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 infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012). It was proved that when the degrees obey a power law with exponent τ∈ (3 , 4) , the sequence of clusters ordered in decreasing size and multiplied through by n^{- ( τ - 2 ) / ( τ - 1 )} converges as n→ ∞ 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 (J Combin Theory Ser B 82(2):237–269, 2001) for the Erdős–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.