Self-avoiding walk on \mathbb{Z}(2) with Yang-Baxter weights: Universality of critical fugacity and 2-point function
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.
- Publication:
-
Annales de L'Institut Henri Poincare Section (B) Probability and Statistics
- Pub Date:
- November 2020
- DOI:
- 10.1214/19-AIHP1024
- arXiv:
- arXiv:1708.00395
- Bibcode:
- 2020AIHPB..56.2281G
- Keywords:
-
- Mathematics - Probability;
- Mathematical Physics;
- Mathematics - Combinatorics;
- 82B41;
- 82B27
- E-Print:
- 25 pages, 10 figures