Exact and approximate results for deposition and annihilation processes on graphs

By Mathew D. Penrose and Aidan Sudbury

We consider random sequential adsorption processes where the initially empty sites of a graph are irreversibly occupied, in random order, either by monomers which block neighbouring sites, or by dimers. We also consider a process where initially occupied sites annihilate their neighbours at random times. We verify that these processes are well-defined on infinite graphs, and derive forward equations governing joint vacancy/occupation probabilities. Using these, we derive exact formulae for occupation probabilities and pair correlations in Bethe lattices. For the blocking and annihilation processes we also prove positive correlations between sites an even distance apart, and for blocking we derive rigorous lower bounds for the site occupation probability in lattices, including a lower bound of 1/3 for Z2. We also give normal approximation results for the number of occupied sites in a large finite graph.