  Consider a simple random walk on a realization of an Erd\H{o}s-R\'enyi graph.
Assume that it is asymptotically almost surely (a.a.s.) connected. Conditional
on an eigenvector delocalization conjecture, we prove a Central Limit Theorem
(CLT) for the average target hitting time. By the latter we mean the expected
time it takes the random walk on average to first hit a vertex $j$ when
starting in a fixed vertex $i$. The average is taken with respect to $\pi_i$,
the invariant measure of the random walk.
