Word Measures on Unitary Groups: Improved Bounds for Small Representations

Let F be a free group of rank r and fix some w∈ F. For any compact group G we can define a measure μ w,G on G by (Haar-)uniformly sampling g₁,...,gᵣ∈ G and evaluating w(g₁,...,gᵣ). In [23], Magee and Puder study the behavior of the moments of μ w,U(n) as a function of n, establishing a connection between their asymptotic behavior and certain algebraic invariants of w, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of w. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under μ w,U(n).