Stochastic Loewner evolution and Dyson's circular ensembles - Cardy, J (2003)
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.