  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 \tau. We investigate the case where $\tau\in(3,4)$, so that the
degrees have finite variance but infinite third moment. The sizes of the
largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting
times of a "thinned" L\'{e}vy 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 \tau>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 Erd\H{o}s-R\'{e}nyi 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].
