  The asymmetric simple exclusion process (ASEP) for N particles on a ring with
L sites may be analyzed using the Bethe ansatz. In this paper, we provide a
rigorous proof that the Bethe ansatz is complete for the periodic ASEP. More
precisely, we show that for all but finitely many values of the hopping rate,
the solutions of the Bethe ansatz equations do indeed yield all L choose N
eigenstates. The proof follows ideas of Langlands and Saint-Aubin, which draw
upon a range of techniques from algebraic geometry, topology and enumerative
combinatorics.
