Nonamenable Liouville Graphs

TL;DR

This paper constructs a class of non-amenable graphs with Liouville property by adding edges to binary trees, showing they admit no non-constant bounded harmonic functions.

Contribution

It introduces a novel method of augmenting binary trees to produce non-amenable Liouville graphs with specific harmonic function properties.

Findings

Constructed non-amenable Liouville graphs from binary trees

Proved these graphs admit no non-constant bounded harmonic functions

Demonstrated the use of uniform expanders in graph augmentation

Abstract

Add to each level of binary tree edges to make the induced graph on the level a uniform expander. It is shown that such a graph admits no non-constant bounded harmonic functions.