  The asymmetric simple exclusion exclusion process (ASEP) is a model of
particles hopping on a one-dimensional lattice of n sites. It was introduced
around 1970, and since then has been extensively studied by researchers in
statistical mechanics, probability, and combinatorics. Recently the ASEP on a
lattice with open boundaries has been linked to Koornwinder polynomials, and
the ASEP on a ring has been linked to Macdonald polynomials. In this article we
study the two-species asymmetric simple exclusion process (ASEP) on a ring, in
which two kinds of particles ("heavy" and "light"), as well as "holes," can hop
both clockwise and counterclockwise (at rates 1 or t depending on the particle
types) on a ring of n sites. We introduce some new tableaux on a cylinder
called cylindric rhombic tableaux (CRT), and use them to give a formula for the
stationary distribution of the two-species ASEP -- each probability is
expressed as a sum over all CRT of a fixed type. When lambda is a partition in
{0,1,2}^n, we then give a formula for the nonsymmetric Macdonald polynomial
E_{lambda} and the symmetric Macdonald polynomial P_{lambda} by refining our
tableaux formulas for the stationary distribution.
