The six-vertex model and Schramm-Loewner evolution

Understand

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 $4<\kappa\leq 12+8\sqrt{2}$.
• For square ice, $\kappa=12$.
• At the &#34;free-fermion point&#34; of the 6-vertex model, $\kappa=8+4\sqrt{3}$.