Critical edge behavior in the modified Jacobi ensemble and the Painlevé V transcendents

摘要

We study the Jacobi unitary ensemble defined by the perturbed Chebyshev weight W (x) = (1-x2)-1/2 (tn 2-x2)α, x e{open} (-1,1),tn > 1, x ∈ (-1, 1), tn > 1. The algebraic singularity coalesces with the end point of the support of the weight function, namely, tn → 1, the hard edge, as n → ∞. We use the Riemann-Hilbert approach (the method of Deift and Zhou) to obtain the asymptotics of the polynomials orthogonal with respect to w(x). We show that the universality property is preserved: the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel. The main result is on the local behavior at the edge of the spectrum. The limit kernel at the edge, when tn - 1 = O(n-2), is described by the ψ -function for a specific solution of the Painlevé V equation. We also show that when tn varies to a fixed t > 1, the Painlevé V kernel transits to the Bessel kernel J-1/2. On the other hand, when tn approaches 1, the Painlevé V kernel can be approximated by another Bessel kernel Jα-1/2.