Weights of uniform spanning forests on nonunimodular transitive graphs

TL;DR

This paper investigates properties of uniform spanning forests on nonunimodular transitive graphs, revealing that trees are light in the wired case and heavy in the free case, with various structural results depending on the graph class.

Contribution

It provides new insights into the structure of uniform spanning forests on nonunimodular transitive graphs, including conditions for trees being light or heavy and their connectivity properties.

Findings

Trees in wired uniform spanning forests are almost surely light.

In certain graphs, free and wired uniform spanning forests coincide.

Trees in free uniform spanning forests are heavy with branching number greater than one.

Abstract

Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.