It has a limit if the argument of the function is expressed in degrees.

If G is undirected, you can do better than finding max-flow min-cut over all pairs of vertices. Since you are guaranteed that any given vertex $s$ is contained in one of the partitions of a minimum cut, you can choose any arbitrary vertex $s$ and compute the max flow/min cut to each other vertex $t$. The minimum of these values is the min cut of the graph with cardinality $lambda(G)$, the edge connectivity of G.

Thanks Thorny - this was very helpful. So the $E(\kappa(H))$ tends to around $\frac{n}{2}$ for $H \in G(n,p)$, which is what I was most interested in.

Yes, you are correct.