Floodings of metric graphs

TL;DR

This paper introduces a continuum framework for analyzing random labelings of finite graphs with a fixed number of peaks, showing convergence to deterministic floodings on metric graphs as partitions become finer.

Contribution

It develops a novel continuum approach linking combinatorial graph labelings to deterministic floodings on metric graphs, with convergence results and explicit examples.

Findings

Random labelings converge to floodings as partition size decreases

Floodings are deterministic functions evolving with maximal entropy

The framework applies to various graph structures and provides qualitative insights

Abstract

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.