Fourier and small ball estimates for word maps on unitary groups

To a non-trivial word w(x₁,...,xᵣ) in a free group Fᵣ on r elements and a group G, one can associate the word map wG:Gʳ→ G that takes an r-tuple (g₁,...,gᵣ) in Gʳ to w(g₁,...,gᵣ). If G is compact, we further associate the word measure τw,G, defined as the distribution of wG(X₁,...,Xᵣ), where X₁,...,Xᵣ are independent and Haar-random elements in G. In this paper we study word maps and word measures on the family of special unitary groups \ SUₙ\ n≥2. Our first result is a small ball estimate for w_SUₙ. We show that for every w∈ Fᵣ\ 1\ there are ε(w),δ(w)>0 such that if Bₙ is a ball of radius at most δ(w)diam(SUₙ) in the Hilbert-Schmidt metric, then τ_w,SUₙ(B)≤(μ_SUₙ(B))ε(w), where μ_SUₙ is the Haar probability measure. Our second main result is about the random walks generated by τ_w,SUₙ. We provide exponential upper bounds on the large Fourier coefficients of τ_w,SUₙ, and as a consequence we show there exists t(w), such that τ_w,SUₙ*t has bounded density for every t≥ t(w) and every n≥2, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character ρ of SUₙ, an exponential upper bound of the form |ρ(g)|<ρ(1)1-ε, for elements g in SUₙ whose eigenvalues are sufficiently spread out on the unit circle in C×.