On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond

We study the behavior of the chemical reactions $A+A\to A+S$ and $A+A\to S+S$ (where the reactive species $A$ and the inert species $S$ are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number $z$ and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the $A$ species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice ($z=2$). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.