Merging percolation and classical random graphs:
Phase transition in dimension 1
Tatyana S. Turova and Thomas Vallier
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2006
ISSN 14039338

Abstract:

We study a random graph model which combines properties of the edge percolation
model on Zd and a classical random graph G(n,c/n). We show that this model,
being a homogeneous random graph, has a natural relation to the socalled
"rank 1 case" of inhomogeneous random graphs. This allows us to use the newly
developed theory of inhomogeneous random graphs to describe completely the
phase diagram in the case d = 1. The phase transition is similar to the classical
random graph, it is of the second order. We also find the scaled size of
the largest connected component above the phase transition.



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