Dependent percolation, critical exponents, random walks, and anchored isoperimetry

We study three problems related to percolation processes and random walks.; In Chapter 1, we consider a dependent bond percolation model on Z2 , introduced by Balint Toth, in which every edge is present with probability 1/2, and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. A more detailed analysis leads to the derivation of the following critical exponents: the tail probability P (diameter of the cycle of the origin > n) ≈ n-gamma, and the expectation E (length of a cycle conditioned on having diameter n) ≈ ndelta. We show that gamma = (5- 17 )/4 = 0.219... and delta = ( 17 + 1)/4 = 1.28... The relation gamma + delta = 3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the Additive Brownian Motion, whose level sets have Hausdorff dimension 3/2. The value of delta comes from the asymptotic solution of a sixth order singular ODE.; Benjamini, Lyons and Schramm (1999) initiated a systematic study of the properties of transitive graphs that are preserved under random perturbations. They showed that simple random walk on any infinite percolation cluster of a non-amenable Cayley graph has positive speed, and conjectured that there is a geometric reason for this: namely, that the clusters have anchored expansion, a property somewhat weaker than non-amenability. And, indeed, Virag proved that anchored expansion implies positive speed, while, extending a result of Chen and Peres, we show that on any graph with anchored expansion, if p satisfies a natural lower bound, then any infinite cluster of Bernoulli(p) percolation also has anchored expansion. The core of the method is to use fast decay of the probability of seeing a finite cluster with large outer edge boundary in supercritical percolation. This is applicable more broadly, e.g. to prove survival of anchored isoperimetric inequalities under percolation with a large enough parameter p on a very large class of graphs, and for all supercritical p in the case of Zd . Using a recent elegant way of Lyons, Morris and Schramm to deduce a certain kind of decay for Green's function from anchored isoperimetry, we also give a short proof of the result of Angel, Benjamini, Berger and Peres (2004): whenever a wedge in Z3 is transient, then the infinite percolation cluster on it is also such. (Abstract shortened by UMI.)...