  We study convergence to equilibrium for a large class of Markov chains in
random environment. The chains are sparse in the sense that in every row of the
transition matrix $P$ the mass is essentially concentrated on few entries.
Moreover, the random environment is such that rows of $P$ are independent and
such that the entries are exchangeable within each row. This includes various
models of random walks on sparse random directed graphs. The models are
generally non reversible and the equilibrium distribution is itself unknown. In
this general setting we establish the cutoff phenomenon for the total variation
distance to equilibrium, with mixing time given by the logarithm of the number
of states times the inverse of the average row entropy of $P$. As an
application, we consider the case where the rows of $P$ are i.i.d. random
vectors in the domain of attraction of a Poisson-Dirichlet law with index
$\alpha\in(0,1)$. Our main results are based on a detailed analysis of the
weight of the trajectory followed by the walker. This approach offers an
interpretation of cutoff as an instance of the concentration of measure
phenomenon.
