Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite $n$ Gaussian Unitary Ensembles

Abstract

In this paper we study the gap probability problem in the Gaussian unitary ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto $\sigma$-form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as $\sigma _n(a)$, $R_n(a)$ and $r_n(a)$. We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned $\sigma$-form and Painlevé IV we are able to show that the quantity $r_n(a)$ satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity $r_n(a)$ satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities $R_n(a)$ and $\sigma _n(a)$.