The Dyson Brownian Minor Process

Adler, Mark; Nordenstam, Eric; Van Moerbeke, Pierre

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 971-1009.

Nous considérons une processus stochastique fourni par une matrice ${\left\{H_t\right\}}_{t\ge 0}$ de taille $n$, dont les éléments évoluent selon un processus d’Ornstein-Uhlenbeck. Les valeurs propres de $H_t$ évoluent selon un mouvement Brownien de Dyson, c’est-à-dire qu’elles décrivent $n$ processus d’Ornstein-Uhlenbeck répulsifs.

Dans cet article, nous considérons non seulement les valeurs propres de la matrice elle-même, mais aussi les valeurs propres combinées avec celles des mineurs principaux  ; c’est-à-dire les valeurs propres des sous-matrices dans le coin supérieur gauche de la matrice $H_t$. Ce processus, projeté sur des chemins “spatiaux” appropriés, est un processus déterminantal dont nous fournissons le noyau  ; en outre, le noyau GUE-mineur et le noyau du processus de Dyson apparaissent tous deux comme des cas particuliers.

La limite dans le “bulk” de ce noyau fournit une généralisation, dépendante du temps, du noyau “bead” de Boutillier.

Nous calculons également le noyau pour un processus de mouvements browniens entrelacés introduit par Warren  ; celui-ci est également un processus déterminantal le long de chemins spatiaux.

Consider an $n\times n$ Hermitean matrix valued stochastic process ${\left\{H_t\right\}}_{t\ge 0}$ where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect.

In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k\times k$ minors in the upper left corner of $H_t$. Projecting this process to a space-like path leads to a determinantal process for which we compute the kernel. This kernel contains the well known GUE minor kernel, and the Dyson Brownian motion kernel as special cases.

In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier’s bead kernel.

We also compute the kernel for a process of intertwined Brownian motions introduced by Warren. That too is a determinantal process along space-like paths.

Classification : 60B20, 60G55, 60J65, 60J10
Keywords: Dyson’s Brownian motion, bead kernel, extended kernels, Gaussian Unitary Ensemble
Mot clés : Mouvement Brownien de Dyson, le noyau “bead”, noyaux étendus, l’ensemble unitaire gaussien (GUE)

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