Hypergeometric SLE: Conformal Markov Characterization and Applications

Abstract: This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves---Hypergeometric SLE---denoted by hSLE$_{\kappa}(\nu)$ for $\kappa\in (0,4]$ and $\nu<\kappa-6$. The proof relies crucially on Dub\'edat's commutation relation [Dub07] and a uniqueness result proved in [MS16b]. The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or proved to be conformal invariant) in topological rectangles with alternating boundary conditions. We also prove various properties of hSLE: continuity, reversibility, target-independence, and conditional law characterization. As by-products, we give two applications of these properties. The first one is about the critical Ising interfaces. We prove the convergence of the Ising interface in rectangles with alternating boundary conditions. This result was first proved by Izyurov in [Izy15], but our proof is new which is based on the properties of hSLE. The second application is the existence of the so-called pure partition functions of multiple SLEs. Such existence was proved for $\kappa\in (0,8)\setminus \mathbb{Q}$ in [KP16], and it was later proved for $\kappa\in (0,4]$ in [PW17]. We give a new proof of the existence for $\kappa\in (0,6]$ using the properties of hSLE.