PDEs for the joint distributions of the Dyson, Airy and Sine processes

Details PDEs for the joint distributions of the Dyson, Airy and Sine processes PDEs for the joint distributions of the Dyson, Airy and Sine processes

2004-03-29

arxiv.org/abs/math/0403504v4

Annals of Probability 2005, Vol. 33, No. 4, 1326-1361

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000107 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000107

In a celebrated paper, Dyson shows that the spectrum of an n\times n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n noncolliding 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 PDEs enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t_1 and t_2, when t_2-t_1\to \infty, as illustrated in this paper for the Airy process. This paper also contains a rigorous proof that the extended Hermite kernel, governing the joint probabilities for the Dyson process, converges to the extended Airy and Sine kernels after the appropriate rescalings.

Affiliated with

Also associated with

No associations

Rating

PDEs for the joint distributions of the Dyson, Airy and Sine processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with PDEs for the joint distributions of the Dyson, Airy and Sine processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and PDEs for the joint distributions of the Dyson, Airy and Sine processes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-78774