Minimum stationary values of sparse random directed graphs

Abstract: 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 $n^{-1}$.