Selfavoiding walk on $\mathbb{Z}^{2}$ with Yang–Baxter weights : Universality of critical fugacity and 2point function

Glazman, Alexander
Tel Aviv university, School of mathematical sciences, Tel Aviv, Israel.

Manolescu, Ioan
Université de Fribourg, 23 Chemin du Musée, CH1700 Fribourg, Switzerland.
Published in:
 Annales de l’Institut Henri Poincaré, Probabilités et Statistiques.  2020, vol. 56, no. 4, p. 2281–2300
English
We consider a selfavoiding walk model (SAW) on the faces of the square lattice Z2. 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 θ∈[π3,2π3] 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 2point function of the walk in the halfplane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic concides with that of the selfavoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles θ equal to π3.For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to 1+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 selfavoiding 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.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/309306
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