Gaussian upper bounds for heat kernels on graphs with unbounded geometry

TL;DR

This paper establishes sharp Gaussian upper bounds for heat kernels on graphs with unbounded geometry, extending known results to more general graph Laplacians and including new polynomial correction terms.

Contribution

It provides the first large-time Gaussian upper bounds for heat kernels on graphs with unbounded geometry, incorporating a new polynomial correction term and handling unbounded Laplacians.

Findings

Gaussian upper bounds hold under Sobolev and volume doubling conditions

Results are new even for normalized Laplacians on graphs

Error terms reflect unbounded vertex degree and measure degeneracy

Abstract

We prove large-time Gaussian upper bounds for continuous-time heat kernels of Laplacians on graphs with unbounded geometry. Our estimates hold for centers of large balls satisfying a Sobolev inequality and volume doubling. Distances are measured with respect to an intrinsic metric with finite distance balls and finite jump size. The Gaussian decay is given by Davies' function which is natural and sharp in the graph setting. Furthermore, we find a new polynomial correction term which does not blow up at zero. Although our main focus is unbounded Laplacians, the results are new even for the normalized Laplacian. In the case of unbounded vertex degree or degenerating measure, the estimates are affected by new error terms reflecting the unboundedness of the geometry.