The Dyson Brownian minor process

Abstract

Consider an $n\times n$ Hermitean matrix valued stochastic process $\{H_t{\}}_{t\ge 0}$ where the matrix 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 $k\times k$ in the upper left corner of $H_t$. If you project this process to a space-like path it is a determinantal process and we compute the kernel. This kernel contains as special cases the well known GUE minor kernel, discovered by Johansson-Nordenstam and Okounkov-Reshetikhin in 2006, and the Dyson Brownian motion kernel discovered by Forrester-Nagao in 1998. In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier's bead kernel.

30 pages, 2 figures

References in corpus (6)

• Discrete polynuclear growth and determinantal processes
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Random matrices and determinantal processes
• The bead model and limit behaviors of dimer models
• Determinantal Correlations for Classical Projection Processes
• Random walks conditioned to stay in Weyl chambers of type C and D

Cited by in corpus (5)

• From interacting particle systems to random matrices
• Cycles and eigenvalues of sequentially growing random regular graphs
• Random matrix minor processes related to percolation theory
• Pfaffian Stochastic Dynamics of Strict Partitions
• Eigenvalue fluctuations for random regular graphs