On the Uniqueness of 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.

34 pages, 8 figures; v2 (final): accommodated referee's comments, rearranged section 4, clarified proofs, updated references; accepted for publication in Ann. Probab

References in corpus (3)

• Multiple SLE type scaling limits: from local to global
• Limits of conformal images and conformal images of limits for planar random curves
• On the convergence of FK-Ising Percolation to SLE$(16/3,16/3-6)$

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• Determinantal Point Processes, Stochastic Log-Gases, and Beyond