SIT measures and Transience

TL;DR

This paper establishes a fundamental equivalence between transience in graphs and the (edge)-SIT property, which involves a probabilistic measure on paths and the finiteness of mutual edges.

Contribution

It proves that transience is exactly characterized by the (edge)-SIT property, linking graph transience to a probabilistic path measure.

Findings

Transience is equivalent to the (edge)-SIT property.

The (edge)-SIT property involves finite mean mutual edges.

The result provides a new characterization of transience in graphs.

Abstract

A graph is called (edge)-SIT if for some probability measure on paths, the number of mutual edges in two independent paths has finite mean. We show that transience is equivalent to the property of (edge)-SIT.