Discrete uniformizing metrics on distributional limits of sphere packings

Abstract

Suppose that $\{G_n\}$ is a sequence of finite graphs such that each $G_n$ is the tangency graph of a sphere packing in ${\mathbb{R}}^d$. Let ${\rho }_n$ be a uniformly random vertex of $G_n$ and suppose that $(G,\rho )$ is the distributional limit of $\{(G_n,{\rho }_n)\}$ in the sense of Benjamini and Schramm. Then the conformal growth exponent of $(G,\rho )$ is at most $d$. In other words, there exists a unimodular "unit volume" weighting of the graph metric on $(G,\rho )$ such that the volume growth of balls in the weighted path metric is bounded by a polynomial of degree $d$. This generalizes to limits of graphs that can be "coarsely" packed in an Ahlfors $d$-regular metric measure space. Using our previous work, this implies that, under moment conditions on the degree of the root $\rho$,the almost sure spectral dimension of $G$ is at most $d$. This fact was known previously only for graphs packed in ${\mathbb{R}}^2$ (planar graphs), and the case of $d>2$ eluded approaches based on extremal length. In the process of bounding the spectral dimension, we establish that the spectral measure of $(G,\rho )$ is dominated by a variant of the $d$-dimensional Weyl law.