Panagiotou, Konstantinos; Stufler, Benedikt; Weller, Kerstin
(2016):
Scaling limits of random graphs from subcritical classes.
In: Annals of Probability, Vol. 44, No. 5: pp. 32913334

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Abstract
We study the uniform random graph Cn with n vertices drawn from a subcritical class of connected graphs. Our main result is that the resealed graph Cn / root n converges to the Brownian continuum random tree Te multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subGaussian tail bounds for the diameter D (Cn) and height H(Cn(center dot)) of the rooted random graph Cn(center dot) 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 Cn, where we also show the convergence to Te under an appropriate rescaling.