Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite n to Double Scaling

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval (-a, a) (0 < a < 1) is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by H-n(a), R-n(a), and r(n)(a). We find that each one satisfies a second-order differential equation. We show that after a double scaling, the large second-order differential equation in the variable a with n as parameter satisfied by H-n(a) can be reduced to the Jimbo-MiwaOkamoto sigma form of the Painleve V equation.