ON THE CRITICAL VALUE FOR PERCOLATION OF MINIMUM-WEIGHT TREES IN THE
MEAN-FIELD DISTANCE MODEL

David J Aldous

Consider the complete $n$-graph with independent exponential (mean $n$)
edge-weights.  Let $M(c,n)$ be the maximal size of subtree for which the 
average edge-weight is at most $c$.
It is shown that $M(c,n)$ transitions from $o(n)$ to $\Omega(n)$
around some critical value $c(0)$, which can be specified in terms
of a fixed point of a mapping on probability distributions.