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

Given an n× n random matrix Xₙ with i.i.d. entries of unit variance, the circular law says that the empirical spectral distribution (ESD) of Xₙ/√n converges to the uniform measure on the unit disk. Let Mₙ be a deterministic matrix that converges in $*$-moments to an operator x. It is known from the work by Śniady and Tao--Vu that the ESD of Xₙ/√n+Mₙ 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.