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January 2018 An upper bound on the number of self-avoiding polygons via joining
Alan Hammond
Ann. Probab. 46(1): 175-206 (January 2018). DOI: 10.1214/17-AOP1182


For $d\geq2$ and $n\in\mathbb{N}$ even, let $p_{n}=p_{n}(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^{d}$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n\in2\mathbb{N}}p_{n}^{1/n}\in(0,\infty)$ is called the connective constant and denoted by $\mu$. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that $p_{n}\mu^{-n}\leq Cn^{-1/2}$ in dimension $d=2$. Here, we establish that $p_{n}\mu^{-n}\leq n^{-3/2+o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d\geq3$, an upper bound of $n^{-2+d^{-1}+o(1)}$ holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.


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Alan Hammond. "An upper bound on the number of self-avoiding polygons via joining." Ann. Probab. 46 (1) 175 - 206, January 2018.


Received: 1 February 2017; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865121
MathSciNet: MR3758729
Digital Object Identifier: 10.1214/17-AOP1182

Primary: 60K35
Secondary: 60D05

Keywords: polygon joining , Self-avoiding walk

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 1 • January 2018
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