Consecutive minors for Dyson's Brownian motions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Adler, Mark¹  Nordenstam, Eric³  van Moerbeke, Pierre^1,2 

[1] Brandeis Univ, Dept Math, Waltham, MA 02454 USA

[2] Univ Louvain, Dept Math, B-1348 Louvain, Belgium

[3] Univ Vienna, Fac Math, A-1090 Vienna, Austria

关键词: Dyson's Brownian motion;    Diffusions on spectra of minors;    Markov process;   

DOI  :  10.1016/j.spa.2014.01.008

来源: Elsevier

【 摘 要 】

In 1962, Dyson (1962) introduced dynamics in random matrix models, in particular into GUE (also for beta = 1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: beta = 1, 2, 4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for beta = 2, 4. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free