The scaling limits of uniform connected graphs with fixed surplus and of random unicellular maps are built from tilted Brownian trees by identifying points at common heights sampled via local time.
Universality for critical heavy-tailed network models: Metric structure of maximal components
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abstract
We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory Relat. Fields 2018]. We develop general principles under which the identical scaling limits as the rank-one case can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.
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On breadth-first constructions of scaling limits of random graphs and random unicellular maps
The scaling limits of uniform connected graphs with fixed surplus and of random unicellular maps are built from tilted Brownian trees by identifying points at common heights sampled via local time.