Resumen de Novel scaling limits for critical inhomogeneous random graphs
Shankar Bhamidi, Remco van der Hofstad, Johan S. H. Van Leeuwaarden
We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent τ. We investigate the case where τ∈(3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n−(τ−2)/(τ−1), converge to hitting times of a “thinned” Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812–854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1–59].
Our results should be contrasted to the case τ>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by n−2/3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812–854] for the Erdős–Rényi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682–1703] and Turova [(2009) Preprint].
