My question involves two constants which I don't know how to compute, but it probably doesn't matter. The constant $C>0$ is rather small, say $C=e^{-1000}$. The constant $K$ is larger than $1$.
I am considering the model $G(n,p)$ of a random graphs on $n$ vertices with each possible edge present with probability $p$.
In a graph on $n$ vertices, I call a vertex "small" if it is part of a connected component of size at least $2$ and at most $Cn$.
Is there a constant $B$ such that the probability of having at least $Bn$ small vertices in $G(n,K/n)$ goes to $1$ when $n$ goes to infinity?
Consider any vertex. There are two ways in which a vertex might not be small. One is that it has no edges leaving it. The other is that it is part of a subgraph of unbounded size.
In the limit, the number of edges connecting it with any other vertex obeys the Poisson distribution with mean $K$. Thus, the probability that a vertex has no edge leaving it is $e^{-K}$.
Next, what is the probability that this vertex is part of a subgraph of unbounded size? Imagine we grow the subgraph from the original vertex, at each turn adding edges to the vertices added in the previous turn. The generating function for the Poisson distribution is
$$ F(z) = e^{-K(1-z)} $$
The probability that the graph is bounded (including it having no edges at all) can then be determined by solving the recursive formula $z = F(z)$ for $z$; that is,
$$ z = e^{-K(1-z)} $$
Since $K > 1$, this formula has a solution $z^* < 1$, which cannot be determined analytically, in general. This is the probability that a vertex is part of a bounded subgraph. The probability that a vertex is small is then given by $z^*-e^{-K}$. For any $B < z^*-e^{-K}$, the probability that at least $Bn$ of the vertices are small, as $n \to \infty$, approaches 1.