Gap probabilities for edgeintervals in finite Gaussian and Jacobi unitary matrix ensembles

Abstract

The probabilities for gaps in the eigenvalue spectrum of the finite-dimensionalN × N random matrix Hermite and Jacobi unitaryensembleson some single and disconnected double intervals are found.These are cases where a reflection symmetry exists and the probabilityfactorsinto two other related probabilities, defined on single intervals.Our investigation uses the system of partial differential equationsarising from the Fredholm determinant expression for the gap probabilityand the differential-recurrence equations satisfied by Hermite and Jacobiorthogonal polynomials.In our study we find second- and third-order nonlinear ordinary differentialequations defining the probabilities in the general N case. For N = 1and 2 the probabilities and thus the solution of the equations aregiven explicitly. An asymptotic expansion for large gap size is obtainedfromthe equation in the Hermite case, and also studied is the scaling at theedge of the Hermite spectrum as N → ∞, and the Jacobi toHermitelimit; these last two studies make correspondence to other cases reportedhere or known previously.Moreover, the differential equation arising in the Hermite ensemble issolved in terms of an explicit rational function of a Painlevé-Vtranscendent and its derivative, and an analogous solution is providedin the two Jacobi cases but this time involving a Painlevé-VItranscendent.