I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows $$ \mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2} $$ where $\mathcal{Z}$ enforces the normalization of the measure. In practice, with respect to the standard Haar distribution, I want to include a row-dependent bias which affects only the first $N<L$ columns. I am interpreting this as a sort of canonical distribution and what I would like is to fix the $\omega_k$'s such that $$ - \partial_{\omega_k} \ln \mathcal{Z}[\omega] = \overline{\sum_{i=1}^N |U_{ki}|^2} = n_k $$ where $n_k$ are some assigned quantities $\in [0,1]$. I have found a way to compute $\mathcal{Z}[\omega]$ using the Itzykson-Zuber formula, but I end up with determinant that is not very useful. I hope that the problem can be simplified in the limit of large $L$ and $N$, with a fixed ratio $N/L = n$.
I had thought that for large $L$ one could simply assume that with the Haar distribution, the entry of $U$ become uncorrelated complex gaussian random variables with zero average and $\overline{U_{ki}}^2 = 1/L$. However, this is certainly not correct because the $\omega_{k}$ would not affect all the $U_{ki}$ for $i > N$, but this is certainly incorrect since one must still have $\sum_{i=1}^L |U_{ki}|^2 = 1$.
Do you have any idea or references where this problem has been studied?
Edit1:
Intuitively, I expect that for large $L$, the entries of the matrix $U$ are weakly correlated. If I make this assumption, still enforcing normalisation of the rows $$ \sum_{i=1}^L |U_{ki}|^2 = 1 $$ I can consider the ensemble made of independent entries $U_{kl}$ with the new partition sum $$ \mathcal{Z}^{\rm Gauss}[\omega, \gamma] = \int \prod_{k,l} dU_{kl} \, e^{- \sum_{k,i =1}^L \gamma_k |U_{ki}|^2} e^{- \sum_{k=1}^L \sum_{i=1}^N \omega_k |U_{k,i}|^2} $$ Now I can enforce the constraints as \begin{align} &\partial_{\omega_k} \ln \mathcal{Z}^{\rm Gauss}[\omega, \gamma] + n_k = 0 \\ &\partial_{\gamma_k} \ln \mathcal{Z}^{\rm Gauss}[\omega, \gamma] + 1 = 0 \end{align}
These equations are solved by \begin{equation} \gamma_k = L \frac{n_k - n}{1- n_k} \;, \qquad \omega_k = L \frac{n_k - n}{(1- n_k)n_k} \end{equation} As expected this reduces to the usual Haar ensemble if $n_k = n$ and thus all the $\omega_k = 0$.
Are these values for $\omega_k$ accurate in the limit $L,N \to \infty$ with $N/L = n$ and non-trivial $n_k$, in the original Haar problem?