Operator Spreading in Random Unitary Circuits

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 \(\sin\pi(x-y)/\pi (x-y)\). 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{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} 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.

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.