2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

We study random walks on supercritical percolation clusters on wedges in Z3, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z2 is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This solves a question of C. Hoffman.

Abstract. This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature” takes on the value − ∞ at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are given by the sum of squares of left invariant vector fields. In particular, “L p-type ” gradient estimates hold for p ∈ (1, ∞), and the

Abstract. We study amenability of algebras and modules (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore for any) field K. Similarly, a G-set X is amenable if and only if its span KX is amenable as a KG-module for some (and therefore for any) field K. 1.

We describe the large time asymptotic behaviors of the probabilities p2t(e; e) of return to the origin associated to nite symmetric generating sets of Abelian-by-cyclic groups. We characterize the dierent asymptotic behaviors by simple algebraic properties of the groups 1 . 1

Abstract. We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators. 1.

this paper, we will study Hardy classes on infinite-sheeted Galois covering spaces of compact Riemann surfaces. Such covering spaces may be thought of as surfaces with a large group of automorphisms. They are infinitely connected, with the obvious exceptions of the disc, the plane and the punctured plane [Gri]. We will mostly assume that the covering group is hyperbolic in the sense of Gromov [Gro1], because only in this case do we have enough information about the action of automorphisms on the Martin boundary, where Hardy functions are represented by measures. The Hardy classes of the unit disc have been studied intensively ever since Hardy introduced them in 1915, leading to many developments in function theory, functional analysis and harmonic analysis. The theory has been extended with considerable success to Riemann surfaces with smooth boundary, non-planar as well as planar, see e.g. [Hei]. On the other hand, very little is known about Hardy classes of infinitely connected surfaces. The exception is the special class of Parreau-Widom surfaces, see [Has], which have been shown to have a Hardy theory similar to that of the disc. Apart from relatively compact domains in open surfaces, this is the only class of surfaces known to have

This article has two parts, one treating random walks, the other di#usions. The two parts are related in many ways, at the level of ideas as well as on firmer mathematical ground, and more so than this article can possibly convey. Behind the di#erence in settings from the symmetric group S n to the Lie group SL n (R) to the infinite dimensional torus T # , there is unity in the problems that are discussed and any substantial progress in one particular context sheds light on the entire subject. Part I: Random walks Let G be a group generated by a finite symmetric set S. That is, s # S implies s -1 # S and G = # # 0 S n . The Cayley graph 1 (G, S) has vertex set G and an edge from x to y if and only if y = xs for some s # S. To capture the basic idea of random walk, imagine a walker whose position is a vertex of this graph. At each stage, the walker takes a step along one of the adjacent edges, choosing uniformly at random from the possibilities. Where will the walker be after n steps? More generally, given a probability measure p on G, the associated random walk (X n ) n#0 proceeds at each step by picking s in G with probability p(s) and moving to X n+1 = X n s. The distribution after n steps is the convolution power p (n) where p # q(x) = # y p(y)q(y -1 x). Shu#ing cards Why would anyone want to study random walks on groups? Maybe simply because everyone uses random walks, just as Moliere's Monsieur Jourdain uses prose without realizing it. Indeed, most card shu#ing methods can be modeled as random walks on the symmetric group S n , n = 52, where the shu#ing mechanism is interpreted as choosing at random among a certain set of permutations. A single question obviously takes center stage: how many shu#es are needed to mix up the cards? Bayer and Di...

An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented. 1