Infinite Products of Random Matrices and Repeated Interaction Dynamics

Description

Infinite Products of Random Matrices and Repeated Interaction Dynamics Laurent Bruneau, Alain Joye and Marco Merkli Prepublication de l'Institut Fourier no 698 (2007) www-fourier.ujf-grenoble.fr/prepublications.html Abstract Let ?n be a product of n independent, identically distributed random matrices M , with the properties that ?n is bounded in n, and that M has a deterministic (constant) invariant vector. Assuming that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish, we show that ?n 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 E[M ] 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 form of the limit states. Keywords: product of random matrices, dynamic quantum systems. Resume Soit ?n le produit de n matrices aleatoires M independantes et identiquement distribuees, avec la propriete que ?n est uniformement borne en n et que les ma- trices M admettent un meme vecteur deterministe invariant.

• random stochastic


• quantum systems


• systemes quantiques d'interactions repetees


• interaction open


• trons des theoremes ergodiques pour la dynamique


• inhomogeneous markov


• repeated interaction


• markov chains

Sujets

• Results
• Bruneau
• Markov chain

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

Infinite Products of Random Matrices and Repeated Interaction Dynamics

Laurent Bruneau, Alain Joye and Marco Merkli

Pr´epublicationdel’InstitutFouriern o 698 (2007) www-fourier.ujf-grenoble.fr/prepublications.html

Abstract

Let Ψ n be a product ofnindependent, identically distributed random matrices M, with the properties that Ψ n is bounded inn, and thatMhas a deterministic (constant) invariant vector. Assuming that the probability ofMhaving only the simple eigenvalue 1 on the unit circle does not vanish, we show that Ψ n is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast asn→ ∞. The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data ofE[M No] associated to 1. additional assumptions are made on the matricesM; 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 form of the limit states. Keywords of random matrices, dynamic quantum systems.: product

Re´sum´e

Soit Ψ n le produit denceriatmseriotae´lasMe´epdnnaetesitedntiquementdni distribue´es,aveclapropri´et´equeΨ n ´fibtromenenn´eestenet que les ma-un orm tricesMnettemdameˆenmturdeuctveppusnasonairnE.tteisvainte´einrmtqueM neposs`edequelavaleurpropre1surlecercleunite´etqu’elleestsimpleavecpro- babilite´positive,nousmontronsqueΨ n est la somme d’un processus fluctuant et d’unprocessusd´ecroissant.Cedernierconvergeversze´ropresquesuˆrement,expo- nentiellement vite sintend vers l’infini. La partie fluctuante converge en moyenne de C´esaroversunelimitequiestcaract´erise´eparlesdonne´esspectralesdel’esp´erance deM. On ne suppose ni que les matricesMveinntsos,leibrsinuqleuesre´´lmeents sontr´eels. Nousappliquonsnosre´sultatsg´en´eraux`adeuxclassesdesyste`mesdynamiques: leschaˆınesdeMarkovinhomoge`nesavecmatricesdetransitional´eatoiresetles syste`mesquantiquesd’interactionsr´epe´t´eesal´eatoires.Danslesdeuxcasnousd´emon- tronsdesthe´ore`mesergodiquespourladynamiqueetde´crivonslaformede´tats s e asymptotiques. Mots-cle´s:lasetae´erioys,sodprtsuimadeictrtnqiqsaueu.smesdst`eiqueynam

2000 Mathematics Subject Classification: 60H25, 37A30.

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1

Introduction

Pr´epublicationdel’InstitutFouriern o 698 – Mars 2007

In this paper we study products of infinitely many independent, identically distributed random matrices. The matrices we consider satisfy two basic properties, reflecting the fact that they describe the dynamics of random quantum or classical dynamical systems. The first property is that the norm of any product of such matrices is bounded uniformly in the number of factors. It reflects the fact that the underlying dynamics is in a certain sense norm-preserving. The second property is that there is a deterministic invariant vector. This represents a normalization of the dynamics. Two important examples of systems falling into this category are inhomogeneous Markov chains withrandom transition matrices of, i.e. productsrandom stochastic matrices, as well asrepeated interaction open quantum systems this paper we present general results on infinite. In products of random matrices, and applications of these results to the two classes of dynamical systems mentioned. Our main results are convergence theorems of the infinite random matrix product, Theorems 1.1, 1.2 and 1.3. They translate into ergodic theorems for the corresponding dynamical systems, Theorems 1.4 and 1.5.

1.1 General Results LetM(ω) be a random matrix onC d , with probability space (Ω,F,p). We say that M(ω) is arandom reduced dynamics operator(RRDO) if (1) There exists a norm||| ∙ |||onC d such that, for allω,M(ω) is a contraction onC d endowed with the norm||| ∙ |||.

(2) There is a vectorψ S , constant inω, such thatM(ω)ψ S =ψ S , for allω. We shall normalizeψ S such thatkψ S k= 1 wherek ∙ kdenotes the euclidean norm. To an RRDOM(ω), we associate the (iid)random reduced dynamics process(RRDP) Ψ n (ω) :=M(ω 1 )∙ ∙ ∙M(ω n ), ω∈Ω N ∗ . We show that Ψ n has a decomposition into an exponentially decaying part and a fluc- tuating part. To identify these parts, we proceed as follows. It follows from (1) and (2) that the spectrum of an RRDOM(ω) must lie inside the closed complex unit disk, and that 1 is an eigenvalue (with eigenvectorψ S ). LetP 1 (ω) denote the spectral projection of M(ω) corresponding to the eigenvalue one (dimP 1 (ω)≥1), and letP 1 ∗ (ω) be its adjoint operator. Define ψ(ω) :=P 1 (ω) ∗ ψ S ,(1)

and set P(ω) =|ψ S ihψ(ω)|. Forψ, φ∈C d , we denote by|ψihφ|the rank-one operator|ψihφ|χ=hφ, χiψ, and our convention is to take the inner products linear in the second factor. We put

Q(ωl)=1−P(ω).

Note that the vectorψ(ω) is normalized ashψ S , ψ(ω)i We= 1. decomposeM(ω) as M(ω) =P(ω) +Q(ω)M(ω)Q(ω) =:P(ω) +M Q (ω).(2)