Convergence of the Laws of Non-Hermitian Sums of Projections
Max Sun Zhou
TL;DR
The paper proves that the empirical spectral distribution of X_n = P_n + i Q_n, with P_n and Q_n independently Haar-rotated Hermitian and having at most two atoms in their spectra, converges almost surely in the vague topology to the Brown measure of X = p + i q where p and q are Hermitian and freely independent. The authors employ the Hermitization technique and exploit the algebraic structure of projections to obtain the necessary minimum singular value bounds, establishing convergence for the two-atom case and extending via limiting arguments. A converse is also shown: any deterministic vague limit of X_n must be the Brown measure of some X = p + i q, thereby characterizing all possible limits in this model. The results connect non-Hermitian random matrix limits with operator-valued free probability and provide a precise limit description in a natural Haar-rotated projections setting, including explicit geometry via the hyperbola and rectangle associated with X = p + i q.
Abstract
We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, τ)$ be a tracial von Neumann algebra and let $p, q \in (M, τ)$, where $p$ and $q$ are Hermitian and freely independent. Our main result is the following convergence result: if the law of $P_n$ converges to the law of $p$ and the law of $Q_n$ converges to the law of $q$, then the empirical spectral distributions of the $X_n$ converges to the Brown measure of $X = p + i q$. To prove this, we use the Hermitization technique introduced by Girko, along with the algebraic properties of projections to prove the key estimate. We also prove a converse statement by using the properties of the Brown measure of $X$.