Infinite products of random matrices and repeated interaction dynamics
Bruneau, Laurent ; Joye, Alain ; Merkli, Marco
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010), p. 442-464 / Harvested from Numdam
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On considère le produit Ψn de n matrices aléatoires M i.i.d. ayant les propriétés suivantes: la suite (Ψn)n est bornée et M possède un vecteur invariant déterministe (constant). On suppose que la probabilité pour que, sur le cercle unité, M possède une unique valeur propre simple en 1 soit non nulle. On montre que Ψn est la somme d'un processus fluctuant et d'un processus décroissant. Ce dernier tend vers zéro presque sûrement et exponentiellement rapidement lorsque n tend vers l'infini. Le terme fluctuant converge en moyenne de Cesaro vers une limite caractérisée explicitement par le vecteur invariant déterministe et par les données spectrales associées à la valeur propre 1 de la matrice . Aucune hypothèse supplémentaire n'est faite sur les matrices M; elles peuvent être à valeurs complexes et pas nécessairement inversibles. On applique les résultats généraux à deux classes de systèmes dynamiques: les chaînes de Markov inhomogènes avec matrices de transition aléatoires (matrices stochastiques), ainsi que les systèmes quantiques avec interactions répétées et aléatoires. Dans les deux cas, on prouve des résultats d'ergodicité pour la dynamique et on détermine les états limites.

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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1214/09-AIHP211
Classification:  60H25,  37A30 
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     author = {Bruneau, Laurent and Joye, Alain and Merkli, Marco},
     title = {Infinite products of random matrices and repeated interaction dynamics},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {46},
     year = {2010},
     pages = {442-464},
     doi = {10.1214/09-AIHP211},
     mrnumber = {2667705},
     zbl = {1208.60064},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2010__46_2_442_0}
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Bruneau, Laurent; Joye, Alain; Merkli, Marco. Infinite products of random matrices and repeated interaction dynamics. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) pp. 442-464. doi : 10.1214/09-AIHP211. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_2_442_0/

[1] L. Arnold. Random Dynamical Systems. Springer, Berlin, 2003. | MR 1374107 | Zbl 0906.34001

[2] S. Attal, A. Joye and C.-A. Pillet (Eds). Open Quantum Systems I-III. Lecture Notes in Mathematics 1880-1882. Springer, Berlin, 2006. | MR 2261249

[3] E. A. Azoff. Borel measurability in linear algebra. Proc. Amer. Math. Soc. 42 (1974) 346-350. | MR 327799 | Zbl 0286.15006

[4] A. Beck and J. T. Schwartz. A vector-valued random ergodic theorem. Proc. Amer. Math. Soc. 8 (1957) 1049-1059. | MR 98162 | Zbl 0084.13702

[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. | Zbl 0905.46046

[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. | MR 1308895 | Zbl 0852.65035

[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] L. Bruneau, A. Joye and M. Merkli. Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239 (2006) 310-344. | MR 2258226 | Zbl 1118.81008

[9] L. Bruneau, A. Joye and M. Merkli. Random repeated interaction quantum systems. Comm. Math. Phys. 284 (2008) 553-581. | MR 2448141 | Zbl 1165.82018

[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. | MR 2099549 | Zbl 1063.60099

[11] P. Filipowicz, J. Javanainen and P. Meystre. Theory of a microscopic maser. Phys. Rev. A 34 (1986) 3077-3087.

[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 2002 255-330. TIFR, Mumbai, 2006. | MR 2213480 | Zbl 1247.60009 | Zbl pre05296024

[13] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-30. | MR 144363 | Zbl 0127.10602

[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. | MR 1889995 | Zbl 0990.82017

[15] H. Kesten and F. Spitzer. Convergence in distribution of products of random matrices. Z. Wahrsch. Verw. Gebiete 67 (1984) 363-386. | MR 761563 | Zbl 0535.60016

[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. | MR 2186245 | Zbl 1130.37301

[17] D. Meschede, H. Walther and G. Müller. One-atom maser. Phys. Rev. Lett. 54 (1985) 551-554.

[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. | MR 2291433 | Zbl 1122.81043

[19] A. Mukherjea. Topics in Products of Random Matrices. TIFR, Mumbai, 2000. | MR 1759920 | Zbl 0980.60013

[20] E. Seneta. Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York, 2000. | MR 2209438 | Zbl 1099.60004

[21] S. Schwarz. Infinite product of doubly stochastic matrices. Acta Math. Univ. Comenian. 39 (1980) 131-150. | MR 619269 | Zbl 0521.15010

[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] T. Wellens, A. Buchleitner, B. Kümmerer and H. Maassen. Quantum state preparation via asymptotic completeness. Phys. Rev. Lett. 85 (2000) 3361-3364.