Random matrices with merging singularities and the Painlevé V equation. (English) Zbl 1338.60013
Summary: We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form \(\frac{1}{Z_n} \big|\mathrm{det} \big( M^2-tI \big)\big|^{\alpha} e^{-n\mathrm{Tr} V(M)}dM\), where \(M\) is an \(n\times n\) Hermitian matrix, \(\alpha>-1/2\) and \(t\in\mathbb R\), in double scaling limits where \(n\to\infty\) and simultaneously \(t\to 0\). If \(t\) is proportional to \(1/n^2\), a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of \(\alpha\), they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 33E17 | Painlevé-type functions |
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