Let $U$ be a $n\times n$ Haar random unitary matrix and $U_{ij}$ be its $(i,j)$ entry. What is the marginal distribution over $U_{11}$ and $U_{12}$ (or more generally $U_{ij}$ and $U_{ik}$ for $j\neq k$ but I suppose they are the same problems)?
My question is to explicitly calculate or tight upper bound $\Pr[|U_{11}|\geq\alpha\wedge |U_{12}|\geq \alpha \wedge \cdots\wedge |U_{1m}|\geq\alpha]$ for integer $m<n$ and real number $\alpha\in(0,1)$. I think understanding the joint distribution of two entries would help.
For $m=1$, I used the fact that $\mathbb{E}[|U_{11}|^2]=1/n$. By Markov's inequality, $\Pr[|U_{11}|\geq\alpha]\leq \frac{1}{\alpha^2 n}$. But I am not sure whether this is a tight upper bound, or a similar idea can be used for $m>1$.