Floodings of metric graphs

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.