Density of random subsets and applications to group theory

TL;DR

This paper rigorously defines the intersection behavior of random subsets with density in finite sets, and applies this to analyze phase transitions and small cancellation conditions in random group presentations.

Contribution

It formalizes Gromov's intersection formula for random subsets and applies it to group theory, revealing phase transitions and improving existing results on random groups.

Findings

Validated Gromov's intersection formula for random subsets.

Identified phase transition at density λ/2 in random group presentations.

Enhanced bounds for small cancellation conditions in random groups.

Abstract

Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov's \textit{intersection formula}: "Random subsets" $A$ and $B$ of a finite set $E$ satisfy $dens (A\cap B) = dens A + dens B -1$. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $\lambda/2$ for any $0<\lambda<1$, characterizing the $C'(\lambda)$-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii from density $0$ to density $0\leq d<\frac{1}{120m^2\ln(2m)}$.