Cluster tails for critical power-law inhomogeneous random graphs

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but in finite third moment degrees was obtained. It was proved that when the degrees obey a power law with exponent $ \tau \in (3,4) $, the sequence of clusters ordered in decreasing size and multiplied through by $ n^{ -(\tau -2) / (\tau -1) } $ converges as $ n \rightarrow \infty $ 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 Erdös-Rényi 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.