UC Berkeley

We study various combinatorial formulas arising in the asymmetric exclusion process, orthogonal polynomials, and Ehrhart theory. In particular, we give combinatorial formulas for polynomials with positive coefficients. Explaining the positivity of such polynomials is an interesting problem by itself, and giving combinatorial formulas is useful as they provide a fast and compact way to compute those polynomials.

The asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of particles on a lattice hopping left and right. This process was introduced in the 1970s independently in the context of biology and mathematics. Since then, this model has many variants and was studied extensively in various fields. The ASEP on a line is a Markov chain on a one- dimensional lattice of length $N$ with open boundaries. A particle can hop to the right with the rate 1 and can hop to the left with rate $q$, as long as the neighboring site is empty. And on each boundary, a particle can enter from the left or right with rate $\alpha$ or $\delta$ respectively and can exit to the left or right with rate $\gamma$ or $\beta$ respectively. Papers of Sasamoto \cite{SS} and subsequently, Uchiyama, Sasamoto, and Wadati \cite{MU} revealed a surprising connection between the ASEP on a line and orthogonal polynomials, in particular the Askey-Wilson polynomials which lie in the top hierarchy of (basic) hypergeometric orthogonal polynomials in the sense that all other polynomials in this hierarchy are limiting cases or specializations of the Askey-Wilson polynomials. In Chapter 2, we give combinatorial formulas for the Al-Salam-Chihara polynomials, which are related to the ASEP when $\gamma=\delta=0$. The totally asymmetric exclusion process (TASEP) on a ring is a Markov chain on a periodic one-dimensional lattice of length $N$ where each lattice site can be either occupied by a particle or empty. A particle can hop to its right (when it is empty) with the rate 1. In Chapter 3, we study the inhomogeneous version of the TASEP and show that many steady-state probabilities are proportional to the product of Schubert polynomials.

In the 1960s, Ehrhart introduced Ehrhart polynomials and Ehrhart series to study the number of lattice points inside polytopes. Since then, there has been a lot of study on Ehrhart polynomials and Ehrhart series of many well-known polytopes. The $(k,n)$-$th$ $hypersimplex$ $\Delta_{k,n}$ is a lattice polytope inside $\mathbb{R}^n$ whose vertices are (0,1)-vectors with exactly $k$ 1's. The hypersimplex can be found in several algebraic and geometric contexts, for example, as a moment polytope for the torus action on the Grassmannian, or as a weight polytope for the fundamental representation of $GL_n$. In Chapter4, we prove the first combinatorial formula for the Ehrhart series of the hypersimplex, proving a conjecture of Early \cite{Early1}.