  The mean weight of a cycle in an edge-weighted graph is the sum of the
cycle's edge weights divided by the cycle's length. We study the minimum
mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge
weights drawn from an exponential distribution with mean 1. We show that the
probability of the min mean weight being at most c/n tends to a limiting
function of c which is analytic for c<=1/e, discontinuous at c=1/e, and equal
to 1 for c>1/e. We further show that if the min mean weight is <=1/(en), then
the length of the relevant cycle is Theta_p(1) (i.e., it has a limiting
probability distribution which does not scale with n), but that if the min mean
weight is >1/(en), then the relevant cycle almost always has mean weight
(1+o(1))/(en) and length at least (2/pi^2-o(1)) log^2 n log log n.
