Thomas Vallier
	
Merging Percolation and Classical Random graphs:
Phase Transition in Dimension 1

Abstract

We study a  random graph model which combines properties of the edge
percolation model on $Z^d$ 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 so-called "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.  I will
in a first part introduce both the percolation and the classical
random graph. In a second part I will present the result on our new
model of graph and give concepts of the proofs.