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26
The CalogeroSutherland Model And Generalized Classical Polynomials
 Comm. Math. Phys
, 1997
"... this paper. The first is the discussion of some mathematical properties relating to the eigenfunctions, while the second is the evaluation of the density in the ground state and the exact solution of (1.6) for certain initial conditions. These problems are in fact interrelated; we find that the den ..."
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Cited by 69 (9 self)
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this paper. The first is the discussion of some mathematical properties relating to the eigenfunctions, while the second is the evaluation of the density in the ground state and the exact solution of (1.6) for certain initial conditions. These problems are in fact interrelated; we find that the density for each system can be written in terms of a certain eigenstate and that a summation theorem for the eigenstates gives an exact solution of (1.6). A feature of the Schrodinger operators (1.2) is that after conjugation with the ground state: \Gamma e i @ \Gamma fi @W (1.7) the resulting differential operator has a complete set of polynomial eigenfunctions. In Section 2 we consider the form of the expansion of these polynomials in terms of some different bases of symmetric functions. We note that in the N = 1 case, after a suitable change of variables, the operator (1.7) with W given by (1.3) is the eigenoperator for the classical Hermite, Laguerre and Jacobi polynomials. Previous studies of the operator for general N in the Jacobi case [1] have established an orthogonality relation. Since the polynomials in the Hermite and Laguerre cases are limiting cases of these generalized Jacobi polynomials, we can obtain the corresponding orthogonality relations via the limiting procedure. The generalized Hermite polynomials, which are the polynomial eigenfunctions of (1.4) with W = W as given by (1.3a), are studied in Section 3. Many higherdimensional analogues of properties of the classical Hermite polynomials are obtained, including a generating function formula, differentiation and integration formulas, a summation theorem and recurrence relations. An analogous study of the generalized Laguerre polynomials is performed in Section 4. In Section 5 we relate the...
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.
Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields
, 2000
"... The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is pos ..."
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Cited by 36 (14 self)
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The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the EulerLagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk" in which there is universal behavior involving the sine kernel, and "edge effects" in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any oneparameter family of external fields V=c the equilibrium measure exhibits this regular behavior, except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials and integrable systems.
Fluctuations of eigenvalues and second order Poincaré inequalities
, 2007
"... Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified t ..."
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Cited by 25 (3 self)
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Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
Bulk Universality and Related Properties of Hermitian Matrix Models
, 2007
"... We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C 2 and locally C 3 function (see Theorem 3.1). The proof as our previous proof in [21] is based on the orthogonal polynomial techn ..."
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Cited by 14 (1 self)
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We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C 2 and locally C 3 function (see Theorem 3.1). The proof as our previous proof in [21] is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sinkernel as a unique solution of a certain nonlinear integrodifferential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper [1] on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.
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 ..."
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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.
Global spectrum fluctuations for the βHermite and βLaguerre ensembles via matrix models
 J. Math. Phys
, 2006
"... ensembles via matrix models ..."
Limiting laws of linear eigenvalue statistics for unitary invariant matrix models
 J. Math. Phys
, 2006
"... We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogona ..."
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Cited by 6 (3 self)
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We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q ≥ 2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n → ∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2 × variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
LIMIT THEOREMS FOR SPECTRA OF RANDOM MATRICES WITH MARTINGALE STRUCTURE
, 2003
"... We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equati ..."
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Cited by 6 (0 self)
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We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semicircular law.