4.6 Article

The smallest eigenvalue distribution of the Jacobi unitary ensembles

Journal

MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Volume 44, Issue 13, Pages 10121-10134

Publisher

WILEY
DOI: 10.1002/mma.7394

Keywords

asymptotic expansions; Bessel kernel; Fredholm determinant; Jacobi unitary ensemble; smallest eigenvalue distribution

Funding

  1. Macau Science and Technology Development Fund [FDCT 023/2017/A1]
  2. National Natural Science Foundation of China [11971492]
  3. University of Macau [MYRG 2018-00125-FST]

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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.
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].

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