abstract: 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.
Ping Zhong (Fri,) studied this question.