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Large n limit of Gaussian random matrices with external source, part III: double scaling limit in the critical case, in preparation
"... Abstract. We continue the study of the Hermitian random matrix ensemble with external source ..."
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Cited by 47 (14 self)
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Abstract. We continue the study of the Hermitian random matrix ensemble with external source
Multiple orthogonal polynomials of mixed type and nonintersecting Brownian motions. Preprint. Available at arXiv:math.CA/0511470
, 2005
"... We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a RiemannHilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoff ..."
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Cited by 14 (2 self)
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We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a RiemannHilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a ChristoffelDarboux formula for these polynomials using the solution of the RiemannHilbert problem. The main motivation for studying these polynomials comes from a model of nonintersecting onedimensional Brownian motions with a given number of starting points and endpoints. The correlation kernel for the positions of the Brownian paths at any intermediate time coincides with the ChristoffelDarboux kernel for the multiple orthogonal polynomials of mixed type with respect to Gaussian weights. 1
Asymptotics of nonintersecting Brownian motions and a 4 × 4 Riemann–Hilbert problem
 J. Approx. Theory
"... We consider n onedimensional Brownian motions, such that n/2 Brownian motions start at time t = 0 in the starting point a and end at time t = 1 in the endpoint b and the other n/2 Brownian motions start at time t = 0 at the point −a and end at time t = 1 in the point −b, conditioned that the n Brow ..."
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Cited by 9 (2 self)
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We consider n onedimensional Brownian motions, such that n/2 Brownian motions start at time t = 0 in the starting point a and end at time t = 1 in the endpoint b and the other n/2 Brownian motions start at time t = 0 at the point −a and end at time t = 1 in the point −b, conditioned that the n Brownian paths do not intersect in the whole time interval (0, 1). The correlation functions of the positions of the nonintersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case ab < 1/2. We find that the limiting mean density of the positions of the Brownian motions is supported on one or two intervals and that the correlation kernel has the usual scaling limits from random matrix theory, namely the sine kernel in the bulk and the Airy kernel near the edges.
Dyson’s nonintersecting Brownian motions with a few outliers
 Comm. Pure Appl. Math., online
, 2008
"... 1 A constrained Brownian motion with a few outliers 8 2 The existence of the rAiry process and the rAiry kernel 15 3 An integrable deformation of Gaussian random ensemble with ..."
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Cited by 6 (3 self)
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1 A constrained Brownian motion with a few outliers 8 2 The existence of the rAiry process and the rAiry kernel 15 3 An integrable deformation of Gaussian random ensemble with
Kuijlaars: A phase transition for nonintersecting Brownian motions, and the Painlevé II equation
, 809
"... We consider n nonintersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of ‘large separation ’ between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between t ..."
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Cited by 5 (2 self)
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We consider n nonintersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of ‘large separation ’ between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the RiemannHilbert formalism. In the case of ‘critical separation’ between the endpoints we are led to a model RiemannHilbert problem associated to the HastingsMcLeod solution of the Painlevé II equation. We show that the Painlevé II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the RiemannHilbert problem.
Lectures on random matrix models. The RiemannHilbert approach
, 2008
"... This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its appli ..."
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Cited by 2 (0 self)
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This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its applications to the problem of universality in random matrix models, the double scaling limits, the large N asymptotics of the partition function, and random matrix models with external source.
Universality of the Pearcey process
, 901
"... 2 Nonintersecting Brownian motions on R, forced to several ..."
H(E) =
, 2009
"... In this paper we prove that the integral of the Wishart ensemble is, in a certain sense, a KP τ function, and generalize the result to other random matrix models, especially the Hermitian matrix model with external source. Besides potential application in multivariate statistics, we obtain some inte ..."
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In this paper we prove that the integral of the Wishart ensemble is, in a certain sense, a KP τ function, and generalize the result to other random matrix models, especially the Hermitian matrix model with external source. Besides potential application in multivariate statistics, we obtain some interesting combinatorial results. 1 Motivition and the main result In random matrix theory, the simplest and most studied model is the Gaussian Unitary Ensemble (GUE) [12], for which we consider the integral