The six-vertex model and Schramm-Loewner evolution

Abstract

Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE${}_{\kappa }$, Schramm--Loewner evolution with parameter $\kappa$, where $4<\kappa \le 12+8\sqrt{2}$. For square ice, $\kappa =12$. At the "free-fermion point" of the 6-vertex model, $\kappa =8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le \kappa \le 8$.

5 pages, 5 figures

References in corpus (1)

• Square ice in graphene nanocapillaries

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