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Pré-Publication, Document De Travail Année : 2009

Universal scaling limits of matrix models, and $(p,q)$ Liouville gravity

Michel Bergere
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Bertrand Eynard

Résumé

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like $y$~$x^{p/q}$, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal $(p,q)$-minimal models kernels. Those $(p,q)$ kernels are written in terms of functions solutions of a linear equation of order $q$, with polynomial coefficients of degree at most $\leq p$. For example, near a regular edge $y$~$x^{1/2}$, the (1,2) kernel is the Airy kernel and we recover the Airy law. Those kernels are associated to the $(p,q)$ minimal model, i.e. the $(p,q)$ reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which $q$=2.
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Dates et versions

hal-00413499 , version 1 (04-09-2009)

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Michel Bergere, Bertrand Eynard. Universal scaling limits of matrix models, and $(p,q)$ Liouville gravity. 2009. ⟨hal-00413499⟩
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