Equivalence of zero entropy and the Liouville property for stationary random graphs

TL;DR

This paper establishes a connection between zero entropy and the Liouville property in stationary random graphs, showing that positive entropy implies a rich structure of harmonic functions under certain conditions.

Contribution

It proves that stationary random graphs with positive entropy and growth conditions have an infinite-dimensional space of bounded harmonic functions, linking entropy to harmonic analysis.

Findings

Positive entropy implies infinite-dimensional harmonic functions.

Applications demonstrated for random planar triangulations and Delaunay graphs.

Growth conditions are crucial for the main results.

Abstract

We prove that any stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to random infinite planar triangulations and Delaunay graphs are given.