Products of irreducible random matrices in the (max,+) algebra

International audienceWe consider the recursive equation ``x(n+1)=A(n)x(n)'' where x(n+1) and x(n) are column vectors of size k and where A(n) is an irreducible random matrix of size k x k. The matrix-vector multiplication in the (max,+) algebra is defined by (A(n)x(n))_i= max_j [ A(n)_{ij} +x(n)_j ]. This type of equation can be used to represent the evolution of Stochastic Event Graphs which include cyclic Jackson Networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence (A(n))_n is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices C such that P { A(0) in C } > 0, and the matrices in C have a unique periodic regime

Localization for a matrix-valued Anderson model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr\"odinger operators, acting on L^2(\R)\otimes \C^N, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the F\"urstenberg group of these operators, valid on an interval $I\subset \R$, they exhibit localization properties on $I$, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the F\"urstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters

On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

Given a discrete-time linear switched system $\Sigma(\mathcal A)$ associated with a finite set $\mathcal A$ of matrices, we consider the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius $\rho_{\mathrm d}(\mathcal A)$ and, on the other hand, its probabilistic joint spectral radii $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for Markov random switching signals with transition matrix $P$ and a corresponding invariant probability $\nu$. Note that $\rho_{\mathrm d}(\mathcal A)$ is larger than or equal to $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for every pair $(\nu, P)$. In this paper, we investigate the cases of equality of $\rho_{\mathrm d}(\mathcal A)$ with either a single $\rho_{\mathrm p}(\nu,P,\mathcal A)$ or with the supremum of $\rho_{\mathrm p}(\nu,P,\mathcal A)$ over $(\nu,P)$ and we aim at characterizing the sets $\mathcal A$ for which such equalities may occur

Orbit measures, random matrix theory and interlaced determinantal processes

A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal