The connective constant of a graph is the exponential growth rate of the number of -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 , , , exist and satisfy 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.
DOI : 10.4171/aihpd/3
Mots-clés : Self-avoiding walk, connective constant, transitive graph, quasi-transitive graph, unimodular graph, growth, branching number
@article{AIHPD_2014__1_1_61_0, author = {Grimmett, Geoffrey R. and Holroyd, Alexander E. and Peres, Yuval}, title = {Extendable self-avoiding walks}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {61--75}, volume = {1}, number = {1}, year = {2014}, doi = {10.4171/aihpd/3}, mrnumber = {3166203}, zbl = {1285.05163}, language = {en}, url = {http://www.numdam.org/articles/10.4171/aihpd/3/} }
TY - JOUR AU - Grimmett, Geoffrey R. AU - Holroyd, Alexander E. AU - Peres, Yuval TI - Extendable self-avoiding walks JO - Annales de l’Institut Henri Poincaré D PY - 2014 SP - 61 EP - 75 VL - 1 IS - 1 UR - http://www.numdam.org/articles/10.4171/aihpd/3/ DO - 10.4171/aihpd/3 LA - en ID - AIHPD_2014__1_1_61_0 ER -
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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