Crossing Probabilities of Multiple Ising Interfaces

We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions given by the pure partition functions of multiple SLE(\kappa) with \kappa=3. In particular, this identifies the scaling limits with ratios of specific correlation functions of conformal field theory.Comment: 30 pages, 1 figure; v3: removed an appendix & other minor improvement

On the Uniqueness of Global Multiple SLEs

This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE process 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 and FK-Ising models.Comment: 30 pages, 4 figure

Large deviations of multichordal SLE0+_{0+}, real rational functions, and zeta-regularized determinants of Laplacians

We prove a strong large deviation principle (LDP) for multiple chordal SLE$_{0+}$ curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, that satisfies PDEs arising as a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equations of level two in conformal field theory with central charge $c \to -\infty$. Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the $\kappa \to 0+$ limit of the multiple SLE$_\kappa$. As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition by a M\"obius transformation.Comment: 66 pages, 4 figures. Mostly minor edits, updates, and clarifications: addressed subtleties in the definition of curve spaces and hulls. To appear in JEM