Global Structural Properties of Random Graphs

TL;DR

This paper investigates global structural properties of random graphs, establishing thresholds for these properties in Erdős–Rényi models and exploring implications for associated Coxeter groups.

Contribution

It provides the first sharp threshold for the AS property and tight bounds for the CFS property in Erdős–Rényi graphs, linking graph properties to geometric group theory.

Findings

Sharp threshold for AS property in G(n,p)

Tight bounds for CFS property thresholds

Almost sure quadratic divergence and thickness of Coxeter groups

Abstract

We study two global structural properties of a graph $\Gamma$, denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erd\"os--R\'enyi random graph model G(n,p), proving a sharp threshold for a random graph to have the AS property asymptotically almost surely, and giving fairly tight bounds for the corresponding threshold for CFS. As an application of our results, we show that for any constant p and any $\Gamma\in G(n,p)$, the right-angled Coxeter group $W_\Gamma$ asymptotically almost surely has quadratic divergence and thickness of order 1, generalizing and strengthening a result of Behrstock--Hagen--Sisto.