UC Berkeley

The Asymmetric Simple Exclusion Process (ASEP) is a process from statistical physics that describes the dynamics of interacting particles hopping right and left on a one-dimensional finite lattice with open boundaries. The ASEP is a Markov chain on $2^n$ states denoted by words of length $n$ in particles and holes with three hopping parameters $\alpha, \beta$, and $q$. Particles may enter at the left with rate $\alpha$, they may exit at the right with rate $\beta$, and in the bulk particles can hop to an empty location to the right with rate 1 and to the left with rate $q$.

A main goal in the study of the ASEP is to discover concrete formulae that compute its steady state probabilities. One can compute these probabilities as sums over combinatorial objects such as the \emph{alternative tableaux}. In Chapter 2, we give a determinantal formula for the weight generating function of these tableaux at $q=0$, and thus explicitly compute the steady state probabilities for the ASEP at $q=0$.

The two-species ASEP is a generalization in which there are two species of particles, \emph{heavy} and \emph{light}. Only the \emph{heavy} particles are able to enter and exit at the left and right of the lattice and with rates $\alpha$ and $\beta$, respectively. If particles of two different species are adjacent, they can swap with rate 1 if the heavier particle is on the left, and rate $q$ if it is on the right. In Chapter 3, we give a combinatorial formula for the steady state probabilities of the two-species ASEP at by introducing the \emph{rhombic alternative tableaux} of Figure \ref{at_examples_abs} (b). We show that the weight generating function of these tableaux gives a formula for the steady state probabilities of the two-species ASEP. We give a second proof of this tableaux formula by constructing a Markov Chain on the rhombic alternative tableaux that projects to the two-species ASEP.

In Chapter 4, we introduce a $k$-species ASEP that generalizes the two-species ASEP. We prove a Matrix Ansatz that expresses the steady state probabilities of states of this $k$-species ASEP as a certain matrix product, which generalizes an analogous result for the two-species ASEP. In this $k$-species ASEP, there are $k$ species of particles of varying heaviness. As with the two-species ASEP, only the heaviest particle is allowed to enter and exit at the boundaries of the lattice, with the same respective rates $\alpha$ and $\beta$. Moreover, adjacent particles of different species can swap with rate 1 if the heavier particle is on the left, and rate $q$ if it is on the right. Using the generalized Matrix Ansatz, we introduce tableaux called the \emph{$k$-rhombic tableaux}, which give a combinatorial formula for the probabilities of the $k$-species ASEP.