The Matrix Product Ansatz from a probabilistic viewpoint

In this thesis, we provide a probabilistic characterization of the class of probability measures that can be represented by the Matrix Product Ansatz (MPA). We introduce a constructive procedure, based on a suitable enlargement of the state space, showing that a probability measure admits a representation in terms of non-negative matrices via the MPA if and only if it can be expressed as a mixture of inhomogeneous product measures, where the mixing law is given by a Markov bridge. We illustrate this construction by applying it to several examples of interacting particle systems. Finally, we exploit the resulting probabilistic structure to derive large deviation principles for this class of measures.