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An upper bound on the number of self-avoiding polygons via joining. (English) Zbl 1434.60294

Summary: 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 \(\mu\). N. Madras [J. Stat. Phys. 78, No. 3–4, 681–699 (1995; Zbl 1080.82541)] 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 1080.82541
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References:

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