Integrable models for the vicious and friendly walkers

Random walks of the essentially different classes of random walkers, namely of the vicious and of the friendly ones, on the one-dimensional lattices with the periodic boundary conditions are considered. The walkers are called vicious since arriving on the same lattice site they annihilate not only one another but all the rest as well. On the contrary, the arbitrary number of the friendly walkers can share the same lattice sites. It is shown that the natural model describing the behavior of the friendly walkers is the integrable model of the boson type. The representation of the generating function for the number of the lattice paths made by the fixed number of the friendly walkers for the certain number of steps is obtained.