Continuity of fixed-choice percolation

A fixed-choice Achlioptas process cannot have its largest component of a graph jump from $o(n)$ to a fixed positive fraction of $n$ within $o(n)$ steps, with high probability. The proof combines forced merging of large components and persistence of unsampled graph components. An assumed jump forces positive vertex mass in each band $[k,Dk)$ slightly before the jump: insufficient smaller components can feed the final large component, while enough components above $Dk$ would merge too early. Persistence carries a fixed positive amount of each band to a common time. Taking sufficiently many disjoint bands with $k=1,D,D^2,\dots$ then counts more than $n$ vertices. The number of offered candidates is fixed throughout this argument. Riordan and Warnke establish this obstruction and broader continuity results in their original research paper.