Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Introduction. Operateurs lineaires associes a un graphe. Resultats fondamentaux. Rayon spectral, fonctions generatrices de marche et mesures spectrales. Croissance et nombre isoperimetrique d'un graphe. Fonctions propres positives. Graphes de groupes, arbres et graphes reguliers en distance. Quelques remarques sur les applications en chimie et en physique

https://www.fmf.uni-lj.si/~mohar/Reprints/1989/BM89_BLMS21_Woess_SurveyontheSpectraofInfiniteGraphs.pdf

A Survey on Spectra of infinite Graphs

The concept of freeness was introduced by Voiculescu in the context of operator algebras. Later it was observed that it is also relevant for large random matrices. We will show how the combination of various free probability results with a linearization trick allows to address successfully the problem of determining the asymptotic eigenvalue distribution of general selfadjoint polynomials in independent random matrices.

Free Probability and Random Matrices

The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

/pdf/the-poisson-formula-for-groups-with-hyperbolic-properties-nbjb97cxsw.pdf

The Poisson formula for groups with hyperbolic properties

Enumeration of the 50 fake projective planes

If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of typeA

n−1 for somen⩾2, and Γ must have a presentation of a simple type. The casen=2, when Δ is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when Δ is the building ofG=PGL(3,K),K a local field, and when Γ⩽G. We exhibit such a group Γ whenK=F
 q
 ((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeA
2.

Groups acting simply transitively on the vertices of a building of type 2, I

/pdf/random-walks-on-the-affine-group-of-local-fields-and-of-4vrvw1k4lj.pdf

Random walks on the affine group of local fields and of homogeneous trees

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y 1Y2 … Yn , where the Y j ’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

/pdf/random-walks-on-free-products-quotients-and-amalgams-7b75zume0t.pdf

Random walks on free products, quotients and amalgams

If (P, L) is a projective plane and ℐ is a ‘triangle presentation compatible with a point-line correspondence λ:P →L’, then ℐ gives rise to a group Γℐ and a thick building Δℐ of typeA 2 on the vertices of which Γℐ acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence λ:P →L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these ℐ, Δℐ is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the ℐ for which this is the case, and in these cases, find embeddings of Γℐ intoG. We also describe the arithmetic nature of these groups.

Donald I. Cartwright

Papers

Enumeration of the 50 fake projective planes

Groups acting simply transitively on the vertices of a building of type 2, I

Random walks on the affine group of local fields and of homogeneous trees

Random walks on free products, quotients and amalgams

Groups acting simply transitively on the vertices of a building of typeÃ 2, II: The casesq=2 andq=3