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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