Critical random graphs: limiting constructions and distributional properties

Details Critical random graphs: limiting constructions and distributional properties Critical random graphs: limiting constructions and distributional properties

2009-08-25

arxiv.org/abs/0908.3629v2

Mathematics

Probability

30 pages, 4 figures

Scientific paper

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.

Affiliated with

Also associated with

No associations

Rating

Critical random graphs: limiting constructions and distributional properties does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Critical random graphs: limiting constructions and distributional properties, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Critical random graphs: limiting constructions and distributional properties will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-233419