- March 5, 2012: the percolation result simplifies and more is true:
given an arbitrary graph G and site percolation with parameter p, the expectation of the number
of times that a graph H is embedded is f(p) = p
^{ord(H)} v_{H}, where
v_{H} is the number of times that the graph H is embedded in G originally.
Similarly for bond percolation, the expectation is g(p) p^{size(H)} v_{H}.
So, the decorrelation argument in the proof of Proposition 1 is not needed.
The argument is very simple, well known and was used also
here:
let K be a specific embedding of H in G. Now look at the
event A_{K} that K is present in the site (rsp bond) Erdoes-Renyi probability space. It is
p^{ord(H)} (rsp. p^{size(H)}). Now the expectation for site percolation
E[ v_{H} ] = sum_{K} p^{ord(K)} = v_{H} p^{ord(H)}.
(rsp. sum_{K} p^{size(K)} = v_{H} p^{size(H)}).
It does not matter that the events A_{K} for different embeddings K are correlated,
when different embeddings overlap.