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Craig A. Tracy

Publications and source records attributed to Craig A. Tracy.

60 records · Page 4Linked to original sources
arXiv1995

Largest eigenvalue distribution in the double scaling limit of matrix models: A Coulomb fluid approach

Using thermodynamic arguments we find that the probability that there are no eigenvalues in the interval (-s,\infty) in the double scaling limit of Hermitean matrix models is O(exp(-s^{2m+1})) as s\to+\infty.Here m=1,2,3.. determine the m^{th} multi-critical point of the level density:σ(x)\sim b[1-(x/b)^2]^{m-1/2} and b^2\sim N.Furthermore,the size of the transition zone where the eigenvalue density becomes vanishingly small at the tail of the spectrum is \sim N^{(m-3/2)/(2m+1)} in agreement with earlier work based on the string equation.

hep-th↗
arXiv1993

Hamiltonian Structure of Equations Appearing in Random Matrices

The level spacing distributions in the Gaussian Unitary Ensemble, both in the ``bulk of the spectrum,'' given by the Fredholm determinant of the operator with the sine kernel ${\sin π(x-y) \over π(x-y)}$ and on the ``edge of the spectrum,'' given by the Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y) - \rm{Ai}(y) \rm{Ai}'(x) \over (x-y)}$, are determined by compatible systems of nonautonomous Hamiltonian equations. These may be viewed as special cases of isomonodromic deformation equations for first order $ 2\times 2 $ matrix differential operators with regular singularities at finite points and irregular ones of Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure is explained within the classical R-matrix framework as the equations induced by spectral invariants on the loop algebra ${\tilde{sl}(2)}$, restricted to a Poisson subspace of its dual space ${\tilde{sl}^*_R(2)}$, consisting of elements that are rational in the loop parameter.

hep-th↗
arXiv1992

Level-Spacing Distributions and the Airy Kernel

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sinπ(x-y)/π(x-y)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. 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, M{ô}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{é} 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.

hep-th↗
arXiv1992

Introduction to Random Matrices

These notes provide an introduction to the theory of random matrices. The central quantity studied is $τ(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/π} {\sinπ(x-y)\over x-y} χ_I(y)$. Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $χ_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in $I$ is equal to $τ(a)$. Also $τ(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$'s are the independent variables) that were first derived by Jimbo, Miwa, M{ô}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{é} V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$.

hep-th↗
arXiv1992

Level-Spacing Distributions and the Airy Kernel

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel $\sinπ(x-y)/π(x-y)$. Similarly a double scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. We announce analogies 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, M{ô}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{é} 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.

hep-th↗