A class of solvable reaction–diffusion processes on a Cayley tree

Considering the most general one-species reaction–diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e. ∘∘→•∘∘∘→•∘, ∘∘→••∘∘→•• and ∘•→••∘•→••, and in the second model, only the diffusion process •∘→∘••∘→∘• exists. For the first model, the probabilities Pl(m;t)Pl(m;t), of finding mm particles on the llth shell of the Cayley tree, have been found exactly, and for the second model, the functions Pl(1;t)Pl(1;t) have been calculated. It has been shown that these are the only integrable models if one restricts consideration to the L+1L+1-shell probabilities P(m0,m1,…,mL;t)P(m0,m1,…,mL;t).