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Scaling level-spacing distribution functions in the " bulk of the spectrum " in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sin π(x − y)/π(x − y). Similarly a scaling limit at the " edge of the spectrum " leads to the Airy kernel [Ai(x)Ai ′ (y) − Ai ′ (x)Ai(y)] /(x… (More)

The focus of this paper is on the probability, Eβ(Q; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N 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… (More)

- Craig A. Tracy
- 1998

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a… (More)

- Craig A. Tracy, Harold Widom, Tracy, H. Widom
- 2004

Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals J= [J™ =1 (βij-u Λ 2J)… (More)

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function h t (x) with corner initialization. We prove, with one exception, that the limiting distribution function of h t (x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In… (More)

- Craig A. Tracy, Harold Widom, Tracy, H. Widom
- 1993

Scaling models of random N x N hermitian matrices and passing to the limit N-» oo leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others' as well, the kernel one obtains by scaling in the "bulk" of the… (More)

The focus of this survey paper is on the distribution function F N β (t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and the Gaussian Symplectic Ensemble (GSE, β = 4) in the edge scaling limit of N → ∞. These limiting distribution functions are expressible in terms of a… (More)

- Craig A. Tracy, Harold Widom
- SIAM J. Matrix Analysis Applications
- 2002

In this article we derive, using standard methods of Toeplitz theory, an asymp-totic formula for certain large minors of Toeplitz matrices. Bump and Diaconis obtained the same asymptotics using representation theory, with an answer having a different form. Our Toeplitz-like matrices are of the form where {p i } and {q i } are sequences of integers… (More)

These notes provide an introduction to the theory of random matrices. The central quantity studied is r(a) = det (1-K) where K is the integral operator with kernel 1 sin r(z-y) XI(Y). 7r x-y Here I = [-Ji (a2j-l,a2J) and XI(Y) is the characteristic function of the set I. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is… (More)

It is now believed that the limiting distribution function of the largest eigenvalue in the three classic random matrix models GOE, GUE and GSE describe new universal limit laws for a wide variety of processes arising in mathematical physics and interacting particle systems. These distribution functions, expressed in terms of a certain Painlevé II function,… (More)