Diffusion-limited one-species reactions in the Bethe lattice

We study the kinetics of diffusion-limited coalescence, A + A. A, and annihilation, A + A -> 0, in the Bethe lattice of coordination number z. Correlations build up over time so that the probability of finding a particle next to another varies from rho(2) (rho is the particle density), initially, when the particles are uncorrelated, to [(z-2)/z]rho(2), in the long-time asymptotic limit. As a result, the particle density decays inversely proportionally to time, rho similar to 1/kt, but at a rate k that slowly decreases to an asymptotic constant value.