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Universality of the double scaling limit in random matrix models. (English) Zbl 1111.35031

The authors consider the unitary random matrix model \[ Z^{-1}_{n,N}\exp(- N\,\text{Tr\,}V(M))\,dM \] defined on Hermitian \(n\times n\) matrices \(M\) in a critical regime where the limiting mean density of eigenvalues vanishes at an interior point. They assume that the confining potential \(V\), \(V: \mathbb{R}\to\mathbb{R}\) is real analytic and that it satisfies the growth condition \[ {V(x)\over\log(1+ x^2)}\to +\infty\quad\text{as }|x|\to+\infty. \] Assuming that there are no other singular interior points \(\widetilde x\), where the limiting mean eigenvalue density vanishes quadratically, the authors show that the correlation kernel has a double scaling limit. Moreover, they show that the limiting kernels are expressed in terms of some function associated with a special solution of the general Painlevé II equation. The main ingredients in the proof of the main result of the authors are equilibrium measures and Riemann-Hilbert problems.

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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