  We consider Markovian dynamics on a typical realization of the so-called
Directed Configuration Model (DCM), that is, a random directed graph with
prescribed in- and out-degrees. In this random geometry, we study the meeting
time of two random walks on a typical realization of the graph starting at
stationarity, the coalescence time for a system of coalescent random walks, and
the consensus time of the voter model. Indeed, it is known that the latter
three quantities are related to each other when the underlying sequence of
graphs satisfies certain mean field conditions. Such conditions can be
summarized by requiring a fast mixing time of the random walk and some
anti-concentration of its stationary distribution: properties that a typical
random directed graph is known to have under natural assumptions on the degree
sequence. In this paper we show that, for a typical large graph from the DCM
ensemble, the distribution of the meeting time is well-approximated by an
exponential random variable and we provide the first-order approximation of its
expectation, showing that the latter is linear in the size of the graph, and
the preconstant depends on some easy statistics of the degree sequence. As a
byproduct, we are able to analyze the effect of the degree sequence in changing
the meeting, coalescence and consensus time. Our approach follows the classical
idea of converting meeting into hitting times of a proper collapsed chain,
which we control by the so-called First Visit Time Lemma.
