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

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 concides 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.