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) The emphasis is on the determinants thought of as functions of the end-points a k. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as φ and φ satisfy a certain type of differentiation formula. The (φ, φ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finite N Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system. An analysis of these equations will lead to explicit representations in terms of Painleve transcendents for the distribution functions of the largest and smallest eigenvalues in the finite N Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables. There is also an exponential variant of the kernel in which the denominator is replaced by e bx — e by, where b is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. If b = i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble of N x N unitary matrices, and then an ODE if J is an arc of the circle.

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 direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions, and spacing distributions in terms of them.

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Abstract. We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1) and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e −V (x) where V is a polynomial, V (x) = κ2mx 2m + · · · , κ2m> 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2, 1.3 below. For a proof of universality in the bulk of the spectrum, for the same class of weights, for unitary ensembles see [DKMVZ2], and for orthogonal and symplectic ensembles see [DG]. Our starting point in the unitary case is [DKMVZ2], and for the orthogonal and symplectic cases we rely on our recent work [DG], which in turn depends on the earlier work of Widom [W] and Tracy and Widom [TW2]. As in [DG], the uniform Plancherel–Rotach type asymptotics for the orthogonal polynomials found in [DKMVZ2] plays a central role. The formulae in [W] express the correlation kernels for β = 1 and 4 as a sum of a Christoffel–Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [DG], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [DG]. 1.

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 spectrum is the "sine kernel" —-- —. Rescaling the GUE at the "edge " of the spectrum leads to the kernel π(x- y) M(x)M'(y)- A.. f A. f., where Ai is the Airy function. In previous work we x-y found several analogies between properties of this "Airy kernel " and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painleve transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely 2(x- y) where J α(z) is the Bessel function of order α. In the cases α = +? these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.

We consider the multi-point equal time height fluctuations of a one-dimensional polynuclear growth model in a half space. For special values of the nucleation rate at the origin, the multi-layer version of the model is reduced to a determinantal process, for which the asymptotics can be analyzed. In the scaling limit, the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory.

Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of Mn(ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices. Our approach is motivated by the general idea of “property testing ” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics

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For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2 × 2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w ′ /w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w ′ /w. General formulas are obtained for these extra terms. We do not use skeworthogonal polynomials in the derivations. 1.

Abstract. Consider a deterministic self-adjoint matrix X n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.