The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

Abstract

We compute the Brown measure of $x_0+i{\sigma }_t$, where ${\sigma }_t$ is a free semicircular Brownian motion and $x_0$ is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region ${\Omega }_t$ in the plane. In ${\Omega }_t,$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $x_0+i{\sigma }_t$ by a certain map $Q_t:{\Omega }_t\to \mathbb{R}$ gives the distribution of $x_0+{\sigma }_t.$ We also establish a similar result relating the Brown measure of $x_0+i{\sigma }_t$ to the Brown measure of $x_0+c_t$, where $c_t$ is the free circular Brownian motion.

arXiv comment50 pages and 9 figures. Minor revisions in this version. To appear on Letters in Mathematical Physics

References in corpus (14)

• Random matrix model of QCD at finite density and the nature of the quenched limit
• Nonhermitean Random Matrix Models : a Free Random Variable Approach
• Dysonian dynamics of the Ginibre ensemble
• On the Law of Addition of Random Matrices
• Subordination for the sum of two random matrices
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method
• Eikonal formulation of large dynamical random matrix models
• A Novel Approach to Non-Hermitian Random Matrix Models
• PDE methods in random matrix theory
• The Brown measure of a family of free multiplicative Brownian motions
• The Brown measure of unbounded variables with free semicircular imaginary part
• Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra
• Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Cited by in corpus (6)

• Real spectra of large real asymmetric random matrices
• Itô's formula for noncommutative $C^2$ functions of free Itô processes
• The Brown measure of a family of free multiplicative Brownian motions
• Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra
• The Brown measure of unbounded variables with free semicircular imaginary part
• The Brown measure of the sum of a self-adjoint element and an elliptic element