Twin peaks

TL;DR

This paper investigates the geometric and probabilistic properties of random graph labelings conditioned on having one or two local maxima, revealing structural patterns and peak distributions across different graph types.

Contribution

It provides new insights into the shape and distribution of peaks in random labelings of graphs, including boundary dimensions and peak proximity behaviors.

Findings

Boundary of level sets has dimension 2 in the asymptotic limit.

Gradient lines in ladder graphs are mostly straight segments.

Two peaks can be arbitrarily close in some tree graphs.

Abstract

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.