  We study the weight and length of the minimum mean-weight cycle in the
stochastic mean-field distance model, i.e., in the complete graph on $n$
vertices with edges weighted by independent exponential random variables.
Mathieu and Wilson showed that the minimum mean-weight cycle exhibits one of
two distinct behaviors, according to whether its mean weight is smaller or
larger than $1/(ne)$; and that both scenarios occur with positive probability
in the limit $n\to\infty$. If the mean weight is $< 1/(ne)$, the length is of
constant order. If the mean weight is $> 1/(ne)$, it is concentrated just above
$1/(n e)$, and the length diverges with $n$. The analysis of Mathieu--Wilson
gives a detailed characterization of the subcritical regime, including the
(non-degenerate) limiting distributions of the weight and length, but leaves
open the supercritical behavior. We determine the asymptotics for the
supercritical regime, showing that with high probability, the minimum mean
weight is $(n e)^{-1}[1 + \pi^2/(2 \log^2 n) + O((\log n)^{-3})]$, and the
cycle achieving this minimum has length on the order of $(\log n)^3$.
