  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.
