Locality of connective constants |
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Authors: | Geoffrey R Grimmett Zhongyang Li |
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Institution: | 1. Statistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WB, UK;2. Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA |
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Abstract: | The connective constant of a quasi-transitive graph 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 is based on 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. |
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Keywords: | Self-avoiding walk Connective constant Vertex-transitive graph Quasi-transitive graph Bridge decomposition Cayley graph Unimodularity |
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