The six-vertex model and Schramm-Loewner evolution

TL;DR

This paper links the six-vertex model, including square ice, to Schramm-Loewner evolution (SLE) curves, proposing that their scaling limits are space-filling SLE curves with specific ppa values outside the classical range.

Contribution

It introduces a novel association between 6-vertex model configurations and space-filling SLE curves, extending the understanding of conformal invariance in these models.

Findings

Square ice's scaling limit is a space-filling SLE with ppa=12.

At the free-fermion point, ppa=8+43.

The ppa values are outside the classical 2 to 8 range.

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 $4<\kappa\leq 12+8\sqrt{2}$. 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 $2\le \kappa\le 8$.