The smallest eigenvalue distribution of the Jacobi unitary ensembles

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight x(alpha)(1 - x)(beta), x is an element of [0, 1], alpha, beta > -1, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval [t, 1] is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval (- a, a), a > 0 is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight (1 - x(2))(beta), x is an element of [- 1, 1].

Automated summary

The study focuses on the probability of eigenvalues of Hermitian matrices from the Jacobi unitary ensemble with a specific weight lying in a certain interval, and provides the asymptotic constant in the determinant of the Bessel kernel. A specialization of the results yields the constant for the probability of eigenvalues within a specific interval in the Jacobi unitary ensemble with a symmetric weight.