Twin peaks

We study random labelings of graphs conditioned on a small number (typically one or two) peaks, i.e., local maxima. We show that the boundaries of level sets of a random labeling of a square with a single peak have dimension 2, in a suitable asymptotic sense. The gradient line of a random labeling of a long ladder graph conditioned on a single peak consists mostly of straight line segments. We show that for some tree-graphs, if a random labeling is conditioned on exactly two peaks then the peaks can be very close to each other. We also study random labelings of regular trees conditioned on having exactly two peaks. Our results suggest that the top peak is likely to be at the root and the second peak is equally likely, more or less, to be any vertex not adjacent to the root.