Off-diagonal upper heat kernel bounds on graphs with unbounded geometry

TL;DR

This paper investigates off-diagonal Gaussian upper bounds for heat kernels on graphs with unbounded geometry, extending existing results and exploring optimal metrics for these bounds.

Contribution

It introduces a version of Grigor'yan's two-point method involving intrinsic metrics and discusses the optimality of Gaussian bounds on graphs with unbounded geometry.

Findings

Unified framework for heat kernel bounds on unbounded graphs

Relation of Davies' universal Gaussian to intrinsic metrics

Characterization of bounds for anti-trees and unbounded Laplacians

Abstract

Results regarding off-diagonal Gaussian upper heat kernel bounds on discrete weighted graphs with possibly unbounded geometry are summarized and related. After reviewing uniform upper heat kernel bounds obtained by Carlen, Kusuoka, and Stroock, the universal Gaussian term on graphs found by Davies is addressed and related to corresponding results in terms of intrinsic metrics. Then we present a version of Grigor'yan's two-point method with Gaussian term involving an intrinsic metric. A discussion of upper heat kernel bounds for graph Laplacians with possibly unbounded but integrable weights on bounded combinatorial graphs preceeds the presentation of compatible bounds for anti-trees, an example of combinatorial graph with unbounded Laplacian. Characterizations of localized heat kernel bounds in terms of intrinsic metrics and universal Gaussian are reconsidered. Finally, the problem of optimality of the Gaussian term is discussed by relating Davies' optimal metric with the supremum over all intrinsic metrics.