  Discovered in the context of card shuffling by Aldous, Diaconis and
Shahshahani, the cutoff phenomenon has since then been established in a variety
of Markov chains. However, proving cutoff remains a delicate affair, which
requires a detailed knowledge of the chain. Identifying the general mechanisms
underlying this phase transition -- without having to pinpoint its precise
location -- remains one of the most fundamental open problems in the area of
mixing times. In the present paper, we make a step in this direction by
establishing cutoff for Markov chains with non-negative curvature, under a
suitably refined product condition. The result applies, in particular, to
random walks on abelian Cayley expanders satisfying a mild degree condition,
hence in particular to \emph{almost all} abelian Cayley graphs. Our proof
relies on a quantitative \emph{entropic concentration principle}, which we
believe to lie behind all cutoff phenomena.
