On the Uniqueness of Global Multiple SLEs

TL;DR

This paper proves the uniqueness of global multiple SLEs, characterizing their properties and demonstrating their convergence as scaling limits of interfaces in critical lattice models like Ising and percolation.

Contribution

It establishes the minimal conditions for the uniqueness of global multiple SLEs and confirms their role as scaling limits in critical lattice models.

Findings

Proves the existence and uniqueness of global multiple SLE measures.

Shows convergence of multiple interfaces in critical Ising, FK-Ising, and percolation models.

Provides a minimal characterization for global multiple SLEs.

Abstract

This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising, FK-Ising, and percolation models.