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

TL;DR

The paper develops a general framework to compute the absolutely continuous part of the Brown measure for $x+iy$ with $x,y$ free and $y$ free Poisson of parameter $p$. Central to the approach is a matrix-valued subordination setup for the hermitized element and a tractable left inverse $H$, yielding a reparametrization $h:\mathscr{D}\to\mathscr{M}$ via $h(\lambda)=H_{12}(\lambda,i\delta_0(\lambda))$ and $\delta_0(\lambda)$ defined by $\mathrm{Im}\,H_{11}(\lambda,i\delta_0)=0$. The main result expresses the density of the absolutely continuous Brown measure on $\mathscr{M}$ as $f(s,t)=\frac{1}{4\pi}\left[\frac{2}{t}\left(\frac{\partial \alpha}{\partial s}+\frac{\partial \beta}{\partial t}\right)-\frac{2}{t}-\frac{2\beta}{t^2}\right]$ with $\alpha+i\beta=h^{-1}(s+it)$, under technical assumptions on $x$ and $p$ that ensure the domain geometry and Jacobians are well-behaved. A detailed worked example with $x$ Bernoulli and $p=1$ demonstrates explicit $H$, $\mathscr{D}$, $h$, $\delta_0$, and the boundary of $\mathscr{M}$, and confirms that the Brown measure support coincides with the spectrum via a Lehner-type resolvent criterion. The work combines matrix-valued subordination, analytic extension, and Jacobian analysis to provide a concrete computable density formula for a broad class of non-normal free-probabilistic models with free Poisson components.

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 absolutely continuous part of the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $Ω$ of the Hermitization of $x + i y$, and on the fact that $Ω$ 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 Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: \[ f(s + i \, t) =\frac{1}{4π}\left[\frac{2}{t}\left(\frac{\partial α}{\partial s} +\frac{\partial β}{\partial t}\right)-\frac{2}{t}-\frac{2β}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $α, β: \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$.

Figures (3)

• Figure 1: The sets $\mathscr{D}$ and $\mathscr{M}$ in the case when $p = 1$ and $x$ has symmetric Bernoulli distribution. One gets $\mathscr{D} = \bigl\{ \lambda \in \mathbb{C} : 1+| \lambda |^2 > | 1 - \lambda ( \lambda - \mathrm{i} ) |^2 \, \bigr\}$, or, in $\mathbb{R} ^2$-coordinates: $\mathscr{D} = \bigl\{ (\alpha,\beta)\in \mathbb{R} ^2 : \left(\alpha ^2+(1-\beta ) \beta +1\right)^2+(\alpha (\beta -1)+\alpha \beta )^2-\alpha ^2-\beta ^2-1<0 \, \bigr\}$. Then $\mathscr{M} =\{(s,t)\in \mathbb{R} ^2:4 \left(s^2-1\right) t^3+2 \left(s^2+1\right) \left(s^2-1\right)^2 t^2+\left(s^2-1\right)^4+\left(s^4-s^2+1\right) t^4<0\}$
• Figure 2: The blue line represents the boundary of $\mathscr{M}$. The red dots are the eigenvalues of the matrix approximation $X_N+ \mathrm{i} Y_n$.
• Figure 3: Density of the Brown measure of $x+ \mathrm{i} y$. Observe that the density has unbounded limits at $\pm1$; this matches how eigenvalues were concentrated around those points in Figure \ref{['fig:2']}.

Theorems & Definitions (53)

• Proposition 1.2
• Definition 1.3
• Remark 1.4
• Proposition 1.5
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Lemma 2.1
• proof
• Lemma 2.2