Random Walks and Geometry
Description
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. The book will provide a valuable source both for active researchers and graduate students in the respective fields.
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Table of contents
Surveys and longer articles
Some Markov chains on abelian groups with applications
Random walks and physical models on infinite graphs: an introduction
The Garden of Eden Theorem for cellular automata and for symbolic dynamical systems
Expander graphs, random matrices and quantum chaos
The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps
Simplicité de spectres de Lyapounov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif
An introduction to the Stochastic Loewner Evolution
A canonical form for automorphisms of totally disconnected locally compact groups
Research communications
On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds
Markov processes on vermiculated spaces
Cactus trees and lower bounds on the spectral radius of vertex-transitive graphs
Equilibrium measure, Poisson kernel and effective resistance on networks
Internal diffusion limited aggregation on discrete groups of polynomial growth
On the physical relevance of random walks: an example of random walks on a randomly oriented lattice
Random walks, entropy and hopfianity of free groups
Growth rates of small cancellation groups
Recurrence properties of random walks on finite volume homogeneous manifolds
On the cohomology of foliations with amenable groupoid
Linear rate of escape and convergence in direction
Remarks on harmonic functions on affine buildings
Random walks, spectral radii, and Ramanujan graphs
Cogrowth of arbitrary graphs
Total variation lower bounds for finite Markov chains: Wilson's lemma
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