Discrete uniformizing metrics on distributional limits of sphere packings

Understand

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 $d$.