The evolution of subcritical Achlioptas processes

In Achlioptas processes, starting from an empty graph, in each step two
potential edges are chosen uniformly at random, and using some rule one of them
is selected and added to the evolving graph. Although the evolution of such
`local' modifications of the Erd{\H o}s--R\'enyi random graph process has
received considerable attention during the last decade, so far only rather
simple rules are well understood. Indeed, the main focus has been on
`bounded-size' rules, where all component sizes larger than some constant $B$
are treated the same way, and for more complex rules very few rigorous results
are known.
In this paper we study Achlioptas processes given by (unbounded) size rules
such as the sum and product rules. Using a variant of the neighbourhood
exploration process and branching process arguments we show that certain key
statistics are tightly concentrated at least until the susceptibility (the
expected size of the component containing a randomly chosen vertex) diverges.
Our convergence result is most likely best possible for certain rules: in the
later evolution the number of vertices in small components may not be
concentrated. Furthermore, we believe that for a large class of rules the
critical time where the susceptibility `blows up' coincides with the
percolation threshold.