Fourier and small ball estimates for word maps on unitary groups

To a non-trivial word $w(x_{1},...,x_{r})$ in a free group $F_{r}$ on $r$ elements and a group $G$, one can associate the word map $w_{G}:G^{r}\rightarrow G$ that takes an $r$-tuple $(g_{1},...,g_{r})$ in $G^{r}$ to $w(g_{1},...,g_{r})$. If $G$ is compact, we further associate the word measure $\tau_{w,G}$, defined as the distribution of $w_{G}(\mathsf{X}_{1},...,\mathsf{X}_{r})$, where $\mathsf{X}_{1},...,\mathsf{X}_{r}$ 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 $\left\{ \mathrm{SU}_{n}\right\} _{n\geq2}$. Our first result is a small ball estimate for $w_{\mathrm{SU}_{n}}$. We show that for every $w\in F_{r}\smallsetminus\left\{ 1\right\}$ there are $\epsilon(w),\delta(w)>0$ such that if $B\subseteq\mathrm{SU}_{n}$ is a ball of radius at most $\delta(w)\mathrm{diam}(\mathrm{SU}_{n})$ in the Hilbert-Schmidt metric, then $\tau_{w,\mathrm{SU}_{n}}(B)\leq(\mu_{\mathrm{SU}_{n}}(B))^{\epsilon(w)}$, where $\mu_{\mathrm{SU}_{n}}$ is the Haar probability measure. Our second main result is about the random walks generated by $\tau_{w,\mathrm{SU}_{n}}$. We provide exponential upper bounds on the large Fourier coefficients of $\tau_{w,\mathrm{SU}_{n}}$, and as a consequence we show there exists $t(w)\in\mathbb{N}$, such that $\tau_{w,\mathrm{SU}_{n}}^{*t}$ has bounded density for every $t\geq t(w)$ and every $n\geq2$, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character $\rho$ of $\mathrm{SU}_{n}$, an exponential upper bound of the form $\left|\rho(g)\right|<\rho(1)^{1-\epsilon}$, for elements $g$ in $\mathrm{SU}_{n}$ whose eigenvalues are sufficiently spread out on the unit circle in $\mathbb{C^{\times}}$.