Graphs at infinity: Liouville theorems, Recurrence and Characterization of Dirichlet forms

TL;DR

This survey explores recent advances in the analysis of graphs and their Laplacians, emphasizing large-scale behavior, Liouville theorems, recurrence, and Dirichlet form characterizations.

Contribution

It compiles and discusses recent results connecting graph behavior at infinity with properties of Laplacians and Dirichlet forms.

Findings

Liouville theorems characterize harmonic functions on graphs.

Recurrence relates to the large-scale structure of graphs.

Boundary terms are key in characterizing Dirichlet forms.

Abstract

We survey recent results on graphs and their Laplacians related to the behavior of the graph at large. In particular, we focus on Liouville theorems, recurrence and characterizations of Dirichlet forms via boundary terms.