Craig A. Tracy

On orthogonal and symplectic matrix ensembles

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 Painlevé II function.

Fredholm Determinants, Differential Equations and Matrix Models

AbstractOrthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(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

Painlevé functions of the third kind

J = \cup _{j = 1}^m (a_{2j - 1 ,{\text{ }}} a_{2j} )

Level-spacing distributions and the Airy kernel

. The emphasis is on the determinants thought of as functions of the end-pointsak.We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDEs as long as ϕ and ψ satisfy a certain type of differentiation formula. The (ϕ, ψ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finiteN Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDEs for these ensembles as special cases of the general system.An analysis of these equations will lead to explicit representations in terms of Painlevé transcendents for the distribution functions of the largest and smallest eigenvalues in the finiteN 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 byebx−eby, whereb is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. Ifb=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 PDEs for Dysons circular ensemble ofN×N unitary matrices, and then an ODE ifJ is an arc of the circle.

The Pearcey Process

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.

A Fredholm Determinant Representation in ASEP

We explicitly construct the one‐parameter family of solutions, η (ϑ;ν,λ), that remain bounded as ϑ→∞ along the positive real ϑ axis for the Painleve equation of third kind w w′′= (w′)2−ϑ−1 w w′+2νϑ−1(w 3−w) +w 4−1, where, as ϑ→∞, η ∼ 1−λΓ (ν+1/2)2−2νϑ−ν−1/2 e −2ϑ. We further construct a representation for ψ (t;ν,λ) =−ln[η (t/2;ν,λ)], where ψ (t;ν,λ) satisfies the differential equation ψ′′+t −1ψ′= (1/2)sinh(2ψ)+2νt −1 sinh(ψ). The small‐ϑ behavior of η (ϑ;ν,λ) is described for ‖λ‖<π−1 by η (ϑ;ν,λ)  ∼ 2σ Bϑσ. The parameters σ and B are given as explicit functions of λ and ν. Finally an identity involving the Painleve transcendent η (ϑ;ν,λ) is proved. These results for the special case ν=0 and λ=π−1 make rigorous the analysis of the scaling limit of the spin–spin correlation function of the two‐dimensional Ising model.

On the distributions of the lengths of the longest monotone subsequences in random words

In previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition, particles beginning at the positive integers. There it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a Fredholm determinant. Here we use this formula to obtain three asymptotic results for the positions of these particles. In one an apparently new distribution function arises and in another the distribution function F2 arises. The latter extends a result of Johansson on TASEP to ASEP, and hence proves KPZ universality for ASEP with step initial condition.

Differential Equations for Dyson Processes

Abstract 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 double scaling limit at the “edge of the spectrum” leads to the Airy kernel [Ai(x)Ai′(y) − Ai′(x)Ai(y)]/(x−y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDEs 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 general n, of the probability that an interval contains precisely n eigenvalues.