment graphs.Our alternative representation contains much more independence than previous repr...

Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N . Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p = 1/2, the mixing time of this card shuffling is O(N 2 ), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d = 1, 2, where x and y are connected with probability ∼ β/ x − y −s . We show that if d < s < 2d then the walk is transient, and if s ≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d ≥ 1, if d < s < 2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d ≥ 2 and d < s < 2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition (T ′ ). We show that for every ǫ > 0 and n large enough, the annealed probability of linear slowdown is bounded from above by exp −(log n) d−ǫ . This bound almost matches the known lower bound of exp −C(log n) d , and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability of slowdown. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the assumption of the same condition.

Abstract. We consider a random walk on Z d in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from x ∈ Z d to nearest neighbor x + e is the same as to nearest neighbor x − e. Assuming that the environment is genuinely d-dimensional and balanced we show a quenched invariance principle: for P almost every environment, the diffusively rescaled random walk converges to a Brownian motion with deterministic non-degenerate diffusion matrix. Within the i.i.d. setting, our result extend both Lawler's uniformly elliptic result [15] and Guo and Zeitouni's elliptic result [12] to the general (non elliptic) case. Our proof is based on analytic methods and percolation arguments.