Open Access
September 2016 Scaling limits of random graphs from subcritical classes
Konstantinos Panagiotou, Benedikt Stufler, Kerstin Weller
Ann. Probab. 44(5): 3291-3334 (September 2016). DOI: 10.1214/15-AOP1048


We study the uniform random graph $\mathsf{C}_{n}$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_{n}/\sqrt{n}$ converges to the Brownian continuum random tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide sub-Gaussian tail bounds for the diameter $\mathrm{D}(\mathsf{C}_{n})$ and height $\mathrm{H}(\mathsf{C}_{n}^{\bullet})$ of the rooted random graph $\mathsf{C}_{n}^{\bullet}$. We give analytic expressions for the scaling factor in several cases, including for example the class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_{n}$, where we also show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling.


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Konstantinos Panagiotou. Benedikt Stufler. Kerstin Weller. "Scaling limits of random graphs from subcritical classes." Ann. Probab. 44 (5) 3291 - 3334, September 2016.


Received: 1 November 2014; Revised: 1 July 2015; Published: September 2016
First available in Project Euclid: 21 September 2016

zbMATH: 1360.60073
MathSciNet: MR3551197
Digital Object Identifier: 10.1214/15-AOP1048

Primary: 60C05 , 60F17
Secondary: 05C80

Keywords: Continuum random tree , Random graphs , scaling limits , subcritical graph classes

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 5 • September 2016
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