  We consider the stationary distribution of the simple random walk on the
directed configuration model with bounded degrees. Provided that the minimum
out-degree is at least $2$, with high probability (whp) there is a unique
stationary distribution. We show that the minimum positive stationary value is
whp $n^{-(1+C+o(1))}$ for some constant $C \ge 0$ determined by the degree
distribution. In particular, $C$ is the competing combination of two factors:
(1) the contribution of atypically "thin" in-neighbourhoods, controlled by
subcritical branching processes; and (2) the contribution of atypically "light"
trajectories, controlled by large deviation rate functions. Additionally, our
proof implies that whp the hitting and the cover time are both $n^{1+C+o(1)}$.
Our results complement those of Caputo and Quattropani who showed that if the
minimum in-degree is at least 2, stationary values have logarithmic
fluctuations around $n^{-1}$.
