Young Tableaux and Walks in the Open and periodic TASEP

The Totally Asymmetric Simple Exclusion Process (TASEP) is a paradigmatic model in out-of-equilibrium statistical physics, admitting exact results for its stationary distribution and, in some settings, for finite-time transition probabilities. Since the stationary distribution of the open-boundary TASEP was explicitly determined by Derrida et al., the model has provided a bridge between enumerative combinatorics and statistical physics.
To date, much of the literature on exact finite-size results has focused on the stationary distribution, while comparatively less is known about the transient dynamics. In this seminar, results concerning the enumeration of transition sequences in the state space of the open TASEP will be presented, extending a correspondence already known in the periodic case between TASEP walks and Young tableaux. Unexpectedly, the combinatorial structures arising in the open-boundary setting have themselves been the subject of recent work in enumerative combinatorics. Results from this literature then lead to nontrivial consequences on the structure of the open TASEP.