Let M be a convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume, and let $M^{\ast }$ denote the interior of the convex core of M. In this paper we show that any geodesic plane in $M^{\ast }$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on the topology of M.

We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption.

Let $M_n$ be an $n\times n$ random matrix with independent and identically distributed $\mathrm{Bernoulli}(p)$ entries. We show that there is a universal constant $C\ge 1$ such that, whenever p and n satisfy $Clogn∕n\le p\le C^{-1}$,

where $o_n(1)$ denotes a quantity which converges to zero as $n\to \mathrm{\infty }$. We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.