## The Annals of Probability

### Novel scaling limits for critical inhomogeneous random graphs

#### Abstract

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” 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 $\tau>4$, 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 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].

#### Article information

Source
Ann. Probab., Volume 40, Number 6 (2012), 2299-2361.

Dates
First available in Project Euclid: 26 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1351258728

Digital Object Identifier
doi:10.1214/11-AOP680

Mathematical Reviews number (MathSciNet)
MR3050505

Zentralblatt MATH identifier
1257.05157

#### Citation

Bhamidi, Shankar; van der Hofstad, Remco; van Leeuwaarden, Johan S. H. Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab. 40 (2012), no. 6, 2299--2361. doi:10.1214/11-AOP680. https://projecteuclid.org/euclid.aop/1351258728

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