Non-stability of Liouville measures under convex combinations

TL;DR

This paper demonstrates that the set of Liouville measures on certain countable groups is not closed under convex combinations, providing new insights into harmonic functions and the structure of these measures.

Contribution

It constructs explicit examples of symmetric probability measures whose convex combinations are non-Liouville, answering a question posed by Kaimanovich.

Findings

Liouville measures are not closed under convex combination.

Convex combinations of certain measures become non-Liouville if at least k measures are involved.

Results apply to specific groups like lamplighter groups and the infinite symmetric group.

Abstract

For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant bounded harmonic function, equivalently, the Poisson boundary is non-trivial) if and only if at least $k$ of them appear in the convex combination. Particularly, our result implies that the set of Liouville measures is not closed under convex combination, which answers a question of Kaimanovich. We also provide a similar result under the additional assumption of finite entropy for those non-hyper-FC-central countable groups with the property that every symmetric, finitely supported probability measure is Liouville. These groups are the only known non-trivial examples of countable groups that admit Liouville measures with finite entropy. Examples include the lamplighter group over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group of finite permutations on $\mathbb{Z}$.