Novel scaling limits for critical inhomogeneous random graphs

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 t. We investigate the case where t¿(3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n-(t-2)/(t-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 t>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 Erdos–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]