Varadhan Asymptotics for the Heat Kernel on Finite Graphs

2018/01/07 by Stefan Steinerberger, Steinerberger, Stefan · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1801.02183

openalex publication_date 2018/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple, finite graph and let p_t(x,y) denote the heat kernel on G. The purpose of this short note is to show that for t → 0⁺ p_t(x,y) = # \paths of length~d(x,y)~between~x~and~y\ \fract^d(x,y)d(x,y)! + O(t^d(x,y)+1), where d(x,y) is the usual Graph distance. This is the discrete analogue of the classical Varadhan asymptotic for the heat kernel on manifolds and refines a result of Keller, Lenz, Münch, Schmidt and Telcs. The asymptotic behavior encapsulates additional geometric information: if the Graph is bipartite, then the next term in the expansion is negative.

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