On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson

[2512.23528] On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson Skip to main content Search Submit Donate Log in Mathematics > Operator Algebras arXiv:2512.23528 (math) [Submitted on 29 Dec 2025 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson Authors: Franz Lehner , Alexandru Nica , Kamil Szpojankowski , Ping Zhong View PDF HTML (experimental) Abstract: Let $x,y$ be freely independent selfadjoint elements in a $W^{*}$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $\Omega$ of the hermitization of $x + i y$, and on the fact that $\Omega$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x + i y$ becomes more approachable when it is reparametrized via a certain change of variable $h : \mathcal{D} \to \mathcal{M}$, with $\mathcal{D}, \mathcal{M}$ open subsets of $\mathbb{C}$, where $\\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{cl} \,(\mathcal{M})$ contains the support of the absolutely continuous part of Brown measure. More precisely, we find (with some conditions on the distribution of $x$) the following formula: \[ f(s + i \, t) =\frac{1}{2 \pi}\left[\frac{1}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{1}{t}-\frac{\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha, \beta : \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$. We show that if $x$ has an atom $\alpha$ with $\mu_x(\alpha)>p$, then the Brown measure of $x+iy$ has an atom of mass $\mu_x(\alpha)-p$ at the same point $\alpha$. Moreover we prove that if $\alpha_1,\ldots,\alpha_k$ is the list of atoms of $x$ with mass bigger than $p$, then the Brown measure of $x+iy$ is supported on $\mathrm{cl}(\mathcal{M})\cup\{\alpha_1,\ldots,\alpha_k\}$. Comments: Major revision: main assumption is changed, in the new version we describe the atoms of the Brown measure Subjects: Operator Algebras (math.OA) ; Mathematical Physics (math-ph); Probability (math.PR) Cite as: arXiv:2512.23528 [math.OA] (or arXiv:2512.23528v3 [math.OA] for this version) https://doi.org/10.48550/arXiv.2512.23528 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Kamil Szpojankowski [ view email ] [v1] Mon, 29 Dec 2025 15:06:59 UTC (465 KB) [v2] Tue, 26 May 2026 14:34:36 UTC (468 KB) [v3] Thu, 27 Aug 2026 18:21:04 UTC (441 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OA < prev | next > new | recent | 2025-12 Change to browse by: math math-ph math.MP math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson#684320
• Characterization on spectra of self-similar measures and the dual spectral set conjecture#973455
• Induced random mixed states are symmetric Dirichlet mixtures#987170
• Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian Free Field#609683
• Quantum Systems as Indivisible Stochastic Processes#687799