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33
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 39 (6 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results
 INT. MATH. RES. NOT
, 2003
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Asymptotics of the partition function for random matrices via RiemannHilbert techniques, and applications to graphical enumeration
 Internat. Math. Research Notices
, 2003
"... Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N ..."
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Cited by 33 (6 self)
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Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.
Multicritical unitary random matrix ensembles and the general Painlevé II equation
, 2005
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Universality for Eigenvalue Correlations at the Origin of the Spectrum
"... We establish universality of local eigenvalue correlations in unitary random matrix ensembles Zn dM near the origin of the spectrum. If V is even, and if the recurrence coe#cients of the orthogonal polynomials associated with have a regular limiting behavior, then it is known from w ..."
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Cited by 17 (7 self)
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We establish universality of local eigenvalue correlations in unitary random matrix ensembles Zn dM near the origin of the spectrum. If V is even, and if the recurrence coe#cients of the orthogonal polynomials associated with have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V . Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of RiemannHilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local RiemannHilbert problem near the origin.
Asymptotics of the partition function of a random matrix model
 Ann. Inst. Fourier (Grenoble
"... Dedicated to Pierre van Moerbeke on his sixtieth birthday. Abstract. We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential V (z). Our approach is based on a deformation τtV (z) of V (z) to z 2, 0 ≤ t < ∞ and on the use ..."
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Cited by 14 (2 self)
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Dedicated to Pierre van Moerbeke on his sixtieth birthday. Abstract. We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential V (z). Our approach is based on a deformation τtV (z) of V (z) to z 2, 0 ≤ t < ∞ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of N −2 of the recurrence coefficients of the related orthogonal polynomials, for a onecut regular V; (2) the existence of a full asymptotic expansion in powers of N −2 of the free energy, for a V, which admits a onecut regular deformation τtV; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of V; (4) the onesided analyticity of the recurrent coefficients and the free energy for a onecut singular V; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial V.
Universality of a double scaling limit near singular edge points in random matrix models
, 2008
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On the finite gap ansatz in the continuum limit of the Toda lattice
 Duke Math. J
"... The continuum limit of the Toda lattice was studied by Deift and McLaughlin in the spirit of the LaxLevermore theory for the zero dispersion limit of the Kortewegde Vries equation. An important role is played by a quadratic minimization problem arising from an asymptotic analysis of the inverse spe ..."
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Cited by 10 (4 self)
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The continuum limit of the Toda lattice was studied by Deift and McLaughlin in the spirit of the LaxLevermore theory for the zero dispersion limit of the Kortewegde Vries equation. An important role is played by a quadratic minimization problem arising from an asymptotic analysis of the inverse spectral transform. The minimum is taken over density functions ψ in the spectral variable satisfying the constraints 0 ≤ ψ ≤ φ where φ is a function determined by the initial conditions. The finite gap ansatz is said to hold if the set where the two constraints are not effective consists of a finite union of intervals. If the finite gap ansatz hold, weak limits are described in terms of the endpoints of the intervals. Using techniques from logarithmic potential theory, we show that the finite gap ansatz holds for real analytic spectral data. This extends a previous result of Deift, Kriecherbauer and McLaughlin for the situation without upper constraint φ. 1
The polynomial method for random matrices
, 2007
"... We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions a ..."
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Cited by 10 (1 self)
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We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the MarčenkoPastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability ” theory. We hope that the tools developed allow researchers to finally harness the power of the infinite random matrix theory.