On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups

We consider first the $n\times n$ random matrices $ H_{n}=A_{n}+U_{n}^{* }B_{n}U_{n}$, where $A_{n}$ and $B_{n}$ are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as $ n\rightarrow \infty$, and $U_{n}$ is unitary uniformly distributed over $ U(n)$. We find the leading term of asymptotic expansion for the covariance of elements of resolvent of $H_{n}$ and establish the Central Limit Theorem for the elements of sufficiently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices $ W_{n}=S_{n}U_{n}^{* }T_{n}U_{n}$, where $U_n $ is as above and $S_n$ and $T_n $ are non-random unitary matrices having limiting NCM's as $n\rightarrow \infty$.