Quasisymmetric uniformization and heat kernel estimates

TL;DR

This paper establishes a deep connection between geometric embeddings of planar graphs and probabilistic heat kernel estimates, providing new insights into fractals and graph theory.

Contribution

It demonstrates that quasisymmetric circle packing embeddings correspond exactly to sub-Gaussian heat kernel estimates with spectral dimension two for certain planar graphs.

Findings

Characterization of quasisymmetric embeddings via heat kernel estimates

Identification of new classes of graphs and fractals satisfying sub-Gaussian estimates

Establishment of links between geometric, probabilistic, and analytic properties of graphs

Abstract

We show that the circle packing embedding in $\mathbb{R}^2$ of a one-ended, planar triangulation with polynomial growth is quasisymmetric if and only if the simple random walk on the graph satisfies sub-Gaussian heat kernel estimate with spectral dimension two. Our main results provide a new family of graphs and fractals that satisfy sub-Gaussian estimates and Harnack inequalities.