  There is still much to discover about the mechanisms and nature of
discontinuous percolation transitions. Much of the past work considers graph
evolution algorithms known as Achlioptas processes in which a single edge is
added to the graph from a set of $k$ randomly chosen candidate edges at each
timestep until a giant component emerges. Several Achlioptas processes seem to
yield a discontinuous percolation transition, but it was proven by Riordan and
Warnke that the transition must be continuous in the thermodynamic limit.
However, they also proved that if the number $k(n)$ of candidate edges
increases with the number of nodes, then the percolation transition may be
discontinuous. Here we attempt to find the simplest such process which yields a
discontinuous transition in the thermodynamic limit. We introduce a process
which considers only the degree of candidate edges and not component size. We
calculate the critical point $t_{c}=(1-\theta(\frac{1}{k}))n$ and rigorously
show that the critical window is of size $O(\frac{n}{k(n)})$. If $k(n)$ grows
very slowly, for example $k(n)=\log n$, the critical window is barely sublinear
and hence the phase transition is discontinuous but appears continuous in
finite systems. We also present arguments that Achlioptas processes with
bounded size rules will always have continuous percolation transitions even
with infinite choice.
