Self-avoiding walk on nonunimodular transitive graphs
Authors
Publication Date
2019-09Journal Title
Annals of Probability
ISSN
0091-1798
Publisher
Institute of Mathematical Statistics
Volume
47
Issue
5
Pages
2801-2829
Type
Article
This Version
VoR
Metadata
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Hutchcroft, T. (2019). Self-avoiding walk on nonunimodular transitive graphs. Annals of Probability, 47 (5), 2801-2829. https://doi.org/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 these results apply in particular to the product $T_k \times \Z^d$ of a $k$-regular tree ($k\geq 3$) with $\Z^d$, for which these results were previously only known for large $k$.
Sponsorship
Microsoft Research
Embargo Lift Date
2100-01-01
Identifiers
External DOI: https://doi.org/10.1214/18-AOP1322
This record's URL: https://www.repository.cam.ac.uk/handle/1810/288163
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