Infinite Products of Random Matrices and Repeated Interaction Dynamics

This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\in\mathbb{N}}$ is an ergodic sequence of positive operators on the space of signed measures on a space $\mathbb{X}$. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form \[ \mu M_{0,n} \simeq \mu(\tilde{h}) r_n \pi_n,\] where $\tilde{h}$ is a random bounded...

Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighborhood of order $\lambda>0$ of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in $\lambda$ and this requires a detailed analysis of the associated random dynamical system on the unit circle and its inv...

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The maximum absolute values of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the...

In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by $n$ square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and $m$ may change with the dimension of the product matrix and diverge. We give a complete...

In this article we consider products of real random matrices with fixed size. Let $A_1,A_2, \dots $ be i.i.d $k \times k$ real matrices, whose entries are independent and identically distributed from probability measure $\mu$. Let $X_n = A_1A_2\dots A_n$. Then it is conjectured that $$\mathbb{P}(X_n \text{ has all real eigenvalues}) \rightarrow 1 \text{ as } n \rightarrow \infty.$$ We show that the conjecture is true when $\mu$ has an atom.

Motivated by the theory of inhomogeneous Markov chains, we determine a sufficient condition for the convergence to 0 of a general product formed from a sequence of real or complex matrices. When the matrices have a common invariant subspace $H$, we give a sufficient condition for the convergence to 0 on $H$ of a general product. Our result is applied to obtain a condition for the weak ergodicity of an inhomogeneous Markov chain. We compare various types of contractions which ...

Let $ \nu $ be a probability distribution over the linear semi-group $ \mathrm{End}(E) $ for $ E $ a finite dimensional vector space over a locally compact field. We assume that $ \nu $ is proximal, strongly irreducible and that $ \nu^{*n}\{0\}=0 $ for all integers $ n\in\mathbb{N} $. We consider the random sequence $ \overline\gamma_n := \gamma_0 \cdots \gamma_{n-1} $ for $ (\gamma_k)_{k \ge 0} $ independents of distribution law $ \nu $. We define the logarithmic singular ga...

The statistical behaviour of a product of independent, identically distributed random matrices in $\text{SL}(2,{\mathbb R})$ is encoded in the generalised Lyapunov exponent $\Lambda$; this is a function whose value at the complex number $2 \ell$ is the logarithm of the largest eigenvalue of the transfer operator obtained when one averages, over $g \in \text{SL}(2,{\mathbb R})$, a certain representation $T_\ell (g)$ associated with the product. We study some products that aris...

The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. We consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence. Ergodicity includes and vastly generalizes random independence. We obtain a theorem which shows that the composition of such a sequence of channels converges exponentially fast to a replacement (r...

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\Pi_n=M_nM_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {\bf 10}, 15 (1965)], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering produc...