Consider a connected graph $G$ with min-cut $c$. Suppose the edges fail (are removed) independently with probability $p$. Then $U(p)$, the probability that $G$ becomes disconnected, is at least $p^c$. Any such must contain at least $m \geq n c/2$ edges (since the vertices defines cuts).
If we know $U(p) \approx p^c$, can we show $m$ is much larger than this? (For example, if $U(p) = p^c$ exactly, then this means that the min-cut is the only cut and (unless $n = 2$) we have $m = \infty$.)
For example, can we show a bound something like $$ m \geq f( U(p)/p^c ) $$
for some appropriate function $f$?
Thanks for any help!