Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

TL;DR

This paper establishes Gaussian heat kernel bounds on graphs with unbounded geometry, particularly anti-trees, by linking Sobolev inequalities, volume doubling, and isoperimetric properties, extending previous results to new graph classes.

Contribution

It introduces Gaussian upper bounds for heat kernels on anti-trees with unbounded degrees, utilizing intrinsic metric-based isoperimetric and Sobolev inequalities.

Findings

Gaussian heat kernel bounds on anti-trees with unbounded degrees

Anti-trees are Ahlfors regular and satisfy higher-dimensional isoperimetric inequalities

New correction term in heat kernel bounds accounts for distance to the origin

Abstract

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our previous work [KR22] is affected by a new correction term measuring the distance to the origin. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees are Ahlfors regular and that they satisfy an isoperimetric inequality of a larger dimension.