Upper bounds for transition probabilities on graphs and isoperimetric inequalities

TL;DR

This paper establishes necessary and sufficient conditions for heat kernel upper bounds of random walks on weighted graphs, linking these bounds to isoperimetric inequalities, thus providing a comprehensive understanding of transition probabilities.

Contribution

It introduces a set of equivalent conditions connecting heat kernel bounds with isoperimetric inequalities on weighted graphs, advancing theoretical understanding.

Findings

Characterization of heat kernel upper bounds via isoperimetric inequalities

Equivalence of various conditions for transition probability estimates

Framework applicable to weighted graphs and random walks

Abstract

In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities.