Fourier and small ball estimates for word maps on unitary groups
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2024
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| author | Avni, Nir Glazer, Itay Larsen, Michael |
| author_facet | Avni, Nir Glazer, Itay Larsen, Michael |
| contents | 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 $τ_{w,G}$, 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 $ε(w),δ(w)>0$ such that if $B\subseteq\mathrm{SU}_{n}$ is a ball of radius at most $δ(w)\mathrm{diam}(\mathrm{SU}_{n})$ in the Hilbert-Schmidt metric, then $τ_{w,\mathrm{SU}_{n}}(B)\leq(μ_{\mathrm{SU}_{n}}(B))^{ε(w)}$, where $μ_{\mathrm{SU}_{n}}$ is the Haar probability measure.
Our second main result is about the random walks generated by $τ_{w,\mathrm{SU}_{n}}$. We provide exponential upper bounds on the large Fourier coefficients of $τ_{w,\mathrm{SU}_{n}}$, and as a consequence we show there exists $t(w)\in\mathbb{N}$, such that $τ_{w,\mathrm{SU}_{n}}^{*t}$ 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 $ρ$ of $\mathrm{SU}_{n}$, an exponential upper bound of the form $\left|ρ(g)\right|<ρ(1)^{1-ε}$, for elements $g$ in $\mathrm{SU}_{n}$ whose eigenvalues are sufficiently spread out on the unit circle in $\mathbb{C^{\times}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_11108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier and small ball estimates for word maps on unitary groups Avni, Nir Glazer, Itay Larsen, Michael Group Theory Probability 20P05 (Primary) 60B15, 60B20, 22E46, 43A25, 14B05 (Secondary) 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 $τ_{w,G}$, 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 $ε(w),δ(w)>0$ such that if $B\subseteq\mathrm{SU}_{n}$ is a ball of radius at most $δ(w)\mathrm{diam}(\mathrm{SU}_{n})$ in the Hilbert-Schmidt metric, then $τ_{w,\mathrm{SU}_{n}}(B)\leq(μ_{\mathrm{SU}_{n}}(B))^{ε(w)}$, where $μ_{\mathrm{SU}_{n}}$ is the Haar probability measure. Our second main result is about the random walks generated by $τ_{w,\mathrm{SU}_{n}}$. We provide exponential upper bounds on the large Fourier coefficients of $τ_{w,\mathrm{SU}_{n}}$, and as a consequence we show there exists $t(w)\in\mathbb{N}$, such that $τ_{w,\mathrm{SU}_{n}}^{*t}$ 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 $ρ$ of $\mathrm{SU}_{n}$, an exponential upper bound of the form $\left|ρ(g)\right|<ρ(1)^{1-ε}$, for elements $g$ in $\mathrm{SU}_{n}$ whose eigenvalues are sufficiently spread out on the unit circle in $\mathbb{C^{\times}}$. |
| title | Fourier and small ball estimates for word maps on unitary groups |
| topic | Group Theory Probability 20P05 (Primary) 60B15, 60B20, 22E46, 43A25, 14B05 (Secondary) |
| url | https://arxiv.org/abs/2402.11108 |