Enumeration of self avoiding trails on a square lattice using a transfer matrix technique

TL;DR

This paper introduces a transfer matrix algebraic method to enumerate self-avoiding trails on a square lattice, providing new evidence for their universality class and estimating key growth constants.

Contribution

A novel transfer matrix technique is developed for enumerating lattice trails, extending enumeration to 31 steps and analyzing their universality class.

Findings

Enumerated trails up to 31 steps.

Estimated growth constant λ = 2.72062 ± 0.000006.

Supported that trails are in the self-avoiding walk universality class.

Abstract

We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like $A \lambda ^n n^{11 \over 32}$, we find $\lambda = 2.72062 \pm 0.000006$ and $A = 1.272 \pm 0.002$.