Random growth on a Ramanujan graph

TL;DR

This paper analyzes a random growth process on various graphs, especially Ramanujan graphs, using spectral graph theory, and compares theoretical predictions with experiments on flip graphs relevant to triangulation enumeration.

Contribution

It introduces a spectral graph theory-based analysis of random growth on Ramanujan graphs and compares theoretical results with experiments on flip graphs.

Findings

Results are strongest for Ramanujan graphs due to their spectral properties.

Theoretical predictions align with computational experiments on flip graphs.

Analysis extends to Erdős–Rényi random graphs.

Abstract

The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. Further, we consider Erd\H{o}s--R\'enyi random graphs and compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.