  It is widely believed that certain simple modifications of the random graph
process lead to discontinuous phase transitions. In particular, starting with
the empty graph on $n$ vertices, suppose that at each step two pairs of
vertices are chosen uniformly at random, but only one pair is joined, namely,
one minimizing the product of the sizes of the components to be joined. Making
explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas,
D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there
exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability
the order of the largest component "jumps" from $o(n)$ to at least $\delta n$
in $o(n)$ steps of the process, a phenomenon known as "explosive percolation."
We give a simple proof that this is not the case. Our result applies to all
"Achlioptas processes," and more generally to any process where a fixed number
of independent random vertices are chosen at each step, and (at least) one edge
between these vertices is added to the current graph, according to any (online)
rule. We also prove the existence and continuity of the limit of the rescaled
size of the giant component in a class of such processes, settling a number of
conjectures. Intriguing questions remain, however, especially for the product
rule described above.
