Abstract
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 $n$th 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 $T_{k}\times\mathbb{Z}^{d}$ of a $k$-regular tree ($k\geq3$) with $\mathbb{Z}^{d}$, for which these results were previously only known for large $k$.
Citation
Tom Hutchcroft. "Self-avoiding walk on nonunimodular transitive graphs." Ann. Probab. 47 (5) 2801 - 2829, September 2019. https://doi.org/10.1214/18-AOP1322
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