arXiv subjects

Harold Widom

Publications and source records attributed to Harold Widom.

75 records · Page 5Linked to original sources

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↗