Extendable self-avoiding walks
Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 1, pp. 61-75.

The connective constant μ of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by μ F , μ B , μ FB , exist and satisfy μ=μ F =μ B =μ FB for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular.

Publié le :
DOI : 10.4171/aihpd/3
Classification : 05-XX, 60-XX, 82-XX
Mots-clés : Self-avoiding walk, connective constant, transitive graph, quasi-transitive graph, unimodular graph, growth, branching number
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     title = {Extendable self-avoiding walks},
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     pages = {61--75},
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     doi = {10.4171/aihpd/3},
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Grimmett, Geoffrey R.; Holroyd, Alexander E.; Peres, Yuval. Extendable self-avoiding walks. Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 1, pp. 61-75. doi : 10.4171/aihpd/3. http://www.numdam.org/articles/10.4171/aihpd/3/

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