Convergence of the Critical Planar Ising Interfaces to Hypergeometric SLE

2016/10/19 by Hao Wu, Wu, Hao · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1610.06113

23 pages, 5 figures. In v3, we improve the previous results and obtain more general conclusion. In v4, we add more discussion on the relation between hypergeometric SLE and SLE with force points

arxiv created 2017/01/26 · arxiv updated 2017/01/27

Abstract

We consider the planar Ising model in rectangle (Ω; x^L, x^R, y^R, y^L) with alternating boundary condition: \ominus along (x^Lx^R) and (y^Ry^L), ξ^R∈\⊕, free\ along (x^Ry^R), and ξ^L∈\⊕, free\ along (y^Lx^L). We prove that the interface of critical Ising model with these boundary conditions converges to the so-called hypergeometric SLE₃. The method developed in this paper does not require constructing new holomorphic observable and the input is the convergence of the interface with Dobrushin boundary condition. This method could be applied to other lattice models, for instance Loop-Erased Random Walk and level lines of discrete Gaussian Free Field.

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• Hypergeometric SLE: Conformal Markov Characterization and Applications

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