The Jacobi polynomial ensemble and the Painlevé VI equation

We consider the Jacobi polynomial ensemble of n×n random matrices. We show that the probability of finding no eigenvalues in the interval [−1,z] for a random matrix chosen from the ensemble, viewed as a function of z, satisfies a second-order differential equation. After a simple change of variable, this equation can be reduced to the Okamoto–Jimbo–Miwa form of the Painlevé VI equation. The result is achieved by a comparison of the Tracy–Widom and the Virasoro approaches to the problem, which both lead to different third-order differential equations. The Virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments, which drastically simplifies the computations.