Random matrices with merging singularities and the Painlevé V equation

Claeys, Tom [UCL] Fahs, Benjamin [UCL]

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1/Zn∣ 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.