Stochastic Loewner Evolution and Dyson's Circular Ensembles
Abstract
Stochastic Loewner Evolution (SLE_kappa) 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 beta=4/kappa. As a result various equilibrium critical models give realisations of circular ensembles with beta 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.
Cite
@article{arxiv.math-ph/0301039,
title = {Stochastic Loewner Evolution and Dyson's Circular Ensembles},
author = {John Cardy},
journal= {arXiv preprint arXiv:math-ph/0301039},
year = {2009}
}
Comments
8 pages. v.2: main result confirmed using conformal field theory; subtle square root elucidated; figures added. v.3: funding acknowledgements added. v.4: erratum to published paper added: this exposes more fully the relation to CFT results