The purpose of this thesis is to study the asymmetric simple exclusion process, its asymptotics and some connections to other stochastic models. The text begins by giving some results on random matrix theory, such as the distribution function of the largest eigenvalue of a given random matrix.
This is followed by a short section on the totally asymmetric simple exclusion process, which is a stochastic model of fermionic particles jumping only in one direction on a one-dimensional lattice. The probability that a given particle has jumped m times is then shown to be equal to the distribution of the largest eigenvalue of a specific type of a random matrix analyzed earlier.
As this hints at some kind of universality, the particle model is then generalized to the asymmetric simple exclusion process, in which the particles can jump left or right. It turns out this model does not have the simple determinantal structure the earlier models had. The asymptotics of the model will then be analyzed, and it turns out there is a large universality class that encompasses all the models analyzed in the text.
The reader is expected to be familiar with basic measure theory and complex analysis.
