  In this paper, we study the critical behavior of percolation on a
configuration model with degree distribution satisfying an infinite
second-moment condition, which includes power-law degrees with exponent $\tau
\in (2,3)$. It is well known that, in this regime, many canonical random graph
models, such as the configuration model, are robust in the sense that the giant
component is not destroyed when the percolation probability stays bounded away
from zero. Thus, the critical behavior is observed when the percolation
probability tends to zero with the network size, despite of the fact that the
average degree remains bounded.
  In this paper, we initiate the study of critical random graphs in the
infinite second-moment regime by identifying the critical window for the
configuration model. We prove scaling limits for component sizes and surplus
edges, and show that the maximum diameter the critical components is of order
$\log n$, which contrasts with the previous universality classes arising in the
literature. This introduces a third and novel universality class for the
critical behavior of percolation on random networks, that is not covered by the
multiplicative coalescent framework due to Aldous and Limic (1998). We also
prove concentration of the component sizes outside the critical window, and
that a unique, complex giant component emerges after the critical window. This
completes the picture for the percolation phase transition on the configuration
model.
