Smirnov's observable for free boundary conditions, interfaces and crossing probabilities

Abstract

We prove convergence results for variants of Smirnov's fermionic observable in the critical Ising model in presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytölä on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and generalization of this result to arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in FK-Ising model with arbitrary number of alternating wired/free boundary arcs. We also deduce a new crossing formula for the spin Ising model.

arXiv comment27 pages, 2 figures. Several proofs expanded, a small gap fixed in the proof of Theorem 2.6, typos corrected

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• A formula for crossing probabilities of critical systems inside polygons
• Towards a conformal field theory for Schramm-Loewner evolutions
• Crossing Probabilities of Multiple Ising Interfaces
• UST branches, martingales, and multiple SLE(2)
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• Correlations of primary fields in the critical Ising model
• Convergence of the Critical Planar Ising Interfaces to Hypergeometric SLE
• Connection Probabilities of Multiple FK-Ising Interfaces
• Hypergeometric SLE with $\kappa =8$: Convergence of UST and LERW in Topological Rectangles
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• Multiple Ising interfaces in annulus and $2N$-sided radial SLE
• Planar UST Branches and $c=-2$ Degenerate Boundary Correlations