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September 2019 Self-avoiding walk on nonunimodular transitive graphs
Tom Hutchcroft
Ann. Probab. 47(5): 2801-2829 (September 2019). DOI: 10.1214/18-AOP1322

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

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Tom Hutchcroft. "Self-avoiding walk on nonunimodular transitive graphs." Ann. Probab. 47 (5) 2801 - 2829, September 2019. https://doi.org/10.1214/18-AOP1322

Information

Received: 1 September 2017; Revised: 1 June 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145303
MathSciNet: MR4021237
Digital Object Identifier: 10.1214/18-AOP1322

Subjects:
Primary: 60K35
Secondary: 05A99, 60B99

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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