We draw a random graph according to Erdös-Rényi scheme $G(n,p)$ where $p=c/n$ for constant $c=1+\varepsilon$, ($\varepsilon>0$).
It is well known that a.a.s. there is a giant component emerging of size $(1-q)n$ where $q$ is the solution to the equation $e^{c(q-1)}=q$ and all other $qn$ vertices are in 'small' components.
Let $v$ be some vertex. I am interested in a distribution of the size of $C(v)$ (the c.c. of $v$ in the graph) conditioned on $v$ not in the giant component. $$\forall k\in\mathbb{N}:P(k)=\Pr[|C(v)|=k|v \text{ not in giant c.c.}]=\ ?$$
I noticed corollary 2.7 from here in pages 10-11. Applying it to our case is as follows: Let $Z$ a GW process with $Bin(n,p)$ as its offspring distribution. Conditionally on the extinction event, $Z$ is distributed as a sub-critical GW process $\widetilde{Z}$ with offspring distribution given by: $$\forall k\in\mathbb{N}\cup \{0\}: Q(k)=q^{k-1}\binom{n}{k}p^k(1-p)^{n-k}=\frac{\binom{n}{k}(pq)^k(1-p)^{n-k}}{q}$$
Is it correct to say that $P(k)$ is the distributed like the total progeny of the process $\widetilde{Z}$ a.a.s. when $n\to\infty$? Is something known about this dist. ?
This seems like the right interpretation; I haven't read the entire 40-page document to check.
Alon and Spencer's Probabilistic Method handles this question differently. As $n \to \infty$, the $\text{Binomial}(n-1, \frac cn)$ distribution converges to the $\text{Poisson}(c)$ distribution, so we can simply study the Poisson branching process. Theorem 11.4.2 in this book says that for any $c>0$, if $T$ is the total size of this branching process, then $$\Pr[T = k] = \frac{e^{-ck} (ck)^{k-1}}{k!}.$$ For $c\le 1$, these probabilities sum to $1$, whereas for $c>1$, they sum to the extinction probability $q$.
For $c>1$, if you want to condition on extinction, you should divide by $q$ to get the probability distribution. Then, $\Pr[T=k]/q$ will give the limiting probability that $|C(v)|=k$, conditioned on $v$ not being in the giant component.