  We prove a cutoff for the random walk on random $n$-lifts of finite weighted
graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift
is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$
is a constant associated to $\mathcal{G}$, namely the entropy of its universal
cover. Moreover, this mixing time is the smallest possible among all $n$-lifts
of $\mathcal{G}$. In the particular case where the base graph is a vertex with
$d/2$ loops, $d$ even, we obtain a cutoff for a $d$-regular random graph (as
did Lubetzky and Sly in \cite{cutoffregular} with a slightly different
distribution on $d$-regular graphs, but the mixing time is the same).
