Combinatorial Bethe ansatz and ultradiscrete Riemann theta function with   rational characteristics
  The U_q(\hat{sl}_2) vertex model at q=0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann theta function with rational characteristics. 
Atsuo Kuniba and Reiho Sakamoto