Random Matrices with Merging Singularities and the Painlevé V Equation

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^α e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $α>-1/2$ 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 $α$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

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