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Universality of Painleve functions in random matrix models

Tuesday 19th September 2006 - 10:00 to 11:00
INI Seminar Room 1

Several types of critical phenomena take place in the unitary random matrix ensembles (1/Z_n) e^{-n Tr V(M)} dM defined on n-by-n Hermitian matrices M in the limit as n tends to infinity.

The first type of critical behavior is associated with the vanishing of the equilibrium measure in an interior point of the spectrum, while the second type is associated with the higher order vanishing at an endpoint. The two types are associated with special solutions of the Painlev\'e II and Painlev\'e I equation, respectively. The quartic potential is the simplest case where this behavior occurs and serves as a model for the universal appearance of Painlev\'e functions in random matrix models.

Related Links

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons