Discrete Uniformizing Metrics on Distributional Limits of Sphere Packings
Discrete Uniformizing Metrics on Distributional Limits of Sphere Packings
复制标题
球堆积分布极限的离散均匀度量
DOI：
10.1007/s00039-018-0442-2
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发表时间：
2017
影响因子：
2.2
通讯作者：
James R. Lee
中科院分区：
文献类型：
--
作者：
James R. Lee
Suppose that {$${{G_n}}$$Gn} is a sequence of finite graphs such that each $${G_n}$$Gn is the tangency graph of a sphere packing in $${\mathbb{R}^d}$$Rd . Let $${\rho_n}$$ρn be a uniformly random vertex of $${G_n}$$Gn and suppose that $${(G,\rho)}$$(G,ρ) is the distributional limit of {$${{(G_n,\rho_n)}}$$(Gn,ρn)} in the sense of Benjamini and Schramm. Then the conformal growth exponent of $${(G,\rho)}$$(G,ρ) is at most d. In other words, there exists a unimodular “unit volume" weighting of the graph metric on $${(G,\rho)}$$(G,ρ) such that the volume growth of balls in the weighted path metric is bounded by a polynomial of degree d. This assertion generalizes to limits of graphs that can be “quasi-packed” in an Ahlfors d-regular metric measure space. It implies that, under moment conditions on the degree of the root ρ, the almost sure spectral dimension of G is at most d. This fact was known previously only for graphs packed in $${\mathbb{R}^2}$$R2 (planar graphs), and the case d > 2 eluded approaches based on extremal length. In the process of bounding the spectral dimension, we establish that the spectral measure of $${(G,\rho)}$$(G,ρ) is dominated by a variant of the d-dimensional Weyl law.
Suppose that {$${{G_n}}$$Gn} is a sequence of finite graphs such that each $${G_n}$$Gn is the tangency graph of a sphere packing in $${\mathbb{R}^d}$$Rd . Let $${\rho_n}$$ρn be a uniformly random vertex of $${G_n}$$Gn and suppose that $${(G,\rho)}$$(G,ρ) is the distributional limit of {$${{(G_n,\rho_n)}}$$(Gn,ρn)} in the sense of Benjamini and Schramm. Then the conformal growth exponent of $${(G,\rho)}$$(G,ρ) is at most d. In other words, there exists a unimodular “unit volume" weighting of the graph metric on $${(G,\rho)}$$(G,ρ) such that the volume growth of balls in the weighted path metric is bounded by a polynomial of degree d. This assertion generalizes to limits of graphs that can be “quasi-packed” in an Ahlfors d-regular metric measure space. It implies that, under moment conditions on the degree of the root ρ, the almost sure spectral dimension of G is at most d. This fact was known previously only for graphs packed in $${\mathbb{R}^2}$$R2 (planar graphs), and the case d > 2 eluded approaches based on extremal length. In the process of bounding the spectral dimension, we establish that the spectral measure of $${(G,\rho)}$$(G,ρ) is dominated by a variant of the d-dimensional Weyl law.