Self-avoiding walk on $\mathbb{Z}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function

TL;DR

This paper demonstrates that the critical fugacity and 2-point function of self-avoiding walks on a class of weighted square lattices are universal, matching those on the hexagonal lattice, regardless of local angle parameters satisfying the Yang-Baxter equation.

Contribution

It proves the universality of critical fugacity and 2-point function for self-avoiding walks with Yang-Baxter weights on square lattices, extending known results from the hexagonal lattice.

Findings

2-point function is invariant under rhombic tiling transformations

Critical fugacity is equal to 1+√2 for all angle parameters

Partition function of self-avoiding bridges tends to zero as strip width increases

Abstract

We consider a self-avoiding walk model (SAW) on the faces of the square lattice $\mathbb{Z}^2$. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$ and satisfy the Yang-Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles. By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles $\theta$ equal to $\frac{\pi}{3}$. For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to $1+\sqrt{2}$. We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of self-avoiding bridges in a strip of the hexagonal lattice tends to 0 as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence.