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arXiv · 2003.04911

The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles

Shulin Lyu·Yang Chen
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Abstract

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^{\alpha}(1-x)^{\beta},~x\in[0,1],~\alpha,\beta>0$, 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 $(1-x^2)^{\beta}, x\in[-1,1]$.

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Shulin Lyu, Yang Chen. 2020-03-10. The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles. https://doi.org/10.1002/mma.7394

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