Let D (n, r) be a random r-out regular directed multigraph on the set of vertices {1,..., n}. In
this work, we establish that for every r= 2, there exists¿ r> 0 such that diam (D (n, r))=(1+¿ r+
o (1)) logr n. The constant¿ r is related to branching processes and also appears in other
models of random undirected graphs. Our techniques also allow us to bound some extremal
quantities related to the stationary distribution of a simple random walk on D (n, r). In
particular, we determine the asymptotic behaviour of pmax and pmin, the maximum and the …

Let $ D (n, r) $ be a random $ r $-out regular directed multigraph on the set of vertices
$\{1,\ldots, n\} $. In this work, we establish that for every $ r\ge 2$, there exists $\eta_r> 0$
such that $\text {diam}(D (n, r))=(1+\eta_r+ o (1))\log_r {n} $. Our techniques also allow us to
bound some extremal quantities related to the stationary distribution of a simple random
walk on $ D (n, r) $. In particular, we determine the asymptotic behaviour of $\pi_ {\max} $
and $\pi_ {\min} $, the maximum and the minimum values of the stationary distribution. We …