11:00-12:00 Tuesday, 6 May 2025 Auditorium, Rectorate
Abstract
A rational model is a probability measure on sequences of symbols that assigns the weights in terms of products of matrices with nonnegative entries and having rows and columns labeled by a finite or countable set. We prove large deviations principles for several empirical measures of rational models. Our approach is based on a simultaneous combination of an enlargement of the state space together with a spectral conjugation that produces a stochastic matrix. As a result we describe a rational model as an hidden Markov measure and can deduce the large deviations asymptotic by contraction from the enlarged Markov sequence. The measure on the enlarged state space is a Markov bridge. The invariant measures of several non equilibrium models of interacting particle systems can be represented by the so called Matrix Product Ansatz that corresponds to a rational model with countable infinite matrices. In the finite case we give a variational formula both for the algebraic and the spatial empirical measures, that can be solved in special cases. For the infinite case we illustrate the method using a solvable example that is the invariant measure of the boundary driven TASEP model. We recover in this way the classic result by Derrida, Lebowitz, Speer in particular we obtain a variational representation of the rate function similar to that recently obtained by Bryc. Our approach is general and can be applied to any measure represented by the matrix product ansatz with matrices having positive entries. Joint work with Federica Iacovissi.