[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

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