Stochastic Loewner evolution and Dyson s circular ensembles

Abstract

Stochastic Loewner evolution (SLEκ) has been introduced as a description of the continuum limit of cluster boundaries in two-dimensional critical systems. We show that the problem of N radial SLEs in the unit disc is equivalent to Dyson's Brownian motion on the boundary of the disc, with parameter β = 4/κ. As a result, various equilibrium critical models give realizations of circular ensembles with β different from the classical values of 1, 2 and 4 which correspond to symmetry classes of random U(N) matrices. Some of the bulk critical exponents are related to the spectrum of the associated Calogero–Sutherland Hamiltonian. The main result is also checked against the predictions of conformal field theory.