Random Walks and Geometry

Recent developments show that probability methods have become a very powerful tool in such different areas as statistical physics, dynamical systems, Riemannian geometry, group theory, harmonic analysis, graph theory and computer science.This volume is an outcome of the special semester 2001 - Random Walks held at the Schrödinger Institute in Vienna, Austria. It contains original research articles with non-trivial new approaches based on applications of random walks and similar processes to Lie groups, geometric flows, physical models on infinite graphs, random number generators, Lyapunov exponents, geometric group theory, spectral theory of graphs and potential theory. Highlights are the first survey of the theory of the stochastic Loewner evolution and its applications to percolation theory (a new rapidly developing and very promising subject at the crossroads of probability, statistical physics and harmonic analysis), surveys on expander graphs, random matrices and quantum chaos, cellular automata and symbolic dynamical systems, and others. The contributors to the volume are the leading experts in the area.

Some Markov chains on abelian groups with applications	p. 3
Random walks and physical models on infinite graphs : an introduction	p. 35
The garden of Eden theorem for cellular automata and for symbolic dynamical systems	p. 73
Expander graphs, random matrices and quantum chaos	p. 109
The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps	p. 141
Simplicite de spectres de Lyapounov et propriete d'isolation spectrale pour une famille d'operateurs de transfert sur l'espace projectif	p. 181
An introduction to the Stochastic Loewner evolution	p. 261
A canonical form for automorphisms of totally disconnected locally compact groups	p. 295
On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds	p. 319
Markov processes on vermiculated spaces	p. 337
Cactus trees and lower bounds on the spectral radius of vertex-transitive graphs	p. 349
Equilibrium measure, Poisson kernel and effective resistance on networks	p. 363
Internal diffusion limited aggregation on discrete groups of polynomial growth	p. 377
On the physical relevance of random walks : an example of random walks on a randomly oriented lattice	p. 393
Random walks, entropy and hopfianity of free groups	p. 413
Growth rates of small cancellation groups	p. 421
Recurrence properties of random walks on finite volume homogeneous manifolds	p. 431
On the cohomology of foliations with amenable groupoid	p. 445
Linear rate of escape and convergence in direction	p. 459
Remarks on harmonic functions on affine buildings	p. 473
Random walks, spectral radii, and Ramanujan graphs	p. 487
Cogrowth of arbitrary graphs	p. 501
Total variation lower bounds for finite Markov chains : Wilson's lemma	p. 515
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