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

Abstract

AbstractIn this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval 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 , , and . 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 can be reduced to the Jimbo–Miwa–Okamoto σ form of the Painlevé V equation.