Quasisymmetric uniformization and heat kernel estimates

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.

38 pages, 2 figures; several typos and minor mistakes were corrected; to appear in Transactions of the AMS