Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$ in entries of $U$ and $V$? More specifically, I am interested in polynomials of the form:
$ \sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}$
or
$ |\sum U_{ij}V_{i'j'} X_{ii'}Y_{jj'}|^2$
where $X$ and $Y$ are some arbitrary matrices and the sum is over all indices. For matrices with i.i.d. Gaussian entries there are well-known concentration bounds for this kind of expressions, I would like to know if there is anything similar for unitary matrices (for this case or for $U=V$).
