Journal article

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

  • Glazman, Alexander Tel Aviv university, School of mathematical sciences, Tel Aviv, Israel.
  • Manolescu, Ioan Université de Fribourg, 23 Chemin du Musée, CH-1700 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 self-avoiding 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 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 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 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.
Faculté des sciences et de médecine
Département de Mathématiques
  • English
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