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October, 2020 Self-avoiding walk on the complete graph
Gordon SLADE
J. Math. Soc. Japan 72(4): 1189-1200 (October, 2020). DOI: 10.2969/jmsj/82588258

Abstract

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.

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Gordon SLADE. "Self-avoiding walk on the complete graph." J. Math. Soc. Japan 72 (4) 1189 - 1200, October, 2020. https://doi.org/10.2969/jmsj/82588258

Information

Received: 7 May 2019; Published: October, 2020
First available in Project Euclid: 17 March 2020

MathSciNet: MR4165929
Digital Object Identifier: 10.2969/jmsj/82588258

Subjects:
Primary: 82B27
Secondary: 33B20, 60K35, 82B41

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 4 • October, 2020
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