Coupling of radial multiple SLE with Gaussian free field, and the hydrodynamic limit

Abstract

Schramm--Loewner evolution (SLE) has been one of the central topics in the probabilistic study of two-dimensional critical systems. It is a random curve in two dimensions to which a cluster interface in a critical lattice system is supposed, or has been proved, to converge. The most archetypical setting for SLE is called chordal, where a random curve evolves in a simply-connected domain from a boundary point to another, whereas in its variant called radial, a random curve evolves from a boundary point to a distinguished interior point. Multiple SLE is a variant to another direction; it deals with multiple random curves, and it is a natural direction as there are certainly multiple cluster interfaces found in critical lattice systems. In this paper, we study multiple SLE in the radial setting, namely, radial multiple SLE. We report two main results. One is regarding the local set coupling between radial multiple SLE and Gaussian free field (GFF). This sort of coupling between SLE and GFF has been extensively studied in the chordal setting, and serves as a foundation for many recent developments. We show that the coupling between radial multiple SLE and GFF occurs if and only if the radial multiple SLE is driven by the circular Dyson Brownian motions. The circular Dyson Brownian motions are a typical example of stochastic log-gases. This fact motivates us to study the hydrodynamic limit of the corresponding radial multiple SLE, which refers to the dynamical law of large numbers, when the number of curves tends to infinity. In our other main result, we provide explicit description of the hydrodynamic limit.