Why giant component of $G_{n,m}$ is Uniform along connected graphs?

Question

I Just noticed that for $G_{n,m }$ with $m=\frac {cn}2 $ for $c>1$ the giant component is uniformly distributed along all connected graphs. More precisely if giant component ,$C$, has $n'$ vertices and $m'$ edges for all connected $H_1,H_2$ with $n',m'$ vertices and edges respectively,

$$\mathbb{P}(C=H_1)=\mathbb{P}(C=H_2)$$

Answer 1

This has really got nothing to do with the giant component at all; it's just true because $G \sim \mathcal G_{n,m}$ is uniformly distributed among all $n$-vertex, $m$-edge graphs.

As a consequence, say we fix a partition $V = V' \cup V''$ with $|V'| = n'$ and $|V''| = n - n'$. Given graphs $H$ on $n'$ vertices and $K$ on $n-n'$ vertices, let $H \cup K$ denote the graph on vertex set $V$ with graph $H$ induced on $V'$, graph $K$ induced on $V''$, and no edges between the two.

Then for any graphs $H_1, H_2$ on $n'$ vertices and $K_1, K_2$ on $n''$ vertices $$\Pr[G = H_1 \cup K_1] = \Pr[G = H_2 \cup K_2]$$ just because both sides specify a unique graph for $G$ to be. Moreover, if we specify that both $H_1$ and $H_2$ have $m'$ edges, and both $K_1$ and $K_2$ have $m-m'$ edges, then the number of choices for $K_1$ and for $K_2$ is the same, so $$\sum_{K_1} \Pr[G = H_1 \cup K_1] = \sum_{K_2} \Pr[G = H_2 \cup K_2].$$ When $H_1, H_2$ are connected, this says that the probability $G[V']$ induces $H_1$ as a connected component is the same as the probability $G[V']$ induces $H_2$ as a connected component.

By symmetry over the choices of $V'$ and $V''$, it's true that the probability that a component of $G$ is isomorphic to $H_1$ is the same as the probability that a component of $G$ is isomorphic to $H_2$.

In particular, $\Pr[C = H_1] = \Pr[C = H_2]$ to the extent that this makes sense. (When $n' < \frac n2$, there's a very small but nonzero probability that $G$ has two components of that size, in which case it's unclear what the "giant component" even is.) This says that, conditioned on the number of vertices and edges in the giant component, their exact arrangement is uniform, as long as it makes the giant component connected.