Hypergeometric SLE: Conformal Markov Characterization and Applications

TL;DR

This paper classifies a family of conformally invariant random curves called Hypergeometric SLE, proves their key properties, and applies these results to critical lattice models and multiple SLE partition functions.

Contribution

It provides the complete classification of hypergeometric SLE curves and establishes their properties, connecting them to critical lattice models and multiple SLEs.

Findings

Hypergeometric SLE is the unique scaling limit of certain lattice model interfaces.

Proved convergence of Ising model interfaces with alternating boundary conditions.

Established existence of pure partition functions for multiple SLEs for ppa in (0,6].

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.