Random Matrices with Merging Singularities and the Painlevé V Equation

Symmetry Integrability and Geometry-Methods and Applications

article

Tom Claeys¹  Benjamin Fahs¹ 

[1] Institut de Recherche en Mathématique et Physique, Université catholique de Louvain

关键词: random matrices;    Painlev´e equations;    Riemann–Hilbert problems;   

DOI  :  10.3842/SIGMA.2016.031

来源: National Academy of Science of Ukraine

【 摘 要 】

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, 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 $1/n^2$, 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.

【 授权许可】

Unknown