Stochastic homogenization of horospheric tree products

TL;DR

This paper develops a method to construct invariant measures for horospheric products of trees, demonstrating their amenability through boundary measure systems, advancing understanding of their structural properties.

Contribution

It introduces a novel construction of invariant measures for horospheric tree products using conformal boundary measures, establishing their amenability.

Findings

Invariant measures for horospheric products constructed

Proves amenability of these tree products

Uses conformal boundary measures in the construction

Abstract

We construct measures invariant with respect to equivalence relations which are graphed by horospheric products of trees. The construction is based on using conformal systems of boundary measures on treed equivalence relations. The existence of such an invariant measure allows us to establish amenability of horospheric products of random trees.