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Stochastic Loewner evolution and Dyson’s circular ensembles. (English) Zbl 1038.82074

J. Phys. A, Math. Gen. 36, No. 24, L379-L386 (2003); corrigendum No. 49, 12343 (2003).
Summary: 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 realizations 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.

MSC:

82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
81R12 Groups and algebras in quantum theory and relations with integrable systems
81T27 Continuum limits in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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