Random Matrices with Merging Singularities and the Painlevé V Equation

Abstract

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1Zn∣∣det(M²−tI)∣∣αe−nTrV(M)dM, where M is an n×n Hermitian matrix, α>−1/2 and t∈R, in double scaling limits where n→∞ and simultaneously t→0. If t is proportional to 1/n², 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.