Random Walks, Boundaries and Spectra

Frontmatter

An Inaccessible Graph

Abstract

An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph.

M. J. Dunwoody

A Local Limit Theorem for Random Walks on the Chambers of Ã2 Buildings

Abstract

In this paper we outline an approach for analysing random walks on the chambers of buildings. The types of walks that we consider are those which are well adapted to the structure of the building: Namely walks with transition probabilities p(c, d) depending only on the Weyl distance d(c, d). We carry through the computations for thick locally finite affine buildings of type Ã₂ to prove a local limit theorem for these buildings. The technique centres around the representation theory of the associated Hecke algebra. This representation theory is particularly well developed for affine Hecke algebras, with elegant harmonic analysis developed by Opdam ([28], [29]). We give an introductory account of this theory in the second half of this paper.

James Parkinson, Bruno Schapira

On Continuity of Range, Entropy and Drift for Random Walks on Groups

Abstract

We study continuity of various characteristics of random walks on groups with respect to strong convergence of measures.

Anna Erschler

Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces

Abstract

Let G be a locally compact group, E a homogeneous space of G. We discuss the relations between recurrence of a random walk on G or E, ergodicity of the corresponding transformations and polynomial growth of G or E. We consider the special case of linear groups over local fields.

Yves Guivarc’h, C. R. E. Raja

Ergodic Theorems for Homogeneous Dilations

Abstract

In this paper we prove a general ergodic theorem for ergodic and measure-preserving actions of \(\mathbb{R}^d\) on standard Borel spaces. In particular, we cover R.L. Jones’ ergodic theorem on spheres. Our main theorem is concerned with almost everywhere convergence of ergodic averages with respect to homogeneous dilations of certain Rajchman measures on \(\mathbb{R}^d\). Applications include averages over smooth submanifolds and polynomial curves.1

Michael Björklund

Boundaries from Inhomogeneous Bernoulli Trials

Abstract

The boundary problem is considered for inhomogeneous increasing random walks on the square lattice \(\mathbb{Z}^2_+\) with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.

Alexander Gnedin

Resistance Boundaries of Infinite Networks

Abstract

A resistance network is a connected graph (G, c). The conductance function \(c_{xy}\) weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form ε produces a Hilbert space structure h _ε on the space of functions of finite energy.

The relationship between the natural Dirichlet form \( \rm{\varepsilon}\)and the discrete Laplace operator \( \rm{\Delta}\) on a finite network is given by \( {{\varepsilon(u,\,v)}}\, = \, {\langle{u},\,\Delta {v}\rangle}2, \) where the latter is the usual l ² inner product. We describe a reproducing kernel v _x for ε which allows one to extend the discrete Gauss-Green identity to infinite networks:

\( {{\varepsilon(u,\,v)}}\, = \, {\sum}_{G}\, {u\Delta v}+{\sum}_{bd\,\,G} \,\,{u}\,\frac{\partial {v}} {\partial {n}},\,\, \)

where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula yields a boundary sum representation for the harmonic functions of finite energy.

Techniques from stochastic integration allow one to make the boundary bdG precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel’fand triple \( {S}\,\, \subseteq \, \, {H_\varepsilon}\,\, \subseteq\,\,{{S}^\prime} {\rm{and}\, {gives}\, {a}\, {probability}\, {measure}\,\mathbb{P}}\, {\rm{and}\, {an}\, {isometric}\, {embedding}\,{of}\,{H_\varepsilon}\,\,{into}}\,\,{{S}^\prime},\,\mathbb{P},\) and yields a concrete representation of the boundary as a set of linear functionals on S.

Palle E. T. Jorgensen, Erin P. J. Pearse

Brownian Motion and Negative Curvature

Abstract

It is well known that on a Riemannian manifold, there is a deep interplay between geometry, harmonic function theory, and the long-term behaviour of Brownian motion. Negative curvature amplifies the tendency of Brownian motion to exit compact sets and, if topologically possible, to wander out to infinity. On the other hand, non-trivial asymptotic properties of Brownian paths for large time correspond with non-trivial bounded harmonic functions on the manifold. We describe parts of this interplay in the case of negatively curved simply connected Riemannian manifolds. Recent results are related to known properties and old conjectures.

Marc Arnaudon, Anton Thalmaier

Stochastically Incomplete Manifolds and Graphs

Abstract

We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth.

Radosław Krzysztof Wojciechowski

Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

Abstract

In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. In particular we prove a version of the Allegretto-Piepenbrink theorem which says that positive (super)-solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol’s theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum.

Sebastian Haeseler, Matthias Keller

A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schrödinger Operators

Abstract

We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schrödinger operators developed in [9–12]. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems.

Richard Froese, David Hasler, Wolfgang Spitzer

Some Spectral and Geometric Aspects of Countable Groups

Abstract

We discuss the relationship between the isospectral profile and the spectral distribution of a Laplace operator on a countable group. In the case of locally finite countable groups, we emphasize the relevance of the metric associated to a natural Markov operator: it is an ultra-metric whose balls are optimal sets for the isospectral profile.

Alexander Bendikov, Barbara Bobikau, Christophe Pittet

Percolation Hamiltonians

Abstract

There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators.

Peter Müller, Peter Stollmann

Survey of Scalings for the Largest Connected Component in Inhomogeneous Random Graphs

Abstract

We review some recent results on the exact asymptotics of the components in the inhomogeneous random graph models of rank 1.We discuss the relevance of these results to the analysis of random walk on random graphs.

Tatyana S. Turova

Partition Functions of the Ising Model on Some Self-similar Schreier Graphs

Abstract

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk’s group of intermediate growth; the iterated monodromy group of the complex polynomial z ²-1 known as the “Basilica group”; and the Hanoi Towers group H ⁽³⁾ closely related to the Sierpinski gasket.

Daniele D’Angeli, Alfredo Donno, Tatiana Nagnibeda

Aspects of Toeplitz Determinants

Abstract

We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painlevé V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results.

Igor Krasovsky