ER random graphs in $G(n,m)$ model are known to have a giant component when $m>n/2$ which grows to a value of $\Theta(n)$ very abruptly. Also the size of the second largest component is known to have a maximum value of $n^{2/3}/2$ during the phase transition for a very short period of time, remaining $O(\log n)$ for most time.

Following is the experimental evaluation of varying component sizes for n=1000.

Graph shows *Size of components Vs Number of edges (m) in the graph, where number of components for a given size is shown by frequency map.*

Is there any literature available about the fraction of vertices in components of size $\Omega(f(n))$ excluding the giant component. From what I understand it should be very small fraction (~$O(n^{2/3})$) even if $f(n)=\log n$ but I couldn't find any concrete literature.

Please guide me to any concrete results available regarding the number of such vertices.