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Exact and approximate results for deposition and annihilation processes
on graphs

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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 *Z*^{2}.
We also give normal approximation results for
the number of occupied sites in a large finite graph.