Laboratoire de Physique Théorique
Toulouse - UMR 5152

Finding statistical descriptions of non-equilibrium stationary state is often an arduous task. For specific systems, like ASEP or 1D diffusion-reaction systems, stationary solutions have been characterized using matrix product representations. These representations generalize the product structure of independent random variable to matrices ; the non-commutativity of matrices generates correlation while preserving many of the algebraic properties of expectation. Depending on the matrix structure, the correlation can vary from short-range to long-range correlation. Moreover, from a signal processing perspective, these matrix-correlated random variables can be recast as specific Hidden Markov Models. In this talk, we propose to investigate the general statistical properties of this mathematical framework, with the long-term hope to improve our understanding of related physical systems. In particular, we shall focus on the statistical properties of sums of such random variables. Do we have a large deviation principle for this sum ? Can we find analogous of the law of large number or the central limit theorem ?