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.
