ADE and SLE

TL;DR

This paper explores the probability laws of domain walls in ADE lattice models and their relation to SLE curves, revealing similarities in simply connected domains and differences in multiply-connected domains.

Contribution

It establishes a conjectural link between ADE lattice model domain walls and SLE curves, enhancing understanding of their scaling limits and differences in complex domains.

Findings

Probability laws match in simply connected domains

Laws differ in multiply-connected domains

Provides insights into multiple curve scaling limits

Abstract

We point out that the probability law of a single domain wall separating clusters in ADE lattice models in a simply connected domain is identical to that of corresponding chordal curves in the lattice O(n) and Q-state Potts models, for suitable n or Q. They are conjectured to be described in the scaling limit by chordal SLE(kappa) with kappa rational and >2. However in a multiply-connected domain the laws can differ from those for the corresponding O(n) or Potts model. The correspondence also sheds light on the scaling limit of multiple curves.