2000 character limit reached
Random Matrices with Merging Singularities and the Painlevé V Equation (1508.06734v2)
Published 27 Aug 2015 in math-ph, math.CA, math.CV, and math.MP
Abstract: We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M2-tI \big)\big|{\alpha} e{-n\operatorname{Tr} V(M)}dM$, 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 $1/n2$, a transition takes place which can be described in terms of a family of solutions to the Painlev\'e V equation. These Painlev\'e 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.