Convergence of the Critical Planar Ising Interfaces to Hypergeometric SLE

Abstract

We consider the planar Ising model in rectangle $(\Omega ;x^L,x^R,y^R,y^L)$ with alternating boundary condition: $\ominus$ along $(x^L{x}^R)$ and $(y^R{y}^L)$, ${\xi }^R\in \{\oplus ,\text{free}\}$ along $(x^R{y}^R)$, and ${\xi }^L\in \{\oplus ,\text{free}\}$ along $(y^L{x}^L)$. We prove that the interface of critical Ising model with these boundary conditions converges to the so-called hypergeometric SLE${}_3$. 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.

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

References in corpus (1)

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