Group representations that resist random sampling

We show that there exists a family of groups Gn and nontrivial irreducible representations ρn such that, for any constant t, the average of ρn over t uniformly random elements g1, . . . , gt ∈ Gn has operator norm 1 with probability approaching 1 as n→∞. More quantitatively, we show that there exist families of finite groups for which Ω(log log |G|) random elements are required to bound the norm of a typical representation below 1. This settles a conjecture of A. Wigderson.