Heat kernel asymptotics for scaling limits of isoradial graphs

TL;DR

This paper investigates the asymptotic behavior of the discrete heat kernel on isoradial graphs as both time and edge lengths approach zero, revealing Gaussian and Poissonian regimes depending on their ratio.

Contribution

It introduces a detailed analysis of heat kernel asymptotics on isoradial graphs, identifying distinct regimes based on the scaling of time and edge lengths.

Findings

Gaussian regime resembles Euclidean space heat kernel behavior

Poissonian regime resembles graph heat kernel behavior

Different asymptotic regimes depend on the ratio of time to edge length

Abstract

We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.