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37
Universality of a double scaling limit near singular edge points in random matrix models
, 2008
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Numerical solution of the small dispersion limit of Kortewegde Vries and Witham equations
 Comm. Pure Appl. Math
"... Abstract. The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ2, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where ..."
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Cited by 32 (12 self)
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Abstract. The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ2, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wavenumber and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of ǫ between 10−1 and 10−3. The numerical results are compatible with a difference of order ǫ within the ‘interior ’ of the Whitham oscillatory zone, of order ǫ 1 3 at the left boundary outside the Whitham zone and of order √ ǫ at the right boundary outside the Whitham zone. 1.
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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Cited by 20 (2 self)
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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 existence of a real polefree solution of the fourth order analogue of the Painleve I equation
"... We establish the existence of a real solution y(x, T) with no poles on the real line of the following fourth order analogue of the Painlevé I equation, ..."
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Cited by 18 (7 self)
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We establish the existence of a real solution y(x, T) with no poles on the real line of the following fourth order analogue of the Painlevé I equation,
Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent
, 2008
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Universality in unitary random matrix ensembles when the soft edge meets the hard edge. arXiv:mathph/0701003
"... Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N TrV (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically va ..."
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Cited by 15 (3 self)
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Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N TrV (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N → ∞ such that n/N − 1 = O(n −2/3). For each value of α> −1 we find a oneparameter family of limiting kernels that we describe in terms of the HastingsMcLeod solution of the Painlevé II equation with parameter α + 1/2.
Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight
, 2008
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A phase transition for nonintersecting Brownian motions, and the Painlevé II equation
, 2008
"... 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 th ..."
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Cited by 15 (5 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.
The birth of a cut in unitary random matrix ensembles
 Int Math Res Notices, 2008(article ID rnm166):40
"... We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It t ..."
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Cited by 13 (3 self)
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We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It turns out that limits of the eigenvalue correlation kernel are given by Hermite kernels corresponding to a finite size Gaussian Unitary Ensemble (GUE). When modifying the double scaling limit slightly, we observe a remarkable transition each time the new cut picks up an additional eigenvalue, leading to a limiting kernel interpolating between GUEkernels for matrices of size k and size k + 1. We prove our results using the RiemannHilbert approach. 1
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 13 (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.