PDE's for the Dyson, Airy and Sine processes
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1835-1846.

In 1962, Dyson showed that the spectrum of a n×n random Hermitian matrix, whose entries (real and imaginary) diffuse according to n 2 independent Ornstein-Uhlenbeck processes, evolves as n non-colliding Brownian particles held together by a drift term. When n, the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend to the so-called Sine process. This lecture derives the distribution of the Airy Process at any given time and a PDE for the joint distribution at two different times. Similarly a PDE is found for the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process at different times t 1 and t 2 , 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 obtained enable one to compute the asymptotic behavior of the joint distribution and the covariances for these processes at different times t 1 and t 2 , when t 2 -t 1 .

En 1962, Dyson montre que le spectre d’une matrice aléatoire n×n, dont les éléments (réels et imaginaires) diffusent selon des processus d’Ornstein-Uhlenbeck indépendants, évolue selon n 2 particules browniennes, forcées à s’éviter. En laissant tendre la taille n des matrices vers l’infini et à l’issue d’un changement d’échelle espace-temps, on trouve que la plus grande valeur propre (edge) évolue selon le “processus d’Airy” et que les valeurs propres du milieu (bulk) évoluent selon le “processus Sinus”. Le processus d’Airy est un processus continu stationnaire non- markovien. Cet exposé décrit la distribution de ces processus en chaque moment t, ainsi que la distribution jointe à des moments différents t 1 et t 2 . La méthode consiste à calculer d’abord une EDP pour les probabilités jointes du processus de Dyson à des moments différents t 1 et t 2 ; ceci est basé sur le calcul de la probabilité jointe des valeurs propres d’une chaîne de deux matrices gaussiennes couplées. Cette équation différentielle est alors soumise à une analyse asymptotique, conformément aux changements d’échelle du bord et du milieu. Ces équations aux dérivées partielles permettent de calculer le comportement asymptotique de la covariance du processus à des moments differents t 1 et t 2 , lorsque t 2 -t 1 tend vers l’infini.

DOI: 10.5802/aif.2143
Classification: 60G60, 60G65, 35Q53, 60G10, 35Q58
Keywords: Dyson's Brownian motion, Airy process, coupled Gaussian hermitian matrices
Mot clés : mouvement Brownien de Dyson, processus d'Airy, matrices gaussiennes hermitiennes couplées
Adler, Mark 1

1 Brandeis University, department of mathematics, Waltham Mass 02454 (USA)
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Adler, Mark. PDE's for the Dyson, Airy and Sine processes. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1835-1846. doi : 10.5802/aif.2143. http://archive.numdam.org/articles/10.5802/aif.2143/

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