  Recently, the scaling limit of cluster sizes for critical inhomogeneous
random graphs of rank-1 type having finite variance but infinite third moment
degrees was obtained (see previous work by Bhamidi, van der Hofstad and van
Leeuwaarden). It was proved that when the degrees obey a power law with
exponent in the interval (3,4), the sequence of clusters ordered in decreasing
size and scaled appropriately converges as n goes to infinity to a sequence of
decreasing non-degenerate random variables.
  Here, we study the tails of the limit of the rescaled largest cluster, i.e.,
the probability that the scaling limit of the largest cluster takes a large
value u, as a function of u. This extends a related result of Pittel for the
Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random
graphs with infinite third moment degrees. We make use of delicate large
deviations and weak convergence arguments.
