  We consider random labelings of finite graphs conditioned on a small fixed
number of peaks. We introduce a continuum framework where a combinatorial graph
is associated with a metric graph and edges are identified with intervals. Next
we consider a sequence of partitions of the edges of the metric graph with the
partition size going to zero. As the mesh of the subdivision goes to zero, the
conditioned random labelings converge, in a suitable sense, to a deterministic
function which evolves as an increasing process of subsets of the metric graph
that grows at rate one while maximizing an appropriate notion of entropy. We
call such functions floodings. We present a number of qualitative and
quantitative properties of floodings and some explicit examples.
