Matrix Product Ansatz, Hidden Markov Chain process and sum of correlated random variables

par Eric Bertin (LIPhy , Univ. Grenoble Alpes)

mardi 11 oct. 2016, 11:00 → 12:30 Europe/Paris

Salle des Sommets (Annecy-le-Vieux)

The Matrix Product Ansatz method, that bears some connections to integrable quantum systems, has proven to be an efficient way to determine the exact stationary probability distribution of classical one-dimensional systems that are driven out of equilibrium by boundary reservoirs or external forces. More generally, the Matrix Product Ansatz can be considered as a convenient parameterization of correlated random variables, and can be studied in itself, without any connection to a specific physical model. In this general setting, we show how the Matrix Product Ansatz can be reformulated in terms of Hidden Markov Chain processes. This reformulation allows one to understand the statistics of the sum of correlated random variables described by a Matrix Product Ansatz. We show in particular that the statistical properties of the sum are deeply connected to the ergodic or non-ergodic character of the associated Hidden Markov Chain. In addition, we also determine the large deviation function of the sum using algebraic methods.