  We establish three remarkable consequences of non-negative curvature for
sparse Markov chains. First, their conductance decreases logarithmically with
the number of states. Second, their displacement is at least diffusive until
the mixing time. Third, they never exhibit the cutoff phenomenon. The first
result provides a nearly sharp quantitative answer to a classical question of
Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for
graphs with non-negative curvature. The third offers a striking counterpoint to
the recently established cutoff for non-negatively curved chains with uniform
expansion.
