Fourier and small ball estimates for word maps on unitary groups

To a non-trivial word w (x₁,. . . , xₑ) in a free group Fₑ on r elements and a group G, one can associate the word map w₆: G^r G that takes an r-tuple (g₁,. . . , gₑ) in G^r to w (g₁,. . . , gₑ). If G is compact, we further associate the word measure ₖ, ₆, defined as the distribution of w₆ (X₁,. . . , Xₑ), where X₁,. . . , Xₑ 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 \ SU₍\ ₍₂. Our first result is a small ball estimate for wₒₔ_₍. We show that for every w Fₑ\ 1\ there are (w), (w) >0 such that if B₍ is a ball of radius at most (w) diam (SU₍) in the Hilbert-Schmidt metric, then ₖ, ₒₔ_₍ (B) (ₒₔ_₍ (B) ) ^ (w), where ₒₔ_₍ is the Haar probability measure. Our second main result is about the random walks generated by ₖ, ₒₔ_₍. We provide exponential upper bounds on the large Fourier coefficients of ₖ, ₒₔ_₍, and as a consequence we show there exists t (w), such that ₖ, ₒₔ_₍^*t has bounded density for every t t (w) and every n2, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character of SU₍, an exponential upper bound of the form | (g) |< (1) ^1-, for elements g in SU₍ whose eigenvalues are sufficiently spread out on the unit circle in C^.