  Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact
group $G$ we can define a measure $\mu_{w,G}$ on $G$ by (Haar-)uniformly
sampling $g_1,...,g_r\in G$ and evaluating $w(g_1,...,g_r)$. In
[arXiv:1802.04862], Magee and Puder studied the case where $G$ is the unitary
group $U(n)$, and analyzed how the moments of $\mu_{w,U(n)}$ behave as a
function of $n$. In particular, they obtained asymptotic bounds on those
moments, related to the commutator length and the stable commutator length of
$w$. We continue their line of work and give a more precise analysis of the
asymptotic behavior of the moments of $\mu_{w, U(n)}$, showing that it is
related to another algebraic invariant of $w$: its primitivity rank.
  In addition, we prove a special case of a conjecture of Hanany and Puder
([arXiv:2009.00897, Conjecture 1.13]) regarding the asymptotic behaviour of
expected values of irreducible characters under $\mu_{w, U(n)}$.
