Current Developments in Mathematics

Volume 2012

Duality, statistical mechanics and random matrices

Pages: 229 – 260

DOI: http://dx.doi.org/10.4310/CDM.2012.v2012.n1.a5

Author

Thomas Spencer (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Abstract

This article will present an informal review of some results and conjectures about the spectral theory of large random matrices and related spin systems in statistical mechanics. A class of lattice spin models provides a dual representation for spectral problems in random matrix theory. Ordered and disordered phases of the spins correspond to different spectral types and quantum time evolutions. In three dimensions, we describe a phase transition for a supersymmetric statistical mechanics system inspired by random matrix theory. This transition has a classical interpretation in terms of a history dependent walk on the lattice. In the ordered phase the walk is diffusive while in the disordered phase it is localized near its starting point.

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