Year: 2014
Type: preprint
Abstract: For a transitive infinite connected graph $G$, let $μ(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of other generators. Assume $G\in\mathbf{\cal G}$ is a Cayley graph of a finitely presented group, and Cayley graph sequence $\{G_n\}_{n=1}^{\infty}\subset \mathbf{\cal G}$ converges locally to $G.$ Then $μ(G_n)$ converges to $μ(G)$ as $n\rightarrow\infty.$ This confirms partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and randomness.} Lect. Notes Math. {\bf 2100}. Springer.] that connective constant is continuous with respect to local convergence of infinite transitive connected graphs.
Source: arXiv (Cornell University)
Authors: Song, He, Xiang, Kai-Nan, Zhu, Song-Chao-Hao
Institution: Nankai University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2296946504