Let x(1) denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x(1) is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a p­variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painleve II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.

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On the distribution of the largest eigenvalue in principal components analysis

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

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Level spacing distributions and the Airy kernel

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (β=1) and Gaussian Symplectic (β=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (β=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painleve II function.

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On orthogonal and symplectic matrix ensembles

Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several variables Equidistribution Monodromy of families of curves Monodromy of some other families GUE discrepancies in various families Distribution of low-lying Frobenius eigenvalues in various families Appendix AD: Densities Appendix AG: Graphs References.

Random matrices, Frobenius eigenvalues, and monodromy

We discuss the status of the Fisher-Hartwig conjecture concerning the asymptotic expansion of a class of Toeplitz determinants with singular generating functions. A counterexample is given for a nonrational generating function; and we formulate a generalized Fisher-Hartwig conjecture.

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The Fisher-Hartwig conjecture and generalizations

The object of this paper is to find an asymptotic formula for determinants of finite dimensional Toeplitz operators generated by a class of functions with singularities. The formula is a generalization of the Strong Szegö Limit Theorem.

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Asymptotic formulas for Toeplitz determinants

The Fisher-Hartwig conjecture and Toeplitz eigenvalues

In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w0(x) (supported on or subsets of ) by w0(x) e−tx. It is a well-known fact that under such a deformation the recurrence coefficients denoted as αn and βn evolve in t according to the Toda equations, giving rise to the time-dependent orthogonal polynomials and time-dependent determinants, using Sogo's terminology. If w0 is the normal density , or the gamma density xα e−x, , α > −1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w0 is the beta density (1 − x)α(1 + x)β, x [ − 1, 1], α, β > −1, the resulting 'time-dependent' Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm–Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painleve V. In particular we show that , where is the coefficient of zn−1 of the monic orthogonal polynomials associated with the 'time-dependent' Jacobi weight, satisfies, up to a translation in t, the Jimbo–Miwa σ-form of the same PV; while a recurrence coefficient αn(t) is up to a translation in t and a linear fractional transformation PV(α2/2, − β2/2, 2n + 1 + α + β, − 1/2). These results are found from combining a pair of nonlinear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painleve equation. The case with α = β = −1/2 arose from a certain integrable system and this was brought to our attention by A P Veselov.

Painlevé V and time-dependent Jacobi polynomials

In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants.

Estelle L. Basor

Papers

The Fisher-Hartwig conjecture and generalizations

Asymptotic formulas for Toeplitz determinants

The Fisher-Hartwig conjecture and Toeplitz eigenvalues

Painlevé V and time-dependent Jacobi polynomials

On a Toeplitz determinant identity of Borodin and Okounkov