# Connective constants and height functions for Cayley graphs

 dc.creator Grimmett, Geoffrey Richard dc.creator Li, Z dc.date.accessioned 2016-12-24 dc.date.accessioned 2018-11-24T23:27:04Z dc.date.available 2017-03-14T15:55:15Z dc.date.available 2018-11-24T23:27:04Z dc.date.issued 2017-03-31 dc.identifier https://www.repository.cam.ac.uk/handle/1810/263104 dc.identifier.uri http://repository.aust.edu.ng/xmlui/handle/123456789/3939 dc.description.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 “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “$\textit{group}$ 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 Cayley graphs support unimodular graph height functions, that are in addition $\textit{harmonic}$ on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, 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 additional 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. dc.language en dc.publisher American Mathematical Society dc.publisher Transactions of the American Mathematical Society dc.rights http://creativecommons.org/licenses/by-nc-nd/4.0/ dc.rights Attribution-NonCommercial-NoDerivatives 4.0 International dc.subject self-avoiding walk dc.subject connective constant dc.subject vertex-transitive graph dc.subject quasi-transitive graph dc.subject bridge decomposition dc.subject Cayley graph dc.subject Higman group dc.subject graph height function dc.subject group height function dc.subject indicability dc.subject harmonic function dc.subject solvable group dc.subject unimodularity dc.title Connective constants and height functions for Cayley graphs dc.type Article
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