## 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

### Keywords:

self-avoiding walk; polygon joining

Zbl 1080.82541
Full Text:

### References:

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