Stochastic Loewner Evolution and Dyson's Circular Ensembles

Details Stochastic Loewner Evolution and Dyson's Circular Ensembles Stochastic Loewner Evolution and Dyson's Circular Ensembles

2003-01-28

arxiv.org/abs/math-ph/0301039v4

J. Phys. A 36, L379, 2003; erratum J.Phys. A, 36, 12343, 2003

Physics

Mathematical Physics

8 pages. v.2: main result confirmed using conformal field theory; subtle square root elucidated; figures added. v.3: funding a

Scientific paper

10.1088/0305-4470/36/49/c01

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.

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