Note on short time behavior of semigroups associated to selfadjoint operators

TL;DR

This paper investigates the short-time behavior of heat kernels on locally finite graphs, revealing a polynomial decay related to combinatorial distance, contrasting with classical manifold results, and highlighting different governing metrics.

Contribution

It provides a simple observation linking short-time heat kernel behavior on graphs to combinatorial distance, contrasting with classical manifold results, and discusses implications for graphs with unbounded degree.

Findings

Heat kernel on graphs behaves like t^d for short times

Short-time and global behaviors are governed by different metrics

Behavior differs significantly from classical manifold cases

Abstract

We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.