Extendable self-avoiding walks

Grimmett, Geoffrey R.; Holroyd, Alexander E.; Peres, Yuval

doi:10.4171/aihpd/3

Grimmett, Geoffrey R. ; Holroyd, Alexander E. ; Peres, Yuval

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

Résumé

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 $μ^{F}$ , $μ^{B}$ , $μ^{F B}$ , exist and satisfy $μ = μ^{F} = μ^{B} = μ^{F B}$ 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 : 2014-02-04

MR Zbl

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},
     volume = {1},
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     doi = {10.4171/aihpd/3},
     mrnumber = {3166203},
     zbl = {1285.05163},
     language = {en},
     url = {http://archive.numdam.org/articles/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://archive.numdam.org/articles/10.4171/aihpd/3/

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