  We describe the effect of power-law initial distributions of clusters on
ordinary percolation and its generalizations, specifically, models of explosive
percolation processes based on local optimization. These aggregation processes
were shown to exhibit continuous phase transitions if the evolution starts from
a set of disconnected nodes. Since the critical exponents of the order
parameter in explosive percolation transitions turned out to be very small,
these transitions were first believed to be discontinuous. In this article we
analyze the evolution starting from clusters of nodes whose sizes are
distributed according to a power law. We show that these initial distributions
change dramatically the position and order of the phase transitions in these
problems. We find a particular initial power-law distribution producing a
peculiar effect on explosive percolation, namely before the emergence of the
percolation cluster, the system is in a "critical phase" with an infinite
generalized susceptibility. This critical phase is absent in ordinary
percolation models with any power-law initial conditions. The transition from
the critical phase is an infinite order phase transition, which resembles the
scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the
critical singularity of susceptibility at this peculiar infinite-order
transition in explosive percolation. It turns out that the susceptibility in
this situation does not obey the Curie-Weiss law.
