[Posted with the kind permission of the sender.]
Dear Doron,
First of all I am very pleased that you are finding my PCM essay
useful for your class, and looking at your website for the course
it looks like it must be a great learning experience.
Regarding your question, and Liyang Diao's writeup, I have to confess
that I have violated Einstein's dictum that everything should be made
as simple as possible, but not simpler. The formulas for N and \chi
on pages 4-5 of my article are not valid for all fixed \epsilon, as they
are unfortunately stated, but rather in the joint limit as n --> \infty
and \epsilon --> 0. This is part of the "appropriate sense" referred
to above the formula for N, but I realise now that the way it is written is
misleading. So in comparison with those Erdos-Renyi formulas, my formulas
are giving just the leading behaviour of the coefficients as \epsilon goes to
zero. On the other hand, \epsilon should not go to zero too quickly,
or we will be inside the critical window and the largest component will
have size n^{2/3}. Specifically, the formulas are valid as long as
\epsilon goes to zero but the product |\epsilon|n^{1/3} goes
to infinity. Thus they are tracking the important crossover to critical
from
above and below --- note that setting \epsilon equal to the borderline
value n^{-1/3} gives the power n^{2/3} in both the subcritical
and supercritical formulas. I am travelling and don't have access to
my books but I believe that these formulas for N can be found in the
book on Random Graphs by Janson, Luczak and Rucinski.
By the way, you and your class may be interested in the book in
preparation by Remco van der Hofstad called Random Graphs and
Complex Networks. The evolving manuscript is available on his
web page (http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf).
In particular, his Theorem 4.8 has a nice characterization of the
size of the largest supercritical cluster (with \epsilon fixed)
in terms of the survival probability for a Poisson branching process.
This seems more transparent than the formulas in the Erdos-Renyi
paper that Liyang Diao mentioned.
Thank you very much for bringing this to my attention.
I hope that there will be the opportunity to correct this in a second
edition of PCM.
With best wishes for the rest of your course, and greetings from Kyoto,
Gordon