A tropical analogue of Fay's trisecant identity and the ultra-discrete   periodic Toda lattice
  We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function. 
Rei Inoue and Tomoyuki Takenawa