The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles

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 $(1-x^2{)}^{\beta },x\in [-1,1]$.

19 pages

References in corpus (4)

• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Asymptotics of Fredholm determinant associated with the Pearcey kernel
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
• Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble