Progress in High-Dimensional Percolation and Random Graphs by Markus Heydenreich & Remco van der Hofstad

Author:Markus Heydenreich & Remco van der Hofstad
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(13.1.2)

and, for any fixed integer ,

(13.1.3)

Equations (13.1.2)–(13.1.3) show that the largest connected component has a size that is concentrated, and all other connected components are much smaller. It is not hard to extend (13.1.2)–(13.1.3) to the (easier) setting where with fixed and , where the limit in (13.1.2) needs to be replaced by the survival probability of a Poisson branching process with mean offspring . When , then , which explains the factor in (13.1.2). The proof of (13.1.2)–(13.1.3) relies on branching process approximations that we explain in some detail below.

In both the sub- and the supercritical phase, the limit of , properly normalized, is deterministic. At the critical point, one can expect that such a scaling limit is random. This is reflected in the following asymptotics:

The Critical Window When for some , for any fixed integer ,

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