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.
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Geoffrey R. Grimmett, Alexander E. Holroyd, Yuval Peres, Extendable self-avoiding walks. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 1, pp. 61–75DOI 10.4171/AIHPD/3