Note on uniformly transient graphs

TL;DR

This paper investigates uniformly transient graphs, characterizing their properties through isoperimetric inequalities, boundary equalities, and solutions to the Dirichlet problem, revealing spectral and compactification features.

Contribution

It provides new characterizations of uniform transience, analyzes boundary and spectral properties, and includes examples like hyperbolic Cayley graphs and high-dimensional lattices.

Findings

Uniformly transient graphs satisfy specific isoperimetric inequalities.

Dirichlet problems on these graphs have unique solutions.

Semigroups and resolvents are ultracontractive and trace class under certain conditions.

Abstract

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have $\ell^p$ independent pure point spectra (for $1 \leq p \leq \infty$). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compacitifcation and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.