Universality for eigenvalue correlations from the modified Jacobi unitary ensemble

By Arno B. J. Kuijlaars, Maarten Vanlessen. 2002

135 citations recorded by the source corpus.

The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^\alpha (1+x)^\beta h(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the orthogonal polynomials with respect to the modified Jacobi weight, and recent results on strong asymptotics derived by K.T-R McLaughlin, W. Van Assche and the authors.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/14e488be0613a2f61966302e3efc0fb9d08304a1

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Two extensions of Lubinsky’s universality theorem (2008, background): “16) come from the random matrix and Riemann–Hilbert literatures; see [15].”
Riemann-Hilbert Analysis for Orthogonal Polynomials (2003, methodology): “For this we refer to the papers [8] and [24] where the universality for the distribution of eigenvalue spacings was obtained from the Riemann-Hilbert method.”
From Random Matrices to Stochastic Operators (2006, unknown): “ar X iv :m at h- ph /0 60 70 38 v2 7 S ep 2”
Universality limits in the bulk for varying measures (2008, unknown): “A (very!) partial list is [2], [3], [4], [8], [9], [10], [11], [12], [13], [14], [16], [25], [27], [28], [29], [38], [57].”
The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] (2001, unknown): “In [ 29 ] this behavior has been used to derive universal behavior for eigenvalue correlations, associated to the modified Jacobi unitary ensemble, in the bulk and at the hard edge of the spectrum.”
Universality limits in the bulk for arbitrary measures on compact sets (2008, background): “Orthogonal polynomials play an important role in random matrix theory [1], [2], [8], [17].”
A New Approach to Universality Limits Involving Orthogonal Polynomials (2007, unknown): “Then uniformly for x ∈ I ′, (1.11) lim n→∞ 1 nr+s+1 K(r,s)n (x, x) = 1 πw (x) (1− x2)(r+s+1)/2 τ r,s.”
Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy (2009, unknown): “[37, 17], we define the functions J1, J2, and J3: J1(λ) = e − 4 3π 1 2 σ3 ( I0(λ ) i πK0(λ ) πiλI ′ 0(λ ) −λ1/2K ′ 0(λ) ) , (3.”
Universality and fine zero spacing on general sets (2009, background): “It has been known that some universality questions in random matrix theory can be expressed in terms of orthogonal polynomials, in particular in the off-diagonal behavior of the reproducing kernel (see [3], [4], [ 9 ] and [13]).”
Large n Limit of Gaussian Random Matrices with External Source, Part I (2004, methodology): “We develop the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert (RH) problems [18], thereby extending the works [3], [4], [16], [17], [25], [26] who treated the unitary invariant case (i.”