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

top
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

top

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

top
  1. [1] L. Arnold. Random Dynamical Systems. Springer, Berlin, 2003. Zbl0906.34001MR1374107
  2. [2] S. Attal, A. Joye and C.-A. Pillet (Eds). Open Quantum Systems I-III. Lecture Notes in Mathematics 1880–1882. Springer, Berlin, 2006. MR2261249
  3. [3] E. A. Azoff. Borel measurability in linear algebra. Proc. Amer. Math. Soc. 42 (1974) 346–350. Zbl0286.15006MR327799
  4. [4] A. Beck and J. T. Schwartz. A vector-valued random ergodic theorem. Proc. Amer. Math. Soc. 8 (1957) 1049–1059. Zbl0084.13702MR98162
  5. [5] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics. Texts and Monographs in Physics 1,2, 2nd edition. Springer, New York, 1996. Zbl0905.46046
  6. [6] R. Bru, L. Elsner and M. Neumann. Convergence of infinite products of matrices and inner–outer iteration schemes. Electron. Trans. Numer. Anal. 2 (1994) 183–193. Zbl0852.65035MR1308895
  7. [7] M. Brune, J. M. Raimond and S. Haroche. Theory of the Rydberg-atom two-photon micromaser. Phys. Rev. A 35 (1987) 154–163. 
  8. [8] L. Bruneau, A. Joye and M. Merkli. Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239 (2006) 310–344. Zbl1118.81008MR2258226
  9. [9] L. Bruneau, A. Joye and M. Merkli. Random repeated interaction quantum systems. Comm. Math. Phys. 284 (2008) 553–581. Zbl1165.82018MR2448141
  10. [10] B. De Saporta, Y. Guivarc’h and E. LePage. On the multidimensional stochastic equation Y(n+1)=a(n)Y(n)+b(n). C. R. Math. Acad. Sci. Paris 339 (2004) 499–502. Zbl1063.60099MR2099549
  11. [11] P. Filipowicz, J. Javanainen and P. Meystre. Theory of a microscopic maser. Phys. Rev. A 34 (1986) 3077–3087. 
  12. [12] Y. Guivarc’h. Limit theorem for random walks and products of random matrices. In Proceedings of the CIMPA-TIFR School on Probability Measures on Groups, Recent Directions and Trends, September 2002255–330. TIFR, Mumbai, 2006. Zbl1247.60009MR2213480
  13. [13] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13–30. Zbl0127.10602MR144363
  14. [14] V. Jaksic and C.-A. Pillet. Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Comm. Math. Phys. 226 (2002) 131–162. Zbl0990.82017MR1889995
  15. [15] H. Kesten and F. Spitzer. Convergence in distribution of products of random matrices. Z. Wahrsch. Verw. Gebiete 67 (1984) 363–386. Zbl0535.60016MR761563
  16. [16] Y. Kifer and P.-D. Liu. Random dynamics. In Handbook of Dynamical Systems 1B 379–499. B. Hasselblatt and A. Katok (Eds). North-Holland, Amsterdam, 2006. Zbl1130.37301MR2186245
  17. [17] D. Meschede, H. Walther and G. Müller. One-atom maser. Phys. Rev. Lett. 54 (1985) 551–554. 
  18. [18] M. Merkli, M. Mück and I. M. Sigal. Instability of equilibrium states for coupled heat reservoirs at different temperatures. J. Funct. Anal. 243 (2007) 87–120. Zbl1122.81043MR2291433
  19. [19] A. Mukherjea. Topics in Products of Random Matrices. TIFR, Mumbai, 2000. Zbl0980.60013MR1759920
  20. [20] E. Seneta. Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York, 2000. Zbl1099.60004MR2209438
  21. [21] S. Schwarz. Infinite product of doubly stochastic matrices. Acta Math. Univ. Comenian. 39 (1980) 131–150. Zbl0521.15010MR619269
  22. [22] M. Weidinger, B. T. H. Varcoe, R. Heerlein and H. Walther. Trapping states in the micromaser. Phys. Rev. Lett. 82 (1999) 3795–3798. 
  23. [23] T. Wellens, A. Buchleitner, B. Kümmerer and H. Maassen. Quantum state preparation via asymptotic completeness. Phys. Rev. Lett. 85 (2000) 3361–3364. 

NotesEmbed ?

top

You must be logged in to post comments.