Self-avoiding polygons on the square lattice

TL;DR

This paper introduces an improved algorithm for enumerating self-avoiding polygons on the square lattice, providing precise estimates of key constants and analyzing their asymptotic behavior with high accuracy.

Contribution

The authors developed a more efficient enumeration algorithm enabling analysis up to perimeter length 90, yielding highly accurate estimates of the connective constant and critical exponent.

Findings

Estimated connective constant μ = 2.63815852927(1)

Critical exponent α ≈ 0.5 with high precision

No evidence of non-analytic correction-to-scaling exponent

Abstract

We have developed an improved algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the connective constant $\mu =2.63815852927(1)$ (biased) and the critical exponent $\alpha = 0.5000005(10)$ (unbiased). The critical point is indistinguishable from a root of the polynomial $581x^4 + 7x^2 - 13 =0.$ An asymptotic expansion for the coefficients is given for all $n.$ There is strong evidence for the absence of any non-analytic correction-to-scaling exponent.