Density of random subsets and applications to group theory

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