Locality of connective constants

2014/11/29 by Grimmett, Geoffrey R., Li, Zhongyang
#05C30 #60K35 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1412.0150

Abstract

The connective constant μ(G) of a quasi-transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. We prove a locality theorem 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 (and a further condition is satisfied). The proof exploits a generalized bridge decomposition of self-avoiding walks, which is valid subject to the assumption that the underlying graph is quasi-transitive and possesses a so-called unimodular graph height function.

Related

• Self-avoiding walks and connective constants
• Strict inequalities for connective constants of transitive graphs
• A general bridge theorem for self-avoiding walks
• Counting self-avoiding walks
• Self-avoiding walks and amenability