[论文解读] Locality of connective constants, II. Cayley graphs

Geoffrey Grimmett, Zhongyang Li|arXiv (Cornell University)|Jan 2, 2015

Geometric and Algebraic Topology参考文献 32被引用 3

一句话总结

本文通过引入与群结构相关的特殊图高函数——群高函数，建立了凯莱图连通常数的局部性。证明了在有限生成且亏格为正、单模结构的群中，此类函数存在，从而通过具有线性增长和周期性差别的调和函数实现局部性结果。

ABSTRACT

Abstract. The connective constant µ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called ‘graph height functions’. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of graph height function termed here a ‘group height function’. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of transitive graphs (and hence Cayley graphs) support graph height functions that are in addition harmonic on the graph. This extends an earlier constructive proof of Grimmett and Li, but subject to an addi-tional condition of unimodularity which is benign in the context of Cayley graphs. It implies the existence of graph height functions for finitely generated solvable groups. The case of non-unimodular graphs may be handled similarly, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an ad-ditional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

研究动机与目标

• 建立凯莱图连通常数的局部性，扩展先前依赖图高函数的研究结果。
• 确定有限生成无限群中群高函数存在的必要与充分条件。
• 证明群高函数是非平凡的调和函数，具有线性增长和周期性差。
• 在单模条件下，将图高函数的存在性扩展至包括凯莱图在内的广泛传递图类。
• 探讨凯莱图上具有线性增长和周期性差别的调和函数的结构意义，超越自避行走理论的范畴。

提出的方法

• 引入群高函数的概念，作为适应凯莱图代数结构的特殊图高函数类型。
• 基于群论性质（如亏格和单模性）建立群高函数存在的必要与充分条件。
• 应用图上调和函数的理论，证明在单模条件下，可构造出图高函数使其为调和函数。
• 利用桥常数将局部图结构与全局连通常数行为联系起来。
• 证明群高函数可产生非平凡调和函数，具有线性增长和周期性差。
• 通过引入单模性作为凯莱图背景下的无害条件，扩展了早期构造性证明。

实验结果

研究问题

• RQ1 在何种群论条件下，凯莱图可容纳用于连通常数局部定理的群高函数？
• RQ2 单模性如何影响凯莱图上图高函数的存在性与调和性？
• RQ3 群高函数与凯莱图上具有线性增长和周期性差别的调和函数之间有何关系？
• RQ4 能否通过群高函数在亏格严格为正的群中建立连通常数的局部性？
• RQ5 在凯莱图上，非平凡调和函数具有线性增长和周期性差时，表现出何种结构性质？

主要发现

• 确立了在有限生成无限群上群高函数存在的必要与充分条件，将其与群亏格和单模性联系起来。
• 由于其单模结构，所有有限生成可解群均存在群高函数。
• 在单模条件下，包括凯莱图在内的大量传递图类支持图高函数，且这些图高函数在图上为调和函数。
• 所得图高函数为非平凡、调和函数，具有线性增长并表现出周期性差。
• 通过群高函数的存在性，建立了亏格严格为正的群的凯莱图上连通常数的局部性。
• 该理论超越自避行走范畴，揭示了凯莱图上具有周期性差性质的内在调和函数结构。