I have a random graph $G = (V, E)$ and each edge is in the graph with probability $p$.
I need to show that the probability that $G$ is $\delta$-edge-expander* when $\delta= \frac{np}{4}$ goes to $1$ as $n=|V|\to\infty$.
*A graph is $\delta$-edge-expander if for every set of nodes $S$ of size at most $\frac{n}{2}$ it holds that the number of edges from $S$ to $S'$ (where $S'$ is all the nodes that don't appear in $S$) is at least $\delta$. You can see the definition in this wiki page.