Discrete uniformizing metrics on distributional limits of sphere packings

TL;DR

This paper proves that distributional limits of sphere packings in -dimensional space have controlled conformal growth and spectral dimension, extending previous results from planar graphs to higher dimensions using unimodular weightings and geometric analysis.

Contribution

It introduces a method to bound the conformal growth and spectral dimension of graph limits from sphere packings in higher dimensions, generalizing prior planar results.

Findings

Conformal growth exponent of graph limits is at most d.

Spectral dimension of graph limits is at most d.

Spectral measure obeys a d-dimensional Weyl law variant.

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.