The Dyson Brownian Minor Process

Mark Adler[1]; Eric Nordenstam[2]; Pierre Van Moerbeke[3]

  • [1] Brandeis University Department of Mathematics Waltham, Mass 02454 (USA)
  • [2] Universität Wien Fakultät für Mathematik Oscar-Morgenstern-Platz 1 1090 Wien (Austria)
  • [3] Université de Louvain Department of Mathematics 1348 Louvain-la-Neuve (Belgium) Brandeis University Waltham, Mass 02454 (USA)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 971-1009
  • ISSN: 0373-0956

Abstract

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Consider an n × n Hermitean matrix valued stochastic process { H t } t ≥ 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 × 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.

How to cite

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Adler, Mark, Nordenstam, Eric, and Van Moerbeke, Pierre. "The Dyson Brownian Minor Process." Annales de l’institut Fourier 64.3 (2014): 971-1009. <http://eudml.org/doc/275428>.

@article{Adler2014,
abstract = {Consider an $n\times n$ Hermitean matrix valued stochastic process $\lbrace H_t\rbrace _\{t\ge 0\}$ 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.},
affiliation = {Brandeis University Department of Mathematics Waltham, Mass 02454 (USA); Universität Wien Fakultät für Mathematik Oscar-Morgenstern-Platz 1 1090 Wien (Austria); Université de Louvain Department of Mathematics 1348 Louvain-la-Neuve (Belgium) Brandeis University Waltham, Mass 02454 (USA)},
author = {Adler, Mark, Nordenstam, Eric, Van Moerbeke, Pierre},
journal = {Annales de l’institut Fourier},
keywords = {Dyson’s Brownian motion; bead kernel; extended kernels; Gaussian Unitary Ensemble; Dyson Brownian minor process; determinantal processes; correlation kernel; point processes},
language = {eng},
number = {3},
pages = {971-1009},
publisher = {Association des Annales de l’institut Fourier},
title = {The Dyson Brownian Minor Process},
url = {http://eudml.org/doc/275428},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Adler, Mark
AU - Nordenstam, Eric
AU - Van Moerbeke, Pierre
TI - The Dyson Brownian Minor Process
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 971
EP - 1009
AB - Consider an $n\times n$ Hermitean matrix valued stochastic process $\lbrace H_t\rbrace _{t\ge 0}$ 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.
LA - eng
KW - Dyson’s Brownian motion; bead kernel; extended kernels; Gaussian Unitary Ensemble; Dyson Brownian minor process; determinantal processes; correlation kernel; point processes
UR - http://eudml.org/doc/275428
ER -

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