Deformed single ring theorems

Given a sequence of deterministic matrices $A=A_N$ and a sequence of deterministic nonnegative matrices $\Sigma ={\Sigma }_N$ such that $A\to a$ and $\Sigma \to \sigma$ in ⁎-distribution for some operators a and σ in a finite von Neumann algebra $A$. Let $U=U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $U\Sigma V^{⁎}+A$ converges to the Brown measure of $T+a$, where $T\in A$ is an R-diagonal operator freely independent from a and $\left|T\right|$ has the same distribution as σ. The assumptions can be removed if A is Hermitian or unitary. By putting $A=0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.