In this talk, we will discuss the extremal values of the stationary distribution of a random walk on a directed random graph with given degrees. While for undirected graphs the stationary distribution is simply determined by the degrees, the graph geometry plays a major role in the directed case. Understanding typical stationary values is key to determine the mixing time of the walk. However, typical results provide no information on extremal values, which are important for many applications. In this talk, we will show that the minimum value may be polynomially smaller than the average value, and that the correct exponents comes from the combination of two factors: (1) the contribution of atypically thin in-neighborhoods, controlled by subcritical branching processes; and (2) the contribution of atypically “light” trajectories, controlled by large deviation rate functions. This result has direct consequences for the understanding of random walks in random digraphs. In particular, we will show how one can obtain an estimate on the cover time of the random digraph. This is joint work with Xing Shi Cai.