Gap probabilities for edge intervals in finite Gaussian and Jacobi unitary matrix ensembles

Abstract

The probabilities for gaps in the eigenvalue spectrum of the finite-dimensional N × N random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second-and third-order nonlinear ordinary differential equations defining the probabilities in th..