Physique statistique, traitement du signal et théorie de l'information
Statistical physics and signal processing in interaction
Interactions between tools and concepts of Statistical Physics and Signal Processing have been explored to see how it can help to analyze and understand situations and models (inspired from Statistical Physics) where usual convergence theorems fail. Studies of independent random variables raised to a power depending on the sample size were shown to yield non standard limit distributions for the maximum. For sums, it provided a link between linearization effect in moment estimation and glass transition in statistical physics. In addition, it formalized the existence of an intrinsic critical moment order for a multifractal process, thus comforting earlier results. A critical moment estimator has been defined and studied for a class of independent (yet with intricate marginal dis- tribution) random variables. A class of random variables with intricate correlation has been studied, whose joint distributions is written as a product of matrices and which can have long range correlations. This model can also be recast into the framework of Hidden Markov Chain models, leading to theoretical design and actual synthesis. The limit behavior of the sum of such random variables has been characterized, both using rescaled limit distributions and large deviations.
Statistical Physics and Information Theory
(N. Fourcaud-Trocme, N. Buonviso, CRNL, Lyon) Entropy creation and exchange in non-equilibrium systems is parallel to Information creation, stockage and interactions in dynamical systems. In particular, biological systems are on one side fluctuating and creating information, while on the other side they interact not only with external thermostat and therefore noise, but also with other (possibly similar) systems. We will develop theoretical and actual practical estimation tools to quantify information as well as information fluxes between biological systems. Notably, we will use these tools to decipher not only functional connectivity maps, but also effective connectivity and produce directed graph of interacting systems.