Tableaux and the Asymmetric Simple Exclusion Process

Various types of tableaux have recently been introduced due to a connection with the asymmetric simple exclusion process (ASEP) and have been the object of study in many recent papers. Relevant to this thesis, there have been several conjectures made regarding two such types of tableaux, namely staircase tableaux and tree--like tableaux. This thesis confirms these conjectures while proving other interesting results. More specifically, Hitczenko and Janson proved that distribution of symbols on the first diagonal of staircase tableaux is asymptotically normal, and they conjectured that other diagonals would be asymptotically Poisson. This thesis proves that conjecture for the kth diagonal where k is fixed. In addition, Laborde Zubieta gave a conjecture on the total number of corners in tree--like tableaux and the total number of corners in symmetric tree--like tableaux. Both conjectures are proven in this thesis. The proofs of these two conjectures are based on bijections with permutation tableaux and type--B permutation tableaux and consequently, results for these tableaux are also given. In addition, the limiting distributions of the number of occupied corners in tree--like tableaux and the number of diagonal boxes in symmetric tree--like tableaux are derived. These theorems extend results of Laborde-Zubieta and Aval et al. respectively.Ph.D., Mathematics -- Drexel University, 201