Minimum stationary values of sparse random directed graphs

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 (uniqueness regime). We show that the minimum positive stationary value is whp n−(1+C+o(1)) for some constant C≥0 determined by the degree distribution, answering a question raised by Bordenave, Caputo and Salez (Probab. Theory Related Fields 170 (2018) 933–960). 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, we give estimates for the expected lower tail of the empirical stationary distribution. As a by-product of our proof, we obtain that the hitting and the cover time are both n1+C+o(1) whp. Our results are in sharp contrast to those of Caputo and Quattropani (Probab. Theory Related Fields 178 (2020) 1011–1066) who showed that under the additional condition of minimum in-degree at least 2 (ergodicity regime), stationary values only have logarithmic fluctuations around n−1.