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

Let G be a locally compact (second countable) group and π:G → U(ℋ) a unitary representation of G on the (separable) Hilbert space ℋ.Of course,ℋ may or may not contain non-trivial vectors invariant under π(G). We wish to define a notion of π almost containing invariant vectors.

Kazhdan’s Property (T)

• completeness of representation in time and frequency domain • some key properties of Fourier transforms • filtering: convolution and deconvolution • practical limitations to deconvolution with applications to microscopy Fourier Analysis

Fourier 해석 입문

Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.

How Groups Grow

We present a survey of results related to the Milnor's problem on group growth We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article is devoted to the intermediate (between polynomial and exponential) growth case A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) is considered, and a number of open problems is suggested

Milnor's Problem on the Growth of Groups and its Consequences

We give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the $L_p$-compression exponent of a group and its wreath product with the cyclic group for $p$ in $[1,2]$.

Speed of random walks, isoperimetry and compression of finitely generated groups

A result of amenability of some automorphism groups of a spherically homogeneous rooted tree of bounded valency is given. It is used to construct uncountably many amenable groups of non-uniform exponential growth. Their Cayley graphs can be made arbitrary close to that of some groups of intermediate growth. Yet those groups are not in the class SG of subexponentially amenable groups.

/pdf/amenability-and-non-uniform-growth-of-some-directed-1unldtffjn.pdf

Amenability and non-uniform growth of some directed automorphism groups of a rooted tree

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq\alpha\leq\beta\leq1$, there is a group $\Gamma $ with measure $\mu $ equidistributed on a finite generating set such that \[\liminf\frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\alpha ,\qquad\limsup \frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\beta .\] The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_{n}$ on such groups are also evaluated, providing an example of group with return probability satisfying \[\liminf\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=\frac{1}{3},\qquad\limsup\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=1\] and drift satisfying \[\liminf\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=\frac{1}{2},\qquad\limsup\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=1.\]

/pdf/behaviors-of-entropy-on-finitely-generated-groups-18wo2d7u0w.pdf

Behaviors of entropy on finitely generated groups

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.

Jérémie Brieussel

Papers

Speed of random walks, isoperimetry and compression of finitely generated groups

Amenability and non-uniform growth of some directed automorphism groups of a rooted tree

Speed of random walks, isoperimetry and compression of finitely generated groups

Behaviors of entropy on finitely generated groups

Random walks on the discrete affine group