Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra

Given an $n\times n$ random matrix $X_n$ with i.i.d. entries of unit variance, the circular law says that the empirical spectral distribution (ESD) of $X_n/\sqrt{n}$ converges to the uniform measure on the unit disk. Let $M_n$ be a deterministic matrix that converges in $*$-moments to an operator ${x}$. It is known from the work by \'{S}niady and Tao--Vu that the ESD of $X_n/\sqrt{n}+M_n$ converges to the Brown measure of ${x}+c$, where $c$ is Voiculescu's circular operator. We obtain a formula for the Brown measure of ${x}+c$ which provides a description of the limit distribution. This answers a question of Biane--Lehner for arbitrary operator ${x}$. Generalizing the case of circular and semi-circular operators, we also consider a family of twisted elliptic operators that are $*$-free from ${x}$. For an arbitrary twisted elliptic operator $g$, possible degeneracy then prevents a direct calculation of the Brown measure of ${x}+g$. We instead show that the whole family of Brown measures are the push-forward measures of the Brown measure of ${x}+c$ under a family of self-maps of the plane, which could possibly be singular. We calculate explicit formula for the case ${x}$ is self-adjoint. In addition, we prove that the Brown measure of the sum of an $R$-diagonal operator and a twisted elliptic element is supported in a deformed ring where the inner boundary is a circle and the outer boundary is an ellipse. These results generalize some known results about free additive Brownian motions where the free random variable ${x}$ is assumed to be self-adjoint. The approach is based on a Hermitian reduction and subordination functions.