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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Universality of Painlev\'e functions in random matrix models

19th September 2006

Author: Arno Kuijlaars (Katholieke Universiteit Leuven)


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.

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