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UNIVERSALITY LIMITS IN THE BULK FOR VARYING MEASURES
"... Abstract. Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form (x) dx in the regi ..."
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Abstract. Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form (x) dx in the region where universality is desired. Wn does not need to be analytic, nor possess more than one derivative and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form W 2n (x) dx, for example, when W = e Q where Q is convex and Q 0 satis…es a Lipschitz condition of some positive order. We also deduce universality for a class of …xed exponential weights on a real interval.
The RiemannHilbert approach to double scaling limit of random matrix eigenvalues near the ”birth of a cut” transition
, 2007
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Nonintersecting squared Bessel paths: critical time and double scaling limit
 Comm. Math. Phys
"... We consider the double scaling limit for a model of n nonintersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a reg ..."
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We consider the double scaling limit for a model of n nonintersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n→ ∞ that intersects the hard edge at x = 0 at a critical time t = t∗. In a previous paper, the scaling limits for the positions of the paths at time t 6 = t ∗ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n→ ∞ of the correlation kernel at critical time t ∗ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued RiemannHilbert problem by the DeiftZhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular thirdorder linear differential equation,
Biorthogonal ensembles with twoparticle interactions
 Nonlinearity
"... We investigate determinantal point processes on [0,+∞) of the form 1 ..."
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We investigate determinantal point processes on [0,+∞) of the form 1
Some recent methods for establishing universality limits
 J. Nonlinear Anal
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Janossy densities for Unitary ensembles at the spectral edge
, 2009
"... For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in ..."
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For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated RiemannHilbert problem. 1
4 GLOBAL ASYMPTOTICS FOR THE CHRISTOFFELDARBOUX KERNEL OF RANDOM MATRIX THEORY
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Critical behavior in Angelesco ensembles
, 2014
"... We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a, 0] and [0, 1], for a < 0. As a → −1 the particles around 0 experience a phase transition. This transition is studied in a double scaling limit, where we let the number of particles of the ensemb ..."
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We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a, 0] and [0, 1], for a < 0. As a → −1 the particles around 0 experience a phase transition. This transition is studied in a double scaling limit, where we let the number of particles of the ensemble tend to infinity while the parameter a tends to −1 at a rate of O(n−1/2). The correlation kernel converges, in this regime, to a new kind of universal kernel, the Angelesco kernel KAng. The result follows from the Deift/Zhou steepest descent analysis, applied to the RiemannHilbert problem for multiple orthogonal polynomials. 1 Introduction and statement of results Multiple orthogonal polynomial (MOP) ensembles [21] form an extension of the more familiar orthogonal polynomial (OP) ensembles [20]. The latter appear as the eigenvalue distributions of unitary random matrix ensembles.
Abstract
, 2008
"... For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in ..."
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For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated RiemannHilbert problem. 1