Self-avoiding walk on nonunimodular transitive graphs
We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product T_k × Z^d of a k-regular tree (k ≥ 3) with Z^d, for which these results were previously only known for large k.
Additional Information© 2019 Institute of Mathematical Statistics. Received: 1 September 2017; Revised: 1 June 2018; Published: September 2019. First available in Project Euclid: 22 October 2019. This work was carried out while the author was an intern at Microsoft Research, Redmond. We thank Omer Angel for improving Lemma 3.4 by a factor of 4. We also thank Tyler Helmuth for helpful discussions, and thank Gordon Slade and Hugo Duminil-Copin for comments on an earlier draft, and thank the anonymous referee for several helpful suggestions.
Accepted Version - 1709.10515.pdf