Questions on the structure of random embeddings of L(mathbb{F}₂)

Motivated by recent developments at the interface of operator algebras and random matrix theory, we propose new conjectures concerning the asymptotic structure of random matrix models of the countable free groups. The first conjecture predicts a random matrix analogue of the Akemann-Ostrand property for free groups, and reveals a succinct approach to recover the Peterson-Thom property for $L(\mathbb{F}_2)$. The second stronger conjecture is motivated by continuous model theory. It predicts that the \emph{random} embedding of the free group factor into a matrix ultraproduct is \emph{existential}. We discuss the interesting relationship between these conjectures.