Abstract
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 tau in (3,4), the sequence of clusters ordered in decreasing size and multiplied through by n^{-(tau -2)/(tau -1)} converges as nrightarrow 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 (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.
Highlights
The Erdos–Rényi random graph G(n, p) on the vertex set [n] := {1, . . . , n} is constructed by including each of the n 2 possible edges with probability p, independently of all other edges
These constants only depend on a, b through c = c/(ab) = (λ + ζ )/(ab), any other dependence disappears since the law of H1(0) only depends on c
Since τ ∈ (3, 4), the sum over i, j such that i + j ≥ 1 is finite, as we can ignore all terms for which τ − 1 − i(τ − 2) − j (τ − 3) < 0
Summary
The Erdos–Rényi random graph G(n, p) on the vertex set [n] := {1, . . . , n} is constructed by including each of the n 2 possible edges with probability p, independently of all other edges. See in particular the recent works [6,34], where it was shown that if the degrees have finite third moment, the scaling limit for the largest critical components in the critical window are essentially the same (up to a trivial rescaling that we explain in more detail below) as for the Erdos–Rényi random graph in Theorem 1.1. In [7] scaling limits were obtained for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent τ ∈ (3, 4). It was shown that the sizes of the largest components, rescaled by n−(τ−2)/(τ−1), converge to hitting times of a thinned Lévy process The latter is a special case of the general multiplicative coalescents studied by Aldous and Limic in [2] and [3].
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