Department of Mathematics, Brown University, 1 Prospect Street, Providence, Rhode Island 02912, USA.
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom.
Phys Rev E. 2017 May;95(5-1):052146. doi: 10.1103/PhysRevE.95.052146. Epub 2017 May 30.
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_{κ}, Schramm-Loewner evolution with parameter κ, where 4<κ≤12+8sqrt[2]. For square ice, κ=12. At the "free-fermion point" of the six-vertex model, κ=8+4sqrt[3]. These unusual values lie outside the classical interval 2≤κ≤8.
方冰是二维冰的统计力学模型，被广泛认为具有共形不变的标度极限。我们将皮亚诺（空间填充）曲线与方冰构型相关联，更一般地与所谓的六顶点模型构型相关联，并认为它的标度极限是随机分形曲线 SLEκ的空间填充版本，其中κ=4<12+8sqrt[2]。对于方冰，κ=12。在六顶点模型的“自由费米子点”，κ=8+4sqrt[3]。这些不寻常的值位于经典区间 2≤κ≤8 之外。
