I am currently take a course on Erdös-Renyi graphs where we have the probability that two vertices are connected is given by $p = \lambda/n$ where n is the number of vertices of the graph G. Then one observe different regimes. The one I am interested at the moment is the supercritical regime, i.e. $\lambda > 1$.
In the lecture we wanted to show that
$$
P_\lambda(|C(1)| \geq k_n) = \zeta_\lambda + O(k_n/n)
$$
where C(1) is the connected component of the vertex 1, $k_n > a \log n, \; a I_\lambda > 1, \; I_\lambda = \lambda - 1 - \log n$ and the subscript denotes that the probability for the random graph is $p=\lambda/n$.
As the proof was only sketched I am trying to work it out on my own. At one step my lecturer defines $\lambda_n = \frac{n-k}{n} \lambda$ and claims that
$$
P_{\lambda_n}(T^* \geq k) = P_\lambda(T^* \geq k) + O(k_n/n)
$$
where T* denotes the total progeny of an associated branching process with offspring distribution $Poi(\lambda)$ respective $Poi(\lambda_n)$. This is the point where I am stuck at the moment.
My first idea was maybe to decompose the Poisson distribution somehow using $Poi(\nu + \mu) = Poi(\nu) + Poi(\mu)$ with two independent R.V. on the right by setting $\nu = \lambda_n$ and $\mu = \frac{\lambda k_n}{n}$.
The lecture orientates on the book "Random Graphs and Complex Networks" (https://www.win.tue.nl/~rhofstad/NotesRGCN.pdf) but follows a different proof strategy. The corresponding result can be found in the book at page 130.
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J. W. Tanner
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