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 six-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE[subscript κ], Schramm-Loewner evolution with parameter κ, where 4 < κ ≤ 12 + 8 √2. For square ice, κ = 12. At the “free-fermion point” of the six-vertex model, κ = 8 + 4√3. These unusual values lie outside the classical interval 2≤κ≤8.