On probabilistic identities and coset identities in pro-$p$ groups

Abstract: It is shown that a probabilistic identity on a $\sigma$-compact $K$-analytic group $G$, $K$ a non-archimedean local field, is a coset identity. As an application, one concludes that compact groups which are linear over a local field and various pro-$p$ groups obtained from free constructions satisfy a probabilistic Tits alternative. By means of Lie-theoretic methods, we also study torsion probabilistic identities in virtually free pro-$p$ and compact $p$-adic analytic groups.