M ar 2 00 4 PDE ’ s for the joint distributions of the Dyson , Airy and Sine processes

In a celebrated paper, Dyson shows that the spectrum of a n× n randomHermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n non-colliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDE’s enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t1 and t2, when t2 − t1 → ∞, as illustrated in this paper for the Airy process. ∗Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: [email protected]. The support of a National Science Foundation grant # DMS01-00782 is gratefully acknowledged. †Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. This work was done while PvM was a member of the Clay Mathematics Institute, One Bow Street, Cambridge, MA 02138, USA. E-mail: [email protected] and @brandeis.edu. The support of a National Science Foundation grant # DMS-01-00782, European Science Foundation, Nato, FNRS and Francqui Foundation grants is gratefully acknowledged. 1