UCLA

Let $(M, \tau)$ be a tracial von Neumann algebra. For $X \in (M, \tau)$, the Brown measure of $X$ is a complex probability measure supported on the spectrum of $X$. It is the spectral measure when $X$ is normal and the empirical spectral distribution when $X$ is a random matrix. We consider operators of the form $X = p + i q$, where $p, q \in (M, \tau)$ are Hermitian, freely independent, and the spectral distributions of $p$ and $q$ consist of finitely many atoms. There is an associated random matrix model $X_n = P_n + i Q_n$, where $P_n, Q_n \in M_n(\mathbb{C})$ are independently Haar-rotated Hermitian matrices. Using a Hermitization technique, we will compute the Brown measure of $X = p + i q$ when the spectral distributions of $p$ and $q$ are $2$ atoms and prove the convergence of the empirical spectral distribution of $X_n$ to the Brown measure of $X$ when the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$. For the general case, we will relate the operator-valued Cauchy transform from the mathematical literature to the Quaternionic analogue of the Cauchy transform in the physics literature. When $X = p + i q$ for $p, q \in (M, \tau)$ Hermitian and freely independent, the Quaternionic Cauchy transform provides heuristics for the boundary and support of the Brown measure of $X$. We verify these heuristics when the spectral distributions of $p$ and $q$ are $2$ atoms, and show that in general, the heuristic implies that the boundary of $X = p + i q$ is an algebraic curve. We conclude by discussing the atoms of the Brown measure.