Brown measures of deformed $L^\infty$-valued circular elements
| Autor: | Alt, Johannes, Krüger, Torben |
|---|---|
| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider the Brown measure of $a+\mathfrak{c}$, where $a$ lies in a commutative tracial von Neumann algebra $\mathcal{B}$ and $\mathfrak{c}$ is a $\mathcal{B}$-valued circular element. Under certain regularity conditions on $a$ and the covariance of $\mathfrak{c}$ this Brown measure has a density with respect to the Lebesgue measure on the complex plane which is real analytic apart from jump discontinuities at the boundary of its support. With the exception of finitely many singularities this one-dimensional spectral edge is real analytic. We provide a full description of all possible edge singularities as well as all points in the interior, where the density vanishes. The edge singularities are classified in terms of their local edge shape while internal zeros of the density are classified in terms of the shape of the density locally around these points. We also show that each of these countably infinitely many singularity types occurs for an appropriate choice of $a$ when $\mathfrak{c}$ is a standard circular element. The Brown measure of $a+\mathfrak{c}$ arises as the empirical spectral distribution of a diagonally deformed non-Hermitian random matrix with independent entries when its dimension tends to infinity. Comment: 49 pages, 5 figures. We provide a full analysis of the regularity and singularities of the Brown measure of deformed B-valued ciruclar elements. Parts of this analysis overlap with the content of our preprint arXiv:2404.17573. The next version of arXiv:2404.17573 will be adjusted to remove this overlap |
| Databáze: | arXiv |
| Externí odkaz: |
|