Infinite products of random matrices and repeated interaction dynamics

Laurent Bruneau; Alain Joye; Marco Merkli

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 2, page 442-464
  • ISSN: 0246-0203

Abstract

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Let Ψn be a product of n independent, identically distributed random matrices M, with the properties that Ψn is bounded in n, and that M has a deterministic (constant) invariant vector. Assume that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish. We show that Ψn is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as n→∞. The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data of associated to 1. No additional assumptions are made on the matrices M; they may have complex entries and not be invertible. We apply our general results to two classes of dynamical systems: inhomogeneous Markov chains with random transition matrices (stochastic matrices), and random repeated interaction quantum systems. In both cases, we prove ergodic theorems for the dynamics, and we obtain the limit states.

How to cite

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Bruneau, Laurent, Joye, Alain, and Merkli, Marco. "Infinite products of random matrices and repeated interaction dynamics." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 442-464. <http://eudml.org/doc/241798>.

@article{Bruneau2010,
abstract = {Let Ψn be a product of n independent, identically distributed random matrices M, with the properties that Ψn is bounded in n, and that M has a deterministic (constant) invariant vector. Assume that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish. We show that Ψn is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as n→∞. The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data of associated to 1. No additional assumptions are made on the matrices M; they may have complex entries and not be invertible. We apply our general results to two classes of dynamical systems: inhomogeneous Markov chains with random transition matrices (stochastic matrices), and random repeated interaction quantum systems. In both cases, we prove ergodic theorems for the dynamics, and we obtain the limit states.},
author = {Bruneau, Laurent, Joye, Alain, Merkli, Marco},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {products of random matrices; random dynamical systems; random stochastic matrix; ergodic theory; linear random dynamical systems; random `stochastic matrix'; ergodic theorems},
language = {eng},
number = {2},
pages = {442-464},
publisher = {Gauthier-Villars},
title = {Infinite products of random matrices and repeated interaction dynamics},
url = {http://eudml.org/doc/241798},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Bruneau, Laurent
AU - Joye, Alain
AU - Merkli, Marco
TI - Infinite products of random matrices and repeated interaction dynamics
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 2
SP - 442
EP - 464
AB - Let Ψn be a product of n independent, identically distributed random matrices M, with the properties that Ψn is bounded in n, and that M has a deterministic (constant) invariant vector. Assume that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish. We show that Ψn is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as n→∞. The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data of associated to 1. No additional assumptions are made on the matrices M; they may have complex entries and not be invertible. We apply our general results to two classes of dynamical systems: inhomogeneous Markov chains with random transition matrices (stochastic matrices), and random repeated interaction quantum systems. In both cases, we prove ergodic theorems for the dynamics, and we obtain the limit states.
LA - eng
KW - products of random matrices; random dynamical systems; random stochastic matrix; ergodic theory; linear random dynamical systems; random `stochastic matrix'; ergodic theorems
UR - http://eudml.org/doc/241798
ER -

References

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