Density of random subsets and applications to group theory

Developing an idea of M. Gromov in [5] 9.A, we study the intersection formula for random subsets with density. The density of a subset A in a finite set E is defined by densA := log|E|(|A|). The aim of this article is to give a precise meaning of Gromov’s intersection formula: "Random subsets" A and B of a finite set E satisfy dens(A∩B) = densA+densB−1. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density λ/2 for any 0 < λ < 1, characterizing the C(λ)-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol’shanskii in [2] from density 0 to density 0 ≤ d < 1 120m2 ln(2m) .