# The Brown measure of unbounded variables with free semicircular imaginary
part

Ching-Wei Ho
Department of Mathematics
Indiana University
Bloomington, IN 47405
<EMAIL_ADDRESS>
###### Abstract

Let $x_{0}$ be a possibly-unbounded self-adjoint random variable,
$\tilde{\sigma}_{\alpha}$ and $\sigma_{\beta}$ are semicircular variables with
variances $\alpha\geq 0$ and $\beta>0$ respectively (when $\alpha=0$,
$\tilde{\sigma}_{\alpha}=0$). Suppose $x_{0}$, $\sigma_{\alpha}$, and
$\tilde{\sigma}_{\beta}$ are all freely independent. We compute the Brown
measure of $x_{0}+\tilde{\sigma}_{\alpha}+i\sigma_{\beta}$, extending the
previous computations which only work for bounded self-adjoint random variable
$x_{0}$. The Brown measure in this unbounded case has the same structure as in
the bounded case; it has connections to the free convolution
$x_{0}+\sigma_{\alpha+\beta}$. We also compute the example where $x_{0}$ is
Cauchy-distributed.

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of results
  2. 2 Background and previous results
    1. 2.1 Free probability
    2. 2.2 Free addition convolution
    3. 2.3 The Brown measure
      1. 2.3.1 A PDE for the Brown measure of $x_{0}+i\sigma_{t}$ with bounded $x_{0}$
    4. 2.4 Noncommutative measure theory
  3. 3 The differential equations for the unbounded case
    1. 3.1 The PDE
    2. 3.2 Solving the PDE
    3. 3.3 Solving the ODEs
  4. 4 The Brown measure of $x_{0}+i\sigma_{t}$
    1. 4.1 The domain
    2. 4.2 Surjectivity
    3. 4.3 The Brown measure computation
  5. 5 The Brown measure of $x_{0}+c_{\alpha,\beta}$
  6. 6 Examples: Cauchy case
    1. 6.1 Adding a circular variable
    2. 6.2 Adding an elliptic variable
    3. 6.3 Adding an imaginary multiple of semicircular variable
  7. 7 Acknowledgments

##  1 Introduction

An elliptic variable $c_{\alpha,\beta}$ is an element in a $W^{*}$-probability
space $(\mathscr{A},\tau)$ of the form
$\tilde{\sigma}_{\alpha}+i\sigma_{\beta}$ where $\tilde{\sigma}_{\alpha}$ and
$\sigma_{\beta}$ are free semicircular random variables with variances
$\alpha>0$ and $\beta>0$ respectively. In this paper, we also consider the
case $\alpha=0$; in this case, $c_{\alpha,\beta}$ is a “degenerate” elliptic
variable, and is just a imaginary multiple of semicircular variable. (We do
not consider the case $\alpha>0$ and $\beta=0$ in this paper. In this case,
$c_{\alpha,\beta}$ is also a degenerate elliptic variable — a semicircular
variable.) The elliptic variable $c_{\alpha,\beta}$ is the limit in
$\ast$-distribution of the Gaussian random matrix
$\sqrt{\alpha}\tilde{X}_{N}+i\sqrt{\beta}X_{N}$ where $\tilde{X}_{N}$ and
$X_{N}$ are independent Gaussian unitary ensembles (GUEs); see [26] and the
book [19]. Also, the empirical eigenvalue distribution of the random matrix
$\sqrt{\alpha}\tilde{X}_{N}+i\sqrt{\beta}X_{N}$ converges to the Brown measure
of the elliptic variable $c_{\alpha,\beta}$ [11]. The Brown measure of
$c_{\alpha,\beta}$ has a simple structure; it is the uniform measure in a
region bounded by a certain ellipse [5].

Let $x_{0}$ be a self-adjoint noncommutative random variable in the
$W^{*}$-probability space $\mathscr{A}$ containing $c_{\alpha,\beta}$, and
that $x_{0}$ is freely independent from $c_{\alpha,\beta}$. In particular,
$x_{0}$ is a _bounded_ random variable. The author computed the Brown measure
of $x_{0}+c_{\alpha,\beta}$ in [15]. In this paper, we extend the computation
to an _unbounded_ self-adjoint random variable $x_{0}$ affiliated with
$\mathscr{A}$. When $x_{0}$ is an unbounded variable, the Brown measure of
$x_{0}+c_{\alpha,\beta}$ has the same structure as in the bounded case; in
particuar, it has direct connections to the Brown measure of $x_{0}+c_{s}$,
where $c_{s}=c_{s/2,s/2}$ is the circular variable with $s=\alpha+\beta$, and
the law of $x_{0}+\sigma_{s}$. We also compute, as an example, the Brown
measure of $x_{0}+c_{\alpha,\beta}$ for $\alpha\geq 0$ and $\beta>0$ when
$x_{0}$ has a Cauchy distribution.

When $x_{0}$ is a bounded random variable and is the limit in distribution of
a sequence $A_{N}$ of Hermitian random matrix independent from the GUEs
$\tilde{X}_{N}$ and $X_{N}$, Śniady [23] shows that the empirical eigenvalue
distribution of

$A_{N}+\sqrt{\alpha}\tilde{X}_{N}+i\sqrt{\beta}X_{N},\quad\alpha>0,\beta>0$

converges to the Brown measure of $x_{0}+c_{\alpha,\beta}$, as $N\to\infty$.
When $x_{0}$ is unbounded, computer simulations show that the Brown measure of
$x_{0}+c_{\alpha,\beta}$ is a reasonable candidate of the limiting eigenvalue
distribution of the random matrix model
$A_{N}+\sqrt{\alpha}\tilde{X}_{N}+i\sqrt{\beta}X_{N}$, where the empirical
eigenvalue distribution of $A_{N}$ converges to the law of $x_{0}$.

The computation of the Brown measure in this paper is based on the PDE method
introduced by Driver, Hall and Kemp in [8], and noncommutative integration
theory introduced by Segal in [22] (also see the books [17, 24] by Jajte and
Takesaki respectively). In [8], Driver, Hall and Kemp computed the Brown
measure of the free multiplicative Brownian motion using the Hamilton–Jacobi
method. Based on this PDE method, Zhong and the author [16] computed the Brown
measures of the circular Brownian motion with arbitrary bounded self-adjoint
initial condition $x_{0}$ and the free multiplicative Brownian motion with
arbitrary unitary initial condition. Later, Hall and the author [13] computed
the Brown measure of the sum of an arbitrary bounded self-adjoint random
variable $x_{0}$ and an imaginary multiple of semicircular variable
$\sigma_{t}$, freely independent from $x_{0}$. Using the results in [13], the
author [15] computed the Brown measure of the sum of a bounded self-adjoint
random variable $x_{0}$ and an elliptic variable $c_{\alpha,\beta}$ that is
freely independent from $x_{0}$. The results in [13, 15, 16] assume the
boundedness of $x_{0}$ in the process of computing the PDE. The key
observation in this paper is to show that the function

$S(t,\lambda,\varepsilon)=\tau[\log((x_{0}+i\sigma_{t}-\lambda)^{*}(x_{0}+i\sigma_{t}-\lambda)+\varepsilon)],\quad
t>0,\lambda\in\mathbb{C},\varepsilon>0$ (1.1)

satisfies the same PDE derived in [13] when $x_{0}$ is an unbounded random
variable affiliated with the $W^{*}$-probability space $\mathscr{A}$, using
noncommutative integration theory. The solution of the PDE yields the Brown
measure of $x_{0}+i\sigma_{t}$. By considering the free additive convolution
$x_{0}+\tilde{\sigma}_{\alpha}$ in place of $x_{0}$, we derive the family of
the Brown measures of $x_{0}+c_{\alpha,\beta}$ for $\alpha\geq 0$ and
$\beta>0$. In particular, by taking $\alpha=\beta>0$, we extend the results in
[16] to unbounded self-adjoint $x_{0}$.

The paper is organized as follows. Section 1.1 lists out all the Brown
measures computed in this paper. These Brown measures are not listed in the
order of being proved in this paper; they are listed according to the
complexity of their structures. Section 2 contains some basic definitions and
background from free probability theory and noncommutative measure theory. It
also contains the PDE computed in [13]. Section 3 contains the crucial theorem
of this paper — showing that we get the same PDE computed in [13] for the
function $S$ defined in (1.1), even when $x_{0}$ is unbounded. In the same
section, we analyze the PDE using the Hamilton–Jacobi method. The PDE is
already solved in [13]; the solution of the PDE is given without proof. In
Section 4, we use the solution of the PDE to compute the Brown measure of
$x_{0}+i\sigma_{t}$. The Brown measure has full measure on an open set
$\Omega_{x_{0},t}$. Since the law of $x_{0}$ now possibly has unbounded
support, we provide a proof whenever the argument of the corresponding
statement in [13] only works for bounded random variable $x_{0}$. In Section
5, we replace the random variable $x_{0}$ in the results in Section 4 by the
free convolution $x_{0}+\tilde{\sigma}_{\alpha}$ to compute the Brown measure
of $x_{0}+c_{\alpha,\beta}$. Finally, in Section 6, we compute the Brown
measure of $x_{0}+c_{\alpha,\beta}$ when $x_{0}$ has a Cauchy distribution.

###  1.1 Statement of results

In this subsection, we list out the Brown measures computed in this paper
according to the complexity of their structures. Let $(\mathscr{A},\tau)$ be a
$W^{*}$-probability space and $\tilde{\mathscr{A}}$ be the algebra of closed
densely defined operators affiliated with $\mathscr{A}$. Let
$x_{0}\in\tilde{\mathscr{A}}$ be self-adjoint satisfying the following
standing assumption throughout the paper.

######  Assumption 1.1.

The law $\mu_{x_{0}}$ of $x_{0}$ is not a Dirac mass at a point on
$\mathbb{R}$, and satisfies

$\int_{1}^{\infty}\log x\,d\mu_{x_{0}}(x)<\infty.$

If $\mu_{x_{0}}$ is a Dirac mass, then the Brown measure of
$x_{0}+c_{\alpha,\beta}$ is just the translation of the elliptic law in the
case $\alpha>0$, or just the translation of the semicircular law on the
imaginary axis in the case $\alpha=0$. We list out the results for $x_{0}$
satisfying Assumption 1.1 below.

We start by Brown measure of $x_{0}+c_{t}$, where $c_{t}=c_{t/2,t/2}$ is the
circular variable, freely independent from $x_{0}$. Recall that $\sigma_{t}$
denotes a semicircular variable of variance $t$, freely independent from
$x_{0}$. The following theorem is proved in Corollaries 5.6 and 5.7(2). When
$x_{0}$ is bounded, this theorem is established in Theorems 3.13 and 3.14 in
[16].

######  Theorem 1.2.

The following statements about the Brown measure of $x_{0}+c_{t}$ hold.

  1. 1.

The Brown measure of $x_{0}+c_{t}$ has full measure on an open set
$\Lambda_{x_{0},t}$ of the form

$\Lambda_{x_{0},t}=\\{\left.u+iv\in\mathbb{C}\right||v|<v_{x_{0},t}(u)\\}$

for a certain function $v_{x_{0},t}$ that appears in the study of the free
convolution $x_{0}+\sigma_{t}$. Then the density of the Brown measure on
$\Lambda_{x_{0},t}$ has the special form

$w_{t}(u+iv)=\frac{1}{\pi
t}\left(1-\frac{t}{2}\frac{d}{du}\int\frac{x\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}\right),\quad
u+iv\in\Lambda_{x_{0},t};$

in particular, the density is strictly positive and constant along vertical
segments in $\Lambda_{x_{0},t}$.

  2. 2.

Consider the function

$\psi_{x_{0},t}(u)=u+t\int\frac{(u-x)\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}},\quad
u\in\mathbb{R}$ (1.2)

that appears in the study of the free convolution $x_{0}+\sigma_{t}$. Then the
push-forward of the Brown measure of $x_{0}+c_{t}$ by the map
$u+iv\mapsto\psi_{x_{0},u}(u)$ is the law of $x_{0}+\sigma_{t}$.

The Brown measure of $x_{0}+c_{s}$ “generates” the family of the Brown
measures of $x_{0}+c_{\alpha,\beta}$ with $\alpha+\beta=s$ in the sense of
push-forward measure. The following theorem is established in Theorem 5.4 and
Corollary 5.7. This result extends Theorems 3.7 and 4.1 in [15] to possibly-
unbounded $x_{0}$ affiliated with $\mathscr{A}$.

######  Theorem 1.3.

The following about the Brown measure of $x_{0}+c_{\alpha,\beta}$, for
$\alpha\geq 0$ and $\beta>0$ holds.

  1. 1.

The Brown measure of $x_{0}+c_{\alpha,\beta}$ has full measure on an open set
$\Omega_{x_{0},t}$ of the form

$\Omega_{x_{0},\alpha,\beta}=\\{\left.u+iv\in\mathbb{C}\right||v|<\varphi_{x_{0},\alpha,\beta}(u)\\}$
(1.3)

for a certain function $\varphi_{x_{0},\alpha,\beta}$. There exists a strictly
increasing homeomorphism $f_{x_{0},\alpha,\beta}$ on $\mathbb{R}$ such that,
by writing $u=f_{x_{0},\alpha,\beta}(u_{0})$, the density of the Brown measure
on $\Omega_{x_{0},\alpha,\beta}$ has the special form

$w_{\alpha,\beta}(u+iv)=\frac{1}{4\pi\beta}\left(1+2\beta\frac{d}{du}\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}}\right),\quad
u+iv\in\Omega_{x_{0},\alpha,\beta},$

where $s=\alpha+\beta$. The density is strictly positive and constant along
vertical segments in $\Omega_{x_{0},\alpha,\beta}$. Furthermore, by writing
$u=f_{x_{0},\alpha,\beta}(u_{0})$ again, the function
$\varphi_{x_{0},\alpha,\beta}$ in (1.3) is given by
$\varphi_{x_{0},\alpha,\beta}(u)=v_{x_{0},s}(u_{0})$ where $v_{x_{0},s}$ is
defined as in Theorem 1.2(1)

  2. 2.

Write $s=\alpha+\beta$. Let
$U_{\alpha,\beta}:\overline{\Lambda}_{x_{0},s}\to\overline{\Omega}_{x_{0},\alpha,\beta}$
be defined by

$U_{\alpha,\beta}(u+iv)=f_{x_{0},\alpha,\beta}(u)+i\frac{2\beta v}{s}.$

Then $U_{\alpha,\beta}$ is a homeomorphism, and the push-forward of the Brown
measure of $x_{0}+c_{s}$ under the map $U_{\alpha,\beta}$ is the Brown measure
of $x_{0}+c_{\alpha,\beta}$.

  3. 3.

Let $s=\alpha=\beta$. Recall the function $\psi_{x_{0},s}$ is defined in
(1.2). The push-forward of the Brown measure of $x_{0}+c_{\alpha,\beta}$ by
the map

$Q_{\alpha,\beta}(u+iv)=\begin{cases}\frac{1}{\alpha-\beta}[su-2\beta
f_{x_{0},\alpha,\beta}^{-1}(u)]\quad&\textrm{if $\alpha\neq\beta$}\\\
\psi_{x_{0},s}(u)\quad&\textrm{if $\alpha=\beta=s/2$}.\end{cases}$

is the law of $x_{0}+\sigma_{s}$.

By putting $\alpha=0$ and $\beta=t$, the above theorem recovers Theorem 4.5 in
this paper which computes the Brown measure of $x_{0}+i\sigma_{t}$.

We close this section by a matrix simulation. Let $A_{N}$ be an $N\times N$
diagonal random matrix whose diagonal entries are independent Cauchy-
distributed random variables; that is, each diagonal entry of $A_{N}$ is a
random variable with density

$\frac{1}{\pi}\frac{1}{x^{2}+1},\quad x\in\mathbb{R}.$

Also, let $X_{N}$ be an $N\times N$ GUE, independent from $A_{N}$. Figure 1
plots a computer simulation of the eigenvalues of $A_{N}+iX_{N}$ with $N=5000$
and the density of the Brown measure of $x_{0}+i\sigma_{1}$ where $x_{0}$ has
the Cauchy distribution. This simulation suggests that the Brown measure is a
reasonable candidate of the limiting eigenvalue distribution even in the
unbounded case.

Figure 1: $5000\times 5000$ matrix simulation of $A_{5000}+iX_{5000}$ and the
Brown measure densty of $x_{0}+i\sigma_{1}$.

##  2 Background and previous results

###  2.1 Free probability

We first introduce noncommutative random variables and freeness (or free
independence).

######  Definition 2.1.

  1. 1.

A $W^{*}$-probability space $(\mathscr{A},\tau)$ is a finite von Neumann
algebra $\mathscr{A}$ with a normal, faithful, tracial state $\tau$.

  2. 2.

The algebra $\tilde{\mathscr{A}}$ affiliated with $\mathscr{A}$ is the set of
closed, densely defined operators such that every $a\in\mathscr{A}$ has a
polar decomposition

$a=u|a|=u\int_{0}^{\infty}\lambda\,dE_{|a|}(\lambda)$

where $u\in\mathscr{A}$ is unitary and the projection-valued spectral measure
$E_{|a|}$ takes values in $\mathscr{A}$. The elements in $\tilde{\mathscr{A}}$
are called (noncommutative) random variables. When we distinguish elements
from $\tilde{\mathscr{A}}$ and $\mathscr{A}$, any $a\in\mathscr{A}$ is called
a bounded random variable and any
$a\in\tilde{\mathscr{A}}\setminus\mathscr{A}$ is called an unbounded random
variable.

  3. 3.

Given any self-adjoint random variable $a\in\tilde{\mathscr{A}}$ with spectral
measure $E_{a}$, the law $\mu_{a}$ of $a$ is the probability measure on
$\mathbb{R}$ defined by

$\mu_{a}(B)=\tau(E_{a}(B))$

for any Borel set $B$.

  4. 4.

The von Neumann subalgebras
$\mathscr{A}_{1},\ldots,\mathscr{A}_{n}\subset\mathscr{A}$ are said to be
free, or freely independent, if given any
$i_{1},\ldots,i_{m}\in\\{1,\ldots,n\\}$ with $i_{k}\neq i_{k+1}$ and
$a_{i_{k}}\in\mathscr{A}_{i_{k}}$ for all $k$ satisfying $\tau(a_{i_{k}})=0$
for all $1\leq k\leq m$, we have

$\tau(a_{i_{1}}\cdots a_{i_{m}})=0.$

The random variables $a_{1},\ldots,a_{m}\in\tilde{\mathscr{A}}$ are free, or
freely independent, if the von Neumann subalgebras $\mathscr{A}_{k}$ generated
by all the spectral projections of $a_{k}$ are free.

In this paper, we denote $\sigma_{t}$ (or $\tilde{\sigma}_{t}$) by the
semicircular variable with variance $t$; that is, the law $\mu_{\sigma_{t}}$
is given by

$\frac{1}{2\pi t}\sqrt{4t-x^{2}}\mathbbm{1}_{|x|\leq 2\sqrt{t}}\,dx.$

One way to study the law of $a\in\tilde{\mathscr{A}}$ is to use the Cauchy
transform $G_{a}$ of $a$ defined by

$G_{a}(z)=\tau[(z-a)^{-1}]=\int\frac{d\mu_{a}(x)}{z-x},\quad
z\in\mathbb{C}^{+},$

where $\mathbb{C}^{+}$ is the upper half plane. The Cauchy transform is
determined by the law $\mu_{a}$ of $a$. Given any finite measure $\mu$ on
$\mathbb{R}$, one can define the Cauchy transform $G_{\mu}$ of $\mu$ by
replacing $\mu_{a}$ in the right hand side of the above equation by $\mu$. We
can recover the law $\mu_{a}$ from $G_{a}$ by the Stieltjes inversion formula

$d\mu_{a}(x)=-\frac{1}{\pi}\lim_{\varepsilon\to
0^{+}}\mathrm{Im}\,G_{a}(x+i\varepsilon)\,dx$

where the limit is in weak sense.

The Cauchy transform $G_{a}$ of $a$ is invertible on some truncated Stolz
angle

$\Gamma_{\alpha,\beta}=\\{\left.x+iy\in\mathbb{C}\right|y>\beta\textrm{ and
}x<\alpha y\\}$

for positive $\alpha$ and $\beta$. The inverse of $G_{a}$ is called the
$K$-transform of $a$ and is denoted by $K_{a}$. The $R$-transform $R_{a}$ is
defined as

$R_{a}(z)=K_{a}(z)-\frac{1}{z}.$

The following fact about $R$-transform was first discovered by Voiculescu [25]
for bounded random variables, then by Maassen [18] for possibly unbounded
random variables with finite variance, and by Bercovici and Voiculescu [2] for
arbitrary unbounded random variables.

######  Theorem 2.2.

If $a,b\in\tilde{\mathscr{A}}$ are self-adjoint freely independent random
variables, then, on the common domain where the $R$-transforms $R_{a}$,
$R_{b}$, and $R_{a+b}$ are defined,

$R_{a+b}=R_{a}+R_{b}.$

###  2.2 Free addition convolution

When $x_{0}\in\mathscr{A}$ is a bounded self-adjoint random variable free from
$\sigma_{t}$, the Brown measure of $x_{0}+i\sigma_{t}$ has direct connections
to $x_{0}+\sigma_{t}$, by [13, 16]. In this paper, we will show that this is
also the case for unbounded self-adjoint random variable $x_{0}$. In this
section, we review the free additive convolution of $x_{0}$ and $\sigma_{t}$,
where $x_{0}$ can be taken to be unbounded.

Biane [3] computed the free additive convolution of a self-adjoint random
variable $x_{0}\in\tilde{\mathscr{A}}$ and the semicircular variable
$\sigma_{t}$. Before we state the results, we first introduce some notations.

######  Definition 2.3.

Denote by $\mu_{x_{0}}$ the law of the self-adjoint random variable
$x_{0}\in\tilde{\mathscr{A}}$.

  1. 1.

Let $H_{x_{0},t}$ be defined by

$H_{x_{0},t}(z)=z+t\int\frac{d\mu_{x_{0}}(x)}{z-x},\quad
z\not\in\mathrm{supp}(\mu_{x_{0}})$ (2.1)

where $\mathrm{supp}(\mu_{x_{0}})$ denotes the support of $\mu_{x_{0}}$.

  2. 2.

Define a nonnegative function on $\mathbb{R}$ by

$v_{x_{0},t}(u)=\inf\left\\{v>0\left|\int\frac{d\mu_{x_{0}}(x)}{(u-x)^{2}+v^{2}}\leq\frac{1}{t}\right.\right\\}.$

This means, for a given $u\in\mathbb{R}$, $v_{x_{0},t}(u)$ is the unique
positive number (if exists) such that
$\int\frac{d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}=\frac{1}{t}$; if
such a number does not exist, $v_{x_{0},t}(u)=0$.

  3. 3.

Let $\Gamma_{x_{0},t}=\\{u+iv\in\mathbb{C}\left|v>v_{x_{0},t}(u)\right.\\}$ be
the region in the upper half plane $\mathbb{C}^{+}$ that is above the graph of
$v_{x_{0},t}$.

Recall that $G_{a}$ denotes the Cauchy transform of $a\in\tilde{\mathscr{A}}$.
The following theorem, due to Biane [3], computes the free additive
convolution of a self-adjoint random variable $x\in\tilde{\mathscr{A}}$ and
the semicircular variable $\sigma_{t}$.

######  Theorem 2.4 ([3]).

  1. 1.

The function $H_{x_{0},t}$ is an injective conformal map from
$\Gamma_{x_{0},t}$ onto the upper half plane $\mathbb{C}^{+}$. The function
extends to a homeomorphism from $\overline{\Gamma}_{x_{0},t}$ onto
$\mathbb{C}^{+}\cup\mathbb{R}$. Hence, for any $u\in\mathbb{R}$,
$H_{x_{0},t}(u+iv_{x_{0},t}(u))$ is real.

  2. 2.

The function $H_{x_{0},t}$ satisfies

$G_{x_{0}+\sigma_{t}}(H_{x_{0},t}(z))=G_{x_{0}}(z).$ (2.2)

  3. 3.

The law of $x_{0}+\sigma_{t}$ is absolutely continuous with respect to the
Lebesgue measure; its density $p_{x_{0},t}$ can be computed using the function
$\psi_{x_{0},t}(u)=H_{x_{0},t}(u+iv_{x_{0},t}(u))$. The function
$\psi_{x_{0},t}:\mathbb{R}\to\mathbb{R}$ is a homeomorphism, and

$p_{x_{0},t}(\psi_{x_{0},t}(u))=\frac{v_{x_{0},t}(u)}{\pi t}.$

When $a$ and $b$ are arbitrary self-adjoint random variables, the free
additive convolution $a+b$ can be studied using Cauchy transforms in a way
similar to (2.2) (the method of subordination functions). The subordination
functions are first discovered by Voiculescu [27] for bounded random variables
$a$ and $b$. Then it is further developed by Biane [4], and again by
Voiculescu [28] to a even more general setting. Belinschi and Bercovici [1]
use a complex analysis approach to show the existence of subordination
functions (where they do not assume the random variables to be bounded).

###  2.3 The Brown measure

In this section, we review the definition of the Brown measure, first
introduced by Brown [7] for operators in a von Neumann algebra with a
faithful, semifinite, normal trace, then extended to a class
$\mathscr{A}^{\Delta}$ of unbounded operators by Haagerup and Schultz [12]. We
first introduce the class $\mathscr{A}^{\Delta}$ of unbounded operators.

######  Definition 2.5.

We denote by $\mathscr{A}^{\Delta}$ the set of operators $a$ in
$\tilde{\mathscr{A}}$ such that

$\tau(\log^{+}|a|)=\int_{0}^{\infty}\log^{+}(\lambda)\,d\mu_{|a|}(\lambda)$

where $\log^{+}$ is defined to be $\log^{+}(x)=\max(\log x,0)$ for $x\geq 0$.

Let $a\in\mathscr{A}^{\Delta}$. Define a function $S$ by

$S(\lambda,\varepsilon)=\tau[\log(|a-\lambda|^{2}+\varepsilon)],\lambda\in\mathbb{C},\varepsilon>0.$

Then

$s(\lambda)=\lim_{\varepsilon\to
0^{+}}S(\lambda,\varepsilon),\quad\lambda\in\mathbb{C}$

is a subharmonic function, with value in $\mathbb{R}\cup\\{-\infty\\}$. The
exponential of $s(\lambda)/2$ is the Fuglede-Kadison determinant of
$a-\lambda$. The Fuglede-Kadison determinant for operators in a finite factor
was introduced by Fuglede and Kadison [10]. The Brown measure of $a$ is
defined to be

$\mathrm{Brown}(a)=\frac{1}{4\pi}\Delta s(\lambda)$

where the Laplacian is in distributional sense.

The first statement of the following theorem characterizes the operators in
$\mathscr{A}^{\Delta}$, and is proved in Lemma 2.4 of [12]. The second
statement is an important property of $\mathscr{A}^{\Delta}$, and is proved in
Propositions 2.5 and 2.6 of [12].

######  Theorem 2.6 ([12]).

  1. 1.

Let $a\in\tilde{\mathscr{A}}$. Then $a\in\mathscr{A}^{\Delta}$ is equivalent
to any of the following.

    1. (a)

There exist $x_{1}\in\mathscr{A}$ and $y_{1}\in\mathscr{A}$ such that
$a=x_{1}y_{1}^{-1}$ and

$\tau[\log|a|]=\tau[\log|x_{1}|]-\tau[\log|y_{1}|].$

    2. (b)

There exist $x_{2}\in\mathscr{A}$ and $y_{2}\in\mathscr{A}$ such that
$a=x_{2}^{-1}y_{2}$ and

$\tau[\log|a|]=\tau[\log|y_{2}|]-\tau[\log|x_{2}|]$

  2. 2.

The set $\mathscr{A}^{\Delta}$ is an algebra; that is, it is a subspace in
$\tilde{\mathscr{A}}$ and is closed under multiplication.

######  Remark 2.7.

An immediate consequence from Theorem 2.6 is that if
$x_{0}\in\mathscr{A}^{\Delta}$, then
$x_{0}+c_{\alpha,\beta}\in\mathscr{A}^{\Delta}$, for all $\alpha\geq 0$ and
$\beta>0$, so that the Brown measure of $x_{0}+c_{\alpha,\beta}$ is defined in
the sense of Haagerup and Schultz.

####  2.3.1 A PDE for the Brown measure of $x_{0}+i\sigma_{t}$ with bounded
$x_{0}$

In [13], Hall and the author computed the Brown measure of
$x_{t}=x_{0}+i\sigma_{t}$ where $x_{0}\in\mathscr{A}$ is a (bounded) self-
adjoint random variable with law $\mu_{x_{0}}$, and $\sigma_{t}\in\mathscr{A}$
is a semicircular variable, freely independent from $x_{0}$. Define

$S(t,\lambda,\varepsilon)=\tau[\log((x_{t}-\lambda)^{*}(x_{t}-\lambda)+\varepsilon)],\quad
t\geq 0,\lambda\in\mathbb{C},\varepsilon>0.$

Then the function $S$ satisfies the following PDE of Hamilton–Jacobi type.

######  Theorem 2.8.

Write $\lambda=u+iv$. Then $S$ satisfies the PDE

$\frac{\partial S}{\partial t}=\varepsilon\left(\frac{\partial
S}{\partial\varepsilon}\right)^{2}+\frac{1}{4}\left(\left(\frac{\partial
S}{\partial u}\right)^{2}-\left(\frac{\partial S}{\partial
v}\right)^{2}\right)$

with initial condition

$S(0,\lambda,\varepsilon)=\int\log((x-\lambda)^{*}(x-\lambda)+\varepsilon)\,d\mu_{x_{0}}(x).$

Hall and the author [13] analyze the function $S$ using the above PDE to
compute the Brown measure of $x_{t}$. The above PDE is obtained by the free
Itô formula developed by Biane and Speicher [6], which only applies to bounded
variable $x_{0}$.

The key observation in this paper is to extend Theorem 2.8 to self-adjoint
unbounded $x_{0}\in\mathscr{A}^{\Delta}$. The solution of this PDE yields the
Brown measure of $x_{0}+i\sigma_{t}$. The proof from some of the other results
in [13] will also need to modified for $\mu_{x_{0}}$ with unbounded support on
$\mathbb{R}$. We do not list all the results of [13] here; rather, we will
state the results in the process of computing the Brown measure, with modified
proof whenever necessary.

###  2.4 Noncommutative measure theory

Given a finite factor $\mathscr{A}$ (which is a subalgebra of the bounded
linear operators of some Hilbert space $H$) with a faithful, normal, tracial
state $\tau$, Segal [22] suggested that we can think of

$x\mapsto\tau(x)$

as a noncommtative version of integration, with the analogy that $\mathscr{A}$
is playing the role of $L^{\infty}$ of a measure space, and $\tau$ is playing
the role of an integral with respect to some measure. In the paper [22], Segal
introduced “pointwise” convergence in a von Neumann algebra (which is called
convergence in strong operator topology) and the theory of noncommutative
integration. The noncommutative version of Egoroff’s theorem that we introduce
in this section is due to Saitô [21]. A more complete introduction of
noncommtative integration can be found, for example, in the books [17, 24].

We start by defining projections and measure topology on $\mathscr{A}$ with
respect to $\tau$. The definition of measure topology can be found in
Definition 2.1 of [24, Chapter 9].

######  Definition 2.9.

  1. 1.

A operator $p\in\mathscr{A}$ is called a (self-adjoint) projection if
$p^{*}=p$ and $p^{2}=p$.

  2. 2.

The measure topology of $\mathscr{A}$ with respect to $\tau$ is the uniform
topology given by a neighborhood system
$\\{x+N(\varepsilon,\delta):\varepsilon,\delta>0\\}$ for $x\in\mathscr{A}$,
where $N(\varepsilon,\delta)$ is the set of $a\in\mathscr{A}$ such that

$\|ap\|<\varepsilon\quad\textrm{and}\quad\tau(1-p)<\delta$

for some projection $p$. The completion of $\mathscr{A}$ with respect to this
measure topology is denoted by $\mathfrak{M}(\mathscr{A})$.

The following is a special case in Theorem 2.13(ii) of [24, Chapter 9].

######  Theorem 2.10.

For any $x\in\mathfrak{M}(M)$, set $\|x\|_{1}=\tau(|x|)$ and

$L^{1}(\mathscr{A},\tau)=\\{\left.x\in\mathfrak{M}(M)\right|\|x\|_{1}<\infty\\}.$

Then $\mathscr{A}$ can be embedded continuously into
$L^{1}(\mathscr{A},\tau)$, and $L^{1}(\mathscr{A},\tau)$ is a Banach space in
which $\mathscr{A}\cap L^{1}(\mathscr{A},\tau)$ is dense. Furthermore,
$L^{1}(\mathscr{A},\tau)$ is invariant under the multiplications of
$\mathscr{A}$ from both sides, and

$\|ax\|_{1}\leq\|a\|\|x\|_{1},\quad\|xa\|_{1}\leq\|a\|\|x\|_{1}$

for each $a\in\mathscr{A}$ and $x\in L^{1}(\mathscr{A},\tau)$.

We now define the strong operator topology on $\mathscr{A}$, which can be
thought as a noncommutative version of pointwise convergence of measurable
functions.

######  Definition 2.11.

Recall that we assume $\mathscr{A}$ is a von Neumann subalgebra of the bounded
linear operators of a Hilbert space $H$. Let $x_{n}$ be a sequence in
$\mathscr{A}$ and $x\in\mathscr{A}$. The sequence $x_{n}$ is said to be
convergent to $x$ in the strong operator topology if, given any $h\in H$,

$\lim_{n\to\infty}x_{n}h=xh.$

Recall that, in measure theory, the Egoroff’s theorem states that, on a finite
measure space, if a sequence of measurable functions $f_{n}$ converges to a
measurable function $f$ pointwise almost everywhere, then, given any
$\varepsilon>0$, the sequence $f_{n}$ converges to $f$ _uniformly_ except on a
measurable set $E$ with measure less than $\varepsilon$. The following
noncommutative version of the Egoroff’s theorem is a weaker version of Theorem
1.3.2 of [17].

######  Theorem 2.12 (Noncommutative Egoroff’s theorem).

Let $x_{n}$ be a sequence in $\mathscr{A}$ and let $x\in\mathscr{A}$. Suppose
that $x_{n}\to x$ in the strong operator topology. Then, for any
$\varepsilon>0$, there exists a projection $p$ in $\mathscr{A}$ and a
subsequence $x_{n_{k}}$ of $x_{n}$ such that $\tau(1-p)<\varepsilon$ and

$\lim_{k\to\infty}\|(x_{n_{k}}-x)p\|=0.$

In analogy to measure theory, the projection $p$ in the above theorem plays
the role of an indicator function of a measurable set that is almost of full
measure, and the norm convergence plays the role of uniform convergence.

##  3 The differential equations for the unbounded case

###  3.1 The PDE

Let $x_{0}\in\mathscr{A}^{\Delta}$ (See Definition 2.5) be self-adjoint and
write $x_{t}=x_{0}+i\sigma_{t}$ where $\sigma_{t}$ is a semicircular variable,
freely independent from $x_{0}$. By Remark 2.7, the Brown measure of
$x_{0}+i\sigma_{t}$ exists in the sense of Haagerup and Schultz [12].

Define

$S(t,\lambda,\varepsilon)=\tau[\log((x_{t}-\lambda)^{*}(x_{t}-\lambda)+\varepsilon)].$

The PDE in Theorem 2.8 was proved in [13] under the assumption that $x_{0}$ is
a bounded self-adjoint random variable in $\mathscr{A}$. In this section, we
extend Theorem 2.8 to arbitrary self-adjoint random variable
$x_{0}\in\mathscr{A}^{\Delta}$.

######  Theorem 3.1.

Write $\lambda=u+iv$. Then $S$ satisfies the PDE

$\frac{\partial S}{\partial t}=\varepsilon\left(\frac{\partial
S}{\partial\varepsilon}\right)^{2}+\frac{1}{4}\left(\left(\frac{\partial
S}{\partial u}\right)^{2}-\left(\frac{\partial S}{\partial
v}\right)^{2}\right)$

with initial condition

$S(0,\lambda,\varepsilon)=\int\log((x-\lambda)^{*}(x-\lambda)+\varepsilon)\,d\mu_{x_{0}}(x).$

Since $x_{0}\in\mathscr{A}^{\Delta}$ is self-adjoint, it has a spectral
decomposition

$x_{0}=\int x\,dE(x)$ (3.1)

where $E$ is the spectral measure of $x_{0}$. In this section, we denote
$p_{n}=E([-n,n])$ as a spectral projection of $x_{0}$, so that $x_{0}p_{n}$ is
a bounded random variable.

We first prove that $S$ is the pointwise limit of a functional on a sequene of
bounded random variables at $t=0$.

######  Lemma 3.2.

For any $\lambda\in\mathbb{C}$, and $\varepsilon>0$, we have

$S(0,\lambda,\varepsilon)=\lim_{n\to\infty}\tau[\log((x_{0}p_{n}-\lambda)^{*}(x_{0}p_{n}-\lambda)+\varepsilon)].$

The convergence is uniform for $(\lambda,\varepsilon)$ in every compact set in
$\mathbb{C}\times(0,\infty)$.

###### Proof.

First, $x_{0}p_{n}$ is self-adjoint because $x_{0}$ and $p_{n}$ commute, and
the spectral decomposition of $x_{0}p_{n}$ is

$x_{0}p_{n}=\int_{[-n,n]}x\,dE(x)$

where the spectral measure $E$ is as in (3.1). So, we have

$\tau[\log((x_{0}p_{n}-\bar{\lambda})(x_{0}p_{n}-\lambda)+\varepsilon)]=\int_{[-n,n]}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x).$

Fix a compact set $K\subset\mathbb{C}\times(0,\infty)$. Choose $n_{0}$ such
that $|x-\lambda|^{2}+\varepsilon>1$ for all $|x|>n_{0}$ whenever
$(\lambda,\varepsilon)\in K$. Then

$\lim_{n\to\infty}\int_{n\geq|x|>n_{0}}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x)=\int_{|x|>n_{0}}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x)$

by the monotone convergence theorem, and

$\displaystyle\quad\lim_{n\to\infty}\tau[\log((x_{0}p_{n}-\lambda)^{*}(x_{0}p_{n}-\lambda)+\varepsilon)]$
$\displaystyle=\int_{[-n_{0},n_{0}]}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x)+\lim_{n\to\infty}\int_{n\geq|x|>n_{0}}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x)$
$\displaystyle=\int_{\mathbb{R}}\log(|x-\lambda|^{2}+\varepsilon)\,d\mu_{x_{0}}(x)$
$\displaystyle=\tau[\log(|x_{0}-\lambda|^{2}+\varepsilon)],$

completing the proof. ∎

We try to compute the partial derivatives of $S$ through several functionals
of $x_{0}p_{n}$. We first introduce some notations.

######  Notation 3.3.

We will use the following notations throughout this section. Recall that
$x_{t}=x_{0}+i\sigma_{t}$.

  * •

For any $t\geq 0$ and $\lambda\in\mathbb{C}$, we write
$x_{n,t,\lambda}=x_{0}p_{n}+i\sigma_{t}-\lambda$ and
$x_{t,\lambda}=x_{t}-\lambda$. Define

$S_{n}(t,\lambda,\varepsilon)=\tau[\log(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)],\quad
t\geq 0,\lambda\in\mathbb{C},\varepsilon>0.$

  * •

Define the sequence of functions

$\displaystyle f_{n}(t,\lambda,\varepsilon)$
$\displaystyle=\tau[(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle g_{n}(t,\lambda,\varepsilon)$
$\displaystyle=\tau[x_{n,t,\lambda}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle h_{n}(t,\lambda,\varepsilon)$
$\displaystyle=\tau[x_{n,t,\lambda}^{*}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]$

for $t\geq 0,\lambda\in\mathbb{C},\varepsilon>0$.

  * •

We also define the functions

$\displaystyle f(t,\lambda,\varepsilon)$
$\displaystyle=\tau[(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle g(t,\lambda,\varepsilon)$
$\displaystyle=\tau[x_{t,\lambda}(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle h(t,\lambda,\varepsilon)$
$\displaystyle=\tau[x_{t,\lambda}^{*}(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}]$

for $t\geq 0$, $\lambda\in\mathbb{C}$ and $\varepsilon>0$.

We will see that $f_{n}$ and $f$ will be used to compute the
$\varepsilon$-derivative of $S$; the pairs $g_{n}$, $g$ and $h_{n}$, $h$ will
be used to compute the $\bar{\lambda}$\- and $\lambda$\- derivatives of $S$
respectively; furthermore, we will show that the function $S$ is well-
approximated by a subsequence of $S_{n}$. In the following, we first establish
that $f$, $g$, and $h$ are the limits of some subsequences $f_{n_{k}}$,
$g_{n_{k}}$, and $h_{n_{k}}$ respectively.

######  Lemma 3.4.

For any subsequence $f_{n_{k}}$ and any compact set
$K\subset[0,\infty)\times\mathbb{C}\times(0,\infty)$, we can further extract a
subsequence, still labeled by $f_{n_{k}}$, such that

$\lim_{k\to\infty}f_{n_{k}}(t,\lambda,\varepsilon)=f(t,\lambda,\varepsilon);$
(3.2)

the convergence is uniform for $(t,\lambda,\varepsilon)\in K$.

###### Proof.

We only prove for the full sequence $f_{n}$ for simpler notations. Let $K$ be
a compact set in $[0,\infty)\times\mathbb{C}\times(0,\infty)$. The function

$\varphi_{n}(t,\lambda,\varepsilon)=\left|f_{n}(t,\lambda,\varepsilon)-f(t,\lambda,\varepsilon)\right|$

is continuous on $K$. There exists $(t_{n},\lambda_{n},\varepsilon_{n})\in K$
such that

$\varphi_{n}(t_{n},\lambda_{n},\varepsilon_{n})=\max_{K}\varphi_{n}(t,\lambda,\varepsilon).$

By replacing $f_{n}$ by a subsequence of $f_{n}$ if necessary, we may assume
$(t_{n},\lambda_{n},\varepsilon_{n})$ converges to a point
$(t_{0},\lambda_{0},\varepsilon_{0})\in K$.

The operators
$(x_{n,t_{n},\lambda_{n}}^{*}x_{n,t_{n},\lambda_{n}}+\varepsilon_{n})^{-1}$
and $(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}$
are in $\mathscr{A}$. Also,
$(x_{n,t_{n},\lambda_{n}}^{*}x_{n,t_{n},\lambda_{n}}+\varepsilon_{n})^{-1}$
converges to
$(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}$ in the
strong operator topology. By the noncommutative Egoroff’s theorem (Theorem
2.12), given any $\delta>0$, there exists a projection $p$ such that
$\tau(1-p)<\delta$ and there exists a subsequence
$(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}$
such that

$\lim_{k\to\infty}\|\\{x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}-x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}\\}p\|=0,$

so that

$\limsup_{k\to\infty}\left|\tau\big{[}\big{\\{}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}-(x_{t_{0},\lambda_{0}}^{*}x_{t,\lambda_{0}}+\varepsilon_{0})^{-1}\big{\\}}p\big{]}\right|=0.$

Hence, by Theorem 2.10, for any $(t,\lambda,\varepsilon)\in K$,

$\displaystyle\limsup_{k\to\infty}\varphi_{n}(t,\lambda,\varepsilon)$
$\displaystyle\leq\limsup_{k\to\infty}\varphi_{n_{k}}(t_{n_{k}},\lambda_{n_{k}},\varepsilon_{n_{k}})$
$\displaystyle=\limsup_{k\to\infty}\left|\tau\big{[}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}-(x_{t_{0},\lambda_{0}}^{*}x_{t,\lambda_{0}}+\varepsilon_{0})^{-1}\big{]}\right|$
$\displaystyle\leq\tau(1-p)\cdot\left(\|(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}\|+\|(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}\|\right)$
$\displaystyle\leq\delta M,$

where $M=\max_{(t,\lambda,\varepsilon)\in K}(1/\varepsilon)<\infty$, since $K$
is compact. Since $\delta>0$ is arbitrary, uniform convergence (3.3) for
$f_{n_{k}}$ on $K$ is established. ∎

######  Lemma 3.5.

Given any compact set $K\subset[0,\infty)\times\mathbb{C}\times(0,\infty)$ and
subsequences $g_{n_{k}}$ and $h_{n_{k}}$, we can further extract subsequences,
still labeled by $g_{n_{k}}$ and $h_{n_{k}}$ (with the same $n_{k}$), such
that

$\lim_{k\to\infty}g_{n_{k}}(t,\lambda,\varepsilon)=g(t,\lambda,\varepsilon)$
(3.3)

and

$\lim_{k\to\infty}h_{n_{k}}(t,\lambda,\varepsilon)=h(t,\lambda,\varepsilon),$

where the convergeces are uniform for $\lambda\in K$.

######  Remark 3.6.

The indices $n_{k}$ of the subsequence can be chosen to be the same for Lemmas
3.4 and 3.5, by applying Lemma 3.5 to $h_{n_{k}}$ and $g_{n_{k}}$, where
$n_{k}$ is chosen from Lemma 3.4 for $f_{n}$. Then we can get the same indices
$n_{k_{j}}$ for the sequences of $f_{n}$, $g_{n}$, and $h_{n}$ such that the
conclusions of Lemmas 3.4 and 3.5 hold.

###### Proof.

We only prove for the full sequences $g_{n}$ and $h_{n}$, for simpler
notations. Let $K$ be a compact set in
$[0,\infty)\times\mathbb{C}\times(0,\infty)$. The function

$\phi_{n}(t,\lambda,\varepsilon)=\left|g_{n}(t,\lambda,\varepsilon)-g(t,\lambda,\varepsilon)\right|$

is continuous on $K$. There exists $(t_{n},\lambda_{n},\varepsilon_{n})\in K$
such that

$\phi_{n}(t_{n},\lambda_{n},\varepsilon_{n})=\max_{K}\phi_{n}(t,\lambda,\varepsilon).$

By replacing $g_{n}$ by a subsequence of $g_{n}$ if necessary, we may assume
$(t_{n},\lambda_{n},\varepsilon_{n})$ converges to a point
$(t_{0},\lambda_{0},\varepsilon_{0})\in K$.

The operators
$x_{n,t_{n},\lambda_{n}}(x_{n,t_{n},\lambda_{n}}^{*}x_{n,t_{0},\lambda_{n}}+\varepsilon_{n})^{-1}$
and
$x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t,\lambda_{0}}+\varepsilon_{0})^{-1}$
are in $\mathscr{A}$; furthermore, the sequence
$x_{n,t_{n},\lambda_{n}}(x_{n,t_{n},\lambda_{n}}^{*}x_{n,t_{n},\lambda_{n}}+\varepsilon_{n})^{-1}$
converges to
$x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}$
in the strong operator topology. By the noncommutative Egoroff’s theorem
(Theorem 2.12), given any $\delta>0$, there exists a projection $p$ such that
$\tau(1-p)<\delta$ and there exists a subsequence
$x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}$
such that

$\lim_{k\to\infty}\|\\{x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t,\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}-x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}\\}p\|=0.$

Hence, by Theorem 2.10, for any $(t,\lambda,\varepsilon)\in K$,

$\displaystyle\limsup_{k\to\infty}\phi_{n}(t,\lambda,\varepsilon)$
$\displaystyle\leq\limsup_{k\to\infty}\phi_{n_{k}}(t_{n_{k}},\lambda_{n_{k}},\varepsilon_{n_{k}})$
$\displaystyle\leq\limsup_{k\to\infty}\left|\tau\big{[}\\{x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}-x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}\\}(1-p)\big{]}\right|$
$\displaystyle\leq\tau(1-p)\cdot(\|x_{n_{k},t,\lambda}(x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}^{*}x_{n_{k},t_{n_{k}},\lambda_{n_{k}}}+\varepsilon_{n_{k}})^{-1}\|+\|x_{t_{0},\lambda_{0}}(x_{t_{0},\lambda_{0}}^{*}x_{t_{0},\lambda_{0}}+\varepsilon_{0})^{-1}\|)$
$\displaystyle\leq\delta M,$

where $M=\max_{(t,\lambda,\varepsilon)\in K}(1/\sqrt{\varepsilon})<\infty$,
since $K$ is compact. Since $\delta>0$ is arbitrary, uniform convergence (3.3)
for $g_{n_{k}}$ on $K$ holds.

The proof for $h_{n}$ is the similar. To obtain the same $n_{k}$ for $g_{n}$
and $h_{n}$, we prove for $h_{n_{k}}$ instead of $h_{n}$, and relabel the
subsequences of $g_{n_{k}}$ and $h_{n_{k}}$ again by $g_{n_{k}}$ and
$h_{n_{k}}$. ∎

The following elementary theorem about uniform convergence of derivatives will
be useful to analyze the function $S$. A proof of the theorem can be found in
Theorem 7.17 of [20].

######  Theorem 3.7 ([20]).

If $u_{n}$ is a sequence of differentiable functions on $[a,b]$ such that
$\lim_{n\to\infty}u_{n}(x_{0})$ exists (in $\mathbb{R}$) for some
$x_{0}\in[a,b]$ and the sequence $u_{n}^{\prime}$ converges uniformly on
$[a,b]$, then $u_{n}$ converges uniformly to a function $u$ on $[a,b]$, and
$u^{\prime}(x)=\lim_{n\to\infty}u_{n}^{\prime}(x)$ for $x\in[a,b]$.

######  Proposition 3.8.

We have

$S(t,\lambda,\varepsilon)=\lim_{k\to\infty}S_{n_{k}}(t,\lambda,\varepsilon)$

for some subsequence $S_{n_{k}}$. Furthermore, the convergence is uniform for
$\lambda$ in any compact set $K\subset\mathbb{C}$ and $\varepsilon$ in any
compact set $I\subset(0,\infty)$; the subsequences $f_{n_{k}}$, $g_{n_{k}}$
and $h_{n_{k}}$ converge uniformly on $[0,t]\times K\times I$.

###### Proof.

Note that $x_{0}p_{n}\in\mathscr{A}$. We can compute the partial derivatives
of $S_{n}$ by applying Lemma 1.1 in Brown’s paper [7], where Brown proves that
the derivative of the trace of the logarithm is given by

$\frac{d}{du}\tau[\log f(u)]=\tau\left[f(u)^{-1}\frac{df}{du}\right].$

Recall that $f_{n}$ and $g_{n}$ are defined in Notation 3.3. We must have

$\displaystyle\frac{\partial S_{n}}{\partial\lambda}$
$\displaystyle=-\tau[x_{n,t,\lambda}^{*}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=-h_{n}(t,\lambda,\varepsilon)$
$\displaystyle\frac{\partial S_{n}}{\partial\bar{\lambda}}$
$\displaystyle=-\tau[x_{n,t,\lambda}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=-g_{n}(t,\lambda,\varepsilon)$
$\displaystyle\frac{\partial S_{n}}{\partial\varepsilon}$
$\displaystyle=-\tau[(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=f_{n}(t,\lambda,\varepsilon).$

By Theorem 2.8, we have

$\frac{\partial S_{n}}{\partial
t}(s,\lambda,\varepsilon)=\varepsilon\,f_{n}(s,\lambda,\varepsilon)^{2}+\mathrm{Re}\left[h_{n}(s,\lambda,\varepsilon)^{2}\right].$
(3.4)

By Lemmas 3.4 and 3.5, there exists a subsequence $S_{n_{k}}$ such that
$\frac{\partial S_{n_{k}}}{\partial t}$ converges uniformly on $[0,t]\times
K\times I$. Also, $S_{n}(0,\lambda,\varepsilon)$ converges to
$S(0,\lambda,\varepsilon)$ uniformly for $(\lambda,\varepsilon)$ in any
compact set as $n\to\infty$ (Lemma 3.2); we must have
$S_{n_{k}}(s,\lambda,\varepsilon)$ converges uniformly to
$S(s,\lambda,\varepsilon)$ for $(s,\lambda,\varepsilon)\in[0,t]\times K\times
I$, by Theorem 3.7. ∎

Now, we are ready to compute the partial derivatives of $S$.

######  Proposition 3.9.

The partial derivatives of $S$ with respect to $\lambda$, $\bar{\lambda}$ and
$\varepsilon$ are given by

$\displaystyle\frac{\partial S}{\partial\lambda}$
$\displaystyle=-\tau[x_{t,\lambda}^{*}(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle\frac{\partial S}{\partial\bar{\lambda}}$
$\displaystyle=-\tau[x_{t,\lambda}(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}]$
$\displaystyle\frac{\partial S}{\partial\varepsilon}$
$\displaystyle=\tau[(x_{t,\lambda}^{*}x_{t,\lambda}+\varepsilon)^{-1}].$

###### Proof.

As in the proof of Proposition 3.8, we must have

$\displaystyle\frac{\partial S_{n}}{\partial\lambda}$
$\displaystyle=-\tau[x_{n,t,\lambda}^{*}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=-h_{n}(t,\lambda,\varepsilon)$
$\displaystyle\frac{\partial S_{n}}{\partial\bar{\lambda}}$
$\displaystyle=-\tau[x_{n,t,\lambda}(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=-g_{n}(t,\lambda,\varepsilon)$
$\displaystyle\frac{\partial S_{n}}{\partial\varepsilon}$
$\displaystyle=-\tau[(x_{n,t,\lambda}^{*}x_{n,t,\lambda}+\varepsilon)^{-1}]=f_{n}(t,\lambda,\varepsilon).$

Fix $\lambda\in\mathbb{C}$ and $\varepsilon>0$. Let $K$ be a compact set in
$\mathbb{C}\times(0,\infty)$ such that $(\lambda,\varepsilon)$ is in the
interior of $K$. By Proposition 3.8, there exist subsequences such that
$f_{n_{k}}$, $g_{n_{k}}$ and $h_{n_{k}}$ (with the same $n_{k}$) converge
uniformly to $f$, $g$ and $h$ on $[0,t]\times K$, in particular, on
$\\{t\\}\times K$. Since $f_{n_{k}}$, $g_{n_{k}}$ and $h_{n_{k}}$ are
$-\partial S_{n_{k}}/\partial\varepsilon$, $-\partial
S_{n_{k}}/\partial\bar{\lambda}$, and $-\partial S_{n_{k}}/\partial\lambda$
respectively, the partial derivatives of $S_{n_{k}}(t,\cdot,\cdot)$ converges
uniformly. Since Proposition 3.8 also concludes $S_{n_{k}}$ converges to $S$
uniformly at $(t,\lambda,\varepsilon)$ for $(\lambda,\varepsilon)\in K$. By
Theorem 3.7, the partial derivatives of $S_{n_{k}}$ converge to the partial
derivatives of $S$; that is, $-f$, $-g$, and $-h$ are the partial derivatives
of $S$ with respect to $\varepsilon$, $\bar{\lambda}$, and $\lambda$
respectively. ∎

###### Proof of Theorem 3.1.

By (3.4), we have, for any subsequence of $S$,

$\frac{\partial S_{n_{k}}}{\partial
t}(s,\lambda,\varepsilon)=\varepsilon\,f_{n_{k}}(s,\lambda,\varepsilon)^{2}+\mathrm{Re}\left[h_{n_{k}}(s,\lambda,\varepsilon)^{2}\right].$

If we write $\lambda=u+iv$, then, by Propositions 3.8 and 3.9,

$\displaystyle\frac{\partial S}{\partial t}$ $\displaystyle=\varepsilon
f(t,\lambda,\varepsilon)^{2}+\mathrm{Re}[h(t,\lambda,\varepsilon)]$
$\displaystyle=\varepsilon\left(\frac{\partial
S}{\partial\varepsilon}\right)^{2}+\frac{1}{4}\left[\left(\frac{\partial
S}{\partial u}\right)^{2}-\left(\frac{\partial S}{\partial
v}\right)^{2}\right],$

which is the desired PDE. The statement about the initial condition is
straightforward. ∎

###  3.2 Solving the PDE

As in [13], we use the Hamilton–Jacobi method to analyze the function $S$
through the PDE in Theorem 3.1. We define the Hamiltonian function
$H:\mathbb{R}^{6}\to\mathbb{R}$ by replacing the partial derivatives of $S$ in
the PDE by the corresponding momentum variables and putting an overall minus
sign. That is,

$H(u,v,\varepsilon,p_{u},p_{v},p_{\varepsilon})=-\frac{1}{4}(p_{u}^{2}-p_{v}^{2})-\varepsilon
p_{\varepsilon}^{2}.$

Then, we introduce the Hamiltonian system (the Hamilton’s equations) for $H$:

$\begin{array}[]{lll}\displaystyle\frac{du}{dt}=\frac{\partial H}{\partial
p_{u}};&\displaystyle\frac{dv}{dt}=\frac{\partial H}{\partial
p_{v}};&\displaystyle\frac{d\varepsilon}{dt}=\frac{\partial H}{\partial
p_{\varepsilon}};\\\\[10.0pt] \displaystyle\frac{dp_{u}}{dt}=-\frac{\partial
H}{\partial u};&\displaystyle\frac{dp_{v}}{dt}=-\frac{\partial H}{\partial
v};&\displaystyle\frac{dp_{\varepsilon}}{dt}=-\frac{\partial
H}{\partial\varepsilon}.\end{array}$ (3.5)

We note that the Hamiltonian function and the system of ODEs are the same as
the one in [13] because the PDE is the same.

Write $\lambda=u+iv$. The initial conditions of $u$, $v$, $\varepsilon$ for
the ODEs (3.5) can be chosen arbitrarily, and these initial conditions are
denoted by $u_{0}$, $v_{0}$, $\varepsilon_{0}$. The initial conditions of
$p_{u}$, $p_{v}$, $p_{\varepsilon}$ are denoted by $p_{u,0}$, $p_{v,0}$, and
$p_{0}$ respectively, and are determined by

$\begin{split}p_{u,0}&=\frac{\partial}{\partial
u_{0}}S(0,\lambda_{0},\varepsilon_{0})=\int\frac{2(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}+\varepsilon_{0}};\\\
p_{v,0}&=\frac{\partial}{\partial
v_{0}}S(0,\lambda_{0},\varepsilon_{0})=\int\frac{2v_{0}\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}+\varepsilon_{0}};\\\
p_{0}&=\frac{\partial}{\partial\varepsilon_{0}}S(0,\lambda_{0},\varepsilon_{0})=\int\frac{d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}+\varepsilon_{0}}.\end{split}$
(3.6)

We have written $\lambda_{0}=u_{0}+iv_{0}$. Note that $S$ is defined with
$\varepsilon>0$. The $\varepsilon_{0}$ that we choose here is also positive.

Using the Hamilton–Jacobi method, the solution $S$ of the PDE can be expressed
in terms of the initial conditions, as stated in the following proposition.
For the proof for the proposition, readers are referred to [13, Proposition
4.2]. For a more general statements, see [8, Section 6.1] and [9].

######  Proposition 3.10.

Suppose that the solution to the Hamiltonian system (3.5) exists on a time
interval $[0,T)$ such that $\varepsilon(t)>0$ for all $t\in[0,T)$. Then we
have, for all $t\in[0,T)$,

$S(t,\lambda(t),\varepsilon(t))=S(0,\lambda_{0},\varepsilon_{0})+tH_{0}$

where $H_{0}=H(u_{0},v_{0},\varepsilon_{0},p_{u,0},p_{v,0},p_{0})$. Also, we
have

$\frac{\partial S}{\partial y}(t,\lambda(t),\varepsilon(t))=p_{y}(t)$

for all $y\in\\{u,v,\varepsilon\\}$ and $t\in[0,T)$.

###  3.3 Solving the ODEs

The PDE in our case (Theorem 3.1) is the same as the one in [13]. In this
section, we briefly summarize the results from [13] about the solutions of the
ODEs (3.5).

The following proposition combines [13, Proposition 4.4] and [13, Definition
4.6].

######  Proposition 3.11.

Suppose that the initial conditions $\lambda_{0}=u_{0}+iv_{0}$,
$\varepsilon_{0}$ are chosen, and initial momenta $p_{u,0}$, $p_{v,0}$,
$p_{0}$ are determined by (3.6). Then the solution of the Hamiltonian system
(3.5) exists up to

$t_{\ast}(\lambda_{0},\varepsilon_{0})=\frac{1}{p_{0}}=\left(\int\frac{d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}+\varepsilon_{0}}\right)^{-1}.$
(3.7)

Up until $t_{\ast}(\lambda_{0},\varepsilon_{0})$, we have

$\begin{array}[]{ll}\displaystyle u(t)=u_{0}-\frac{t}{2}p_{u,0};&\displaystyle
p_{u}(t)=p_{u,0};\\\\[10.0pt] \displaystyle
v(t)=v_{0}+\frac{t}{2}p_{v,0};&\displaystyle p_{v}(t)=p_{v,0};\\\\[10.0pt]
\displaystyle\varepsilon(t)=\varepsilon_{0}(1-tp_{0})^{2};&\displaystyle
p_{\varepsilon}(t)=\frac{p_{0}}{1-tp_{0}}.\end{array}$

For a fixed $\lambda_{0}\in\mathbb{C}$,
$t_{\ast}(\lambda_{0},\varepsilon_{0})$ defined in (3.7) is strictly
decreasing in $\varepsilon_{0}$. For each $\lambda_{0}\in\mathbb{C}$, the
lifetime of the solution path in the limit $\varepsilon_{0}\to 0^{+}$ is given
by

$T(\lambda_{0})=\lim_{\varepsilon_{0}\to
0^{+}}\left(\int\frac{d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}}\right)^{-1}.$
(3.8)

By Proposition 3.11, informally there are two ways to make $\varepsilon(t)=0$.
The first one is to take $\varepsilon_{0}=0$; the second one is to choose
$\varepsilon_{0}$ such that $p_{0}=1/t$. The first scheme works only if the
lifetime of the solution path remains greater than $t$ in the limit
$\varepsilon_{0}\to 0^{+}$; that is, if $T(\lambda_{0})\geq t$. If the first
scheme does not work (that is, if $T(\lambda_{0})<t$), one needs to use the
second scheme to achieve $\varepsilon(t)=0$.

We now identify the set of points $\lambda_{0}\in\mathbb{C}$ such that
$T(\lambda_{0})<t$. This set of points plays a crucial role in [13, 16]. For
proof, see [13, Proposition 5.2].

######  Proposition 3.12.

The set

$\Lambda_{x_{0},t}=\\{\lambda_{0}\in\mathbb{C}\left|T(\lambda_{0})<t\right.\\}$
(3.9)

can be identified as

$\Lambda_{x_{0},t}=\left.\\{u_{0}+iv_{0}\in\mathbb{C}\right|\left|v_{0}\right|<v_{x_{0},t}(u_{0})\\}$

where $v_{x_{0},t}$ is defined in Definition 2.3(2). In particular,
$\Lambda_{x_{0},t}\cap\mathbb{R}=\left.\\{u_{0}\in\mathbb{R}\right|v_{x_{0},t}(u_{0})>0\\}$.
Furthermore, we have
$\mathrm{supp}\,\mu_{x_{0}}\subset\overline{\Lambda_{x_{0},t}}$.

For notational convenience, we also define

$\begin{split}\varepsilon_{0}^{t}(u_{0}+iv_{0})&=v_{x_{0},t}(u_{0})^{2}-v_{0}^{2}\\\
p_{1}&=\int\frac{x\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{0}^{2}+\varepsilon_{0}}.\end{split}$
(3.10)

Note that $\varepsilon_{0}^{t}(\lambda_{0})>0$ for
$\lambda_{0}\in\Lambda_{x_{0},t}$ and $\varepsilon_{0}^{t}(\lambda_{0})=0$ for
$\lambda_{0}\in\partial\Lambda_{x_{0},t}$. Then we have the following solution
at time $t$.

######  Proposition 3.13.

For initial conditions $\lambda_{0}\in\mathbb{C}$ and $\varepsilon_{0}>0$,
denote by $\lambda(t;\lambda_{0},\varepsilon_{0})$ and
$\varepsilon(t;\lambda_{0},\varepsilon_{0})$ the solution to system (3.5) at
time $t$.

  1. 1.

If $\lambda_{0}\in\Lambda_{x_{0},t}^{c}$, then
$t_{\ast}(\lambda_{0},\varepsilon_{0})>t$ for all $\varepsilon_{0}>0$, and

$\displaystyle\lim_{\varepsilon_{0}\to
0^{+}}\lambda(t;\lambda_{0},\varepsilon_{0})$
$\displaystyle=\lambda_{0}-t\int\frac{d\mu_{x_{0}}(x)}{\lambda_{0}-x}$
$\displaystyle\lim_{\varepsilon_{0}\to
0^{+}}\varepsilon(t;\lambda_{0},\varepsilon_{0})$ $\displaystyle=0.$

  2. 2.

If $\lambda_{0}\in\Lambda_{x_{0},t}$, then
$\varepsilon_{0}^{t}(\lambda_{0})>0$ and
$t_{\ast}(\lambda_{0},\varepsilon_{0}^{t}(\lambda_{0}))=t$. Furthermore, we
have

$\displaystyle\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}\lambda(t;\lambda_{0},\varepsilon_{0})$
$\displaystyle=tp_{1}+2iv_{0}$
$\displaystyle\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}\varepsilon(t;\lambda_{0},\varepsilon_{0})$
$\displaystyle=0.$

###### Proof.

Points 1 and 2 are Propositions 4.7 and 4.5 in [13] respectively. The choice
$\varepsilon_{0}^{t}(\lambda_{0})$ can be found in Section 7.3 of [13], right
before Lemma 7.7. ∎

##  4 The Brown measure of $x_{0}+i\sigma_{t}$

###  4.1 The domain

Define the holomorphic map

$H_{x_{0},-t}(z)=z-tG_{x_{0}}(\lambda_{0}),\quad\lambda_{0}\not\in\mathrm{supp}\,\mu_{x_{0}}$
(4.1)

and

$\Omega_{x_{0},t}=[H_{x_{0},-t}(\Lambda_{t}^{c})]^{c}.$

We note that the notation $H_{x_{0},-t}$ is consistent with the one in
Definition 2.3(1). We will show in the later section that, as in [13], the
Brown measure of $x_{0}+i\sigma_{t}$ has mass $1$ on the open set
$\Omega_{x_{0},t}$.

Most of the statements in the following theorem follow the same proof in [13,
Proposition 5.5]. We only supply the proof for the statements that requires a
new proof.

######  Theorem 4.1.

The following statements hold.

  1. 1.

The function $H_{x_{0},-t}$ is continuous and injective on
$\overline{\Gamma}_{t}$. (See Definition 2.3 for the definition of
$\Gamma_{t}$.)

  2. 2.

Define the function $f_{x_{0},t}:\mathbb{R}\to\mathbb{R}$ by

$f_{x_{0},t}(u_{0})=\mathrm{Re}[H_{x_{0},-t}(u_{0}+iv_{x_{0},t}(u_{0}))].$
(4.2)

Then at any point $u_{0}$ such that $v_{x_{0},t}(u_{0})>0$, the function
$f_{x_{0},t}$ is differentiable and

$0<\frac{df_{x_{0},t}}{du_{0}}<2.$

  3. 3.

The function $f_{x_{0},t}$ is continuous and strictly increasing. It maps
$\mathbb{R}$ onto $\mathbb{R}$. In particular, the inverse $f_{x_{0},t}^{-1}$
of $f_{x_{0},t}$ exists.

  4. 4.

Define the function

$\varphi_{x_{0},t}(u)=2v_{x_{0},t}(f_{x_{0},t}^{-1}(u)),\quad u\in\mathbb{R}.$
(4.3)

Then $\varphi_{x_{0},t}(u)\to 0$ as $|u|\to\infty$ and the map $H_{x_{0},-t}$
maps the graph of $v_{x_{0},t}$ to the graph of $\varphi_{x_{0},t}$.

  5. 5.

The map $H_{x_{0},-t}$ takes the region above the graph of $v_{x_{0},t}$ onto
the region above the graph of $\varphi_{x_{0},t}$.

  6. 6.

The set $\Omega_{x_{0},t}$ can be identified as

$\Omega_{x_{0},t}=\left.\\{u+iv\in\mathbb{C}\right|\left|v\right|<\varphi_{x_{0},t}(u)\\}.$

###### Proof.

We only need to prove that $f_{x_{0},t}$ maps $\mathbb{R}$ _onto_
$\mathbb{R}$, $H_{x_{0},-t}$ takes the region above the graph of $v_{x_{0},t}$
onto the region above the graph of $\varphi_{x_{0},t}$, and
$\varphi_{x_{0},t}(u)\to 0$ as $|u|\to\infty$. All the other parts follow the
same proof as in [13, Proposition 5.5]. Note that, since $f_{x_{0},t}$ is
continuous and strictly increasing, by the same argument in [13, Proposition
5.5], $H_{x_{0},-t}$ maps the graph of $v_{x_{0},t}$ to the graph of a
function $\varphi_{x_{0},t}$ defined on the range of $f_{x_{0},t}$ using the
same formula in (4.3) (except that we define $\varphi_{x_{0},t}$ only on the
range of $f_{x_{0},t}$ instead of $\mathbb{R}$).

We first show that $v_{x_{0},t}(u_{0})\to 0$ as $|u_{0}|\to\infty$. Suppose,
on the contrary, that there exist $\delta>0$ and a sequence
$u_{n}\in\mathbb{R}$, such that $|u_{n}|\to\infty$ but
$v_{x_{0},t}(u_{n})>\delta$ for all $n$. For all of these $u_{n}$, since
$v_{x_{0},t}(u_{n})>\delta>0$, we must have

$\int\frac{d\mu_{x_{0}}(x)}{(u_{n}-x)^{2}+\delta^{2}}\geq\int\frac{d\mu_{x_{0}}(x)}{(u_{n}-x)^{2}+v_{x_{0},t}(u_{n})^{2}}=\frac{1}{t}.$

Letting $n\to\infty$, by dominated convergence theorem, we have $0\geq 1/t>0$,
which is a contradiction. Also, we have

$\lim_{y\to\infty}H_{x_{0},-t}(iy)=\infty.$

Let $a=\lim_{u_{0}\to-\infty}f_{x_{0},t}(u_{0})$ and
$b=\lim_{u_{0}\to\infty}f_{x_{0},t}(u_{0})$. Then $\varphi_{x_{0},t}$ can be
defined on $(a,b)$ using the formula in (4.3). The map $H_{x_{0},-t}$ takes
the graph of $v_{x_{0},t}$ to the graph of $\varphi_{x_{0},t}$. The region
above the graph of $v_{x_{0},t}$ is bounded by the Jordan curve formed by the
graph of $v_{x_{0},t}$ and the point $\infty$ on the Riemann sphere.
Furthermore, the map $H_{x_{0},-t}$ is an injective conformal map on the
region above the graph of $v_{x_{0},t}$ and is extended to a homoemorphism on
the _closed_ region bounded by the graph of $v_{x_{0},t}$ and the point
$\infty$ on the Riemann sphere. Since
$\lim_{y\to\infty}H_{x_{0},-t}(iy)=\infty$, the point $\infty$ is on the
boundary of the image of the region above the graph of $v_{x_{0},t}$ under
$H_{x_{0},-t}$. Thus, the graph of $\varphi_{x_{0},t}$ and the point $\infty$
form the boundary of the image of the region above $v_{x_{0},t}$ under the map
$H_{x_{0},-t}$.

If $a\neq-\infty$ or $b\neq\infty$, by the preceding paragraph,
$\varphi_{x_{0},t}(u)\to 0$ as $u\to a^{+}$ or $u\to b^{-}$. In this case, the
union of the graph of $\varphi_{x_{0},t}$ and _any_ point on the Riemann
sphere does not enclose a region in the Riemann sphere. This contradicts the
preceding paragraph that the graph of $\varphi_{x_{0},t}$ and the point
$\infty$ form the boundary of the image of $H_{x_{0},-t}$. Thus,
$f_{x_{0},t}(u_{0})\to\pm\infty$ as $u_{0}\to\pm\infty$.

Finally, $\varphi_{x_{0},t}(u)\to 0$ as $|u|\to\infty$ follows from the fact
that $f_{x_{0},t}^{-1}(u)\to\pm\infty$ as $u\to\infty$, and that
$v_{x_{0},t}(u_{0})\to 0$ as $|u_{0}|\to\infty$. ∎

###  4.2 Surjectivity

We can write $f_{x_{0},t}$ into the form

$f_{x_{0},t}(u_{0})=\mathrm{Re}[H_{x_{0},-t}(u_{0}+iv_{x_{0},t}(u_{0})]=t\int\frac{x\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},t}(u_{0})^{2}}.$
(4.4)

Denote by $\lambda(t;\lambda_{0},\varepsilon_{0})$ and
$\varepsilon(t;\lambda_{0},\varepsilon_{0})$ be the solution of (3.5), with
initial conditions $\lambda_{0},\varepsilon_{0}$. By (4.4) and Proposition
3.13, for any $\lambda_{0}=u_{0}+iv_{0}\in\Lambda_{x_{0},t}$,

$\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}\lambda(t;\lambda_{0},\varepsilon_{0})=f_{x_{0},t}(u_{0})+2iv_{0},$
(4.5)

which is in $\Omega_{x_{0},t}$ since
$2|v_{0}|<\varphi_{x_{0},t}(f_{x_{0},t}(u_{0}))$, by Point 4 of Theorem 4.1.
The following theorem shows that every point in $\Omega_{x_{0},t}$ can be
obtained from $\Lambda_{x_{0},t}$ through the formula (4.5). The proof in [13,
Theorem 7.3] still works for our case.

######  Theorem 4.2.

Let

$U_{t}(u_{0}+iv_{0})=f_{x_{0},t}(u_{0})+2iv_{0},\quad
u_{0}+iv_{0}\in\Lambda_{x_{0},t}$

which is the same as the formula (4.5). Then we have the following results.

  1. 1.

The map $U_{t}$ extends continuously to $\overline{\Lambda}_{x_{0},t}$. This
extension is the unique continuous map of $\overline{\Lambda}_{x_{0},t}$ into
$\overline{\Omega}_{x_{0},t}$ that agrees with $H_{x_{0},-t}$ on
$\partial\Lambda_{x_{0},t}$ and maps each vertical segment in
$\overline{\Lambda}_{x_{0},t}$ linearly to a vertical segment in
$\overline{\Omega}_{x_{0},t}$.

  2. 2.

The map $U_{t}$ is a homeomorphism from $\Lambda_{x_{0},t}$ onto
$\Omega_{x_{0},t}$.

If $x_{0}\in\mathscr{A}^{\Delta}$ is an unbounded variable, the support of the
law $\mu_{x_{0}}$ of $x_{0}$ is unbounded. The support of $\mu_{x_{0}}$ can be
bounded from below but unbounded from above, bounded from above but unbounded
from below, or unbounded from above and below. The following proposition gives
bounds on the real parts of points in $\Omega_{x_{0},t}$.

######  Proposition 4.3.

Let $M=\sup\mathrm{supp}(\mu)$ and $m=\inf\mathrm{supp}(\mu)$. Then every
point $\lambda\in\overline{\Omega}_{x_{0},t}$ satisfies

$m<\mathrm{Re}\lambda<M.$

###### Proof.

The proof of [13, Proposition 7.4] works for this proposition. The difference
from [13, Proposition 7.4] is that $M$ and $m$ may not be both finite — could
be one infinite and one finite, or both infinite. ∎

###  4.3 The Brown measure computation

In this section, we compute the Brown measure of $x_{0}+i\sigma_{t}$ by taking
the Laplacian of $S$ in $\Omega_{x_{0},t}$. By the definition of Brown measure
(See Section 2.3), the Laplacian is in distributional sense. The following
regularity result tells us that we can indeed take the ordinary Laplacian to
compute the density of the Brown measure.

######  Proposition 4.4.

Define the function $\tilde{S}$ by

$\tilde{S}(t,\lambda,z)=S(t,\lambda,z^{2})$ (4.6)

for $z>0$. Then for any $t>0$ and $\lambda^{*}\in\Omega_{x_{0},t}$, the
function

$(\lambda,z)\mapsto\tilde{S}(t,\lambda,z)$

extends to a real analytic function in a neighborhood of $(\lambda^{*},0)$ in
$\mathbb{C}\times\mathbb{R}$.

Let

$s_{t}(\lambda)=\lim_{\varepsilon\to
0^{+}}S(t,\lambda,\varepsilon)=\lim_{\varepsilon\to
0^{+}}\tilde{S}(t,\lambda,\sqrt{\varepsilon})$

where $\tilde{S}$ is defined in (4.6). The computation of the Laplacian
$s_{t}$ is the same as in [13]. Since the computation of the Laplacian is
crucial for the computation of the Brown measure, for completeness, we also
include the computation of the Laplacian here.

Write $\lambda(t;\lambda_{0},\varepsilon_{0})$ be the solution of the system
(3.5) with initial conditions $\lambda_{0}$ and $\varepsilon_{0}$. By (4.5)
and Theorem 4.2, for any $\lambda=u+iv\in\Lambda_{x_{0},t}$,
$\lambda_{0}=u_{0}+iv_{0}=U_{t}^{-1}(\lambda)$ satisfies

$\lim_{\varepsilon_{0}\to\lambda_{0}^{t}(\lambda_{0})^{+}}\lambda(t;\lambda_{0},\varepsilon_{0})=\lambda.$

Then, by Propositions 3.10 and 4.4, for any $\lambda=u+iv\in\Omega_{x_{0},t}$,

$\frac{\partial}{\partial u}s_{t}(\lambda)=\lim_{\varepsilon\to
0^{+}}\frac{\partial}{\partial
u}\tilde{S}(t,\lambda,\varepsilon)=\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}p_{u}(t)=\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}p_{u,0}.$

Using (3.6), (4.4), and Proposition 3.13(2).

$\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}p_{u,0}=\int\frac{2(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},t}(u_{0})^{2}}=\frac{2u_{0}}{t}-\frac{2}{t}\cdot
t\int\frac{x\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},t}(u_{0})^{2}}=\frac{2f_{x_{0},t}^{-1}(u)}{t}-\frac{2u}{t}.$

Hence, we have

$\frac{\partial s_{t}}{\partial
u}=\frac{2f_{x_{0},t}^{-1}(u)}{t}-\frac{2u}{t},\quad\lambda=u+iv\in\Omega_{x_{0},t}.$
(4.7)

Similarly, since $v=2v_{0}$ by (4.5),

$\frac{\partial}{\partial
v}s_{t}(\lambda)=\lim_{\varepsilon_{0}\to\varepsilon_{0}^{t}(\lambda_{0})^{+}}p_{v,0}=\int\frac{2v_{0}\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},t}(u_{0})^{2}}=\frac{v}{t},\quad\lambda=u+iv\in\Omega_{x_{0},t}.$
(4.8)

Combining (4.7) and (4.8), we have

$\Delta
s_{t}(\lambda)=\frac{2}{t}\frac{d}{du}f_{x_{0},t}^{-1}(u)-\frac{1}{t}=\frac{2}{t}\left(\frac{d}{du}f_{x_{0},t}^{-1}(u)-\frac{1}{2}\right)$

We arrive at the Brown measure of $x_{0}+i\sigma_{t}$ as well as a push-
forward result of the Brown measure.

######  Theorem 4.5.

The open set $\Omega_{x_{0},t}$ is a set of full measure for the Brown measure
of $x_{0}+i\sigma_{t}$. The Brown measure is absolutely continuous with a
strictly positive density $w_{t}$ on the open set $\Omega_{x_{0},t}$ by

$w_{t}(\lambda)=\frac{1}{2\pi
t}\left(\frac{d}{du}f_{x_{0},t}^{-1}(u)-\frac{1}{2}\right),\quad\lambda=u+iv\in\Omega_{x_{0},t}.$

In particular, the density $w_{t}$ is constant along vertical segments inside
$\Omega_{x_{0},t}$. Moreover, the push-forward of the Brown measure of
$x_{0}+i\sigma_{t}$ by

$Q_{t}(u+iv)=2f_{x_{0},t}^{-1}(u)-u,\quad u+iv\in\Omega_{x_{0},t}$

is the law of $x_{0}+\sigma_{t}$.

###### Proof.

These results follow from the same proof as for Theorems 7.9 and 8.2 of [13]
since Biane’s Theorem (Theorem 2.4) on the free convolution with a
semicircular variable holds for unbounded self-adjoint random variable
$x_{0}\in\mathscr{A}^{\Delta}$. ∎

Theorem 8.2 of [13] also establishes a push-forward property of the Brown
measure of $x_{0}+i\sigma_{t}$ to the Brown measure of $x_{0}+c_{t}$ where
$c_{t}$ is the circular variable with variance $t$. We have not computed the
Brown measure of $x_{0}+c_{t}$ for unbounded $x_{0}$. We will establish this
analogous push-forward result to $x_{0}+c_{t}$ in the next section.

##  5 The Brown measure of $x_{0}+c_{\alpha,\beta}$

Recall that an elliptic variable has the form

$c_{\alpha,\beta}=\tilde{\sigma}_{\alpha}+i\sigma_{\beta}$

where $\tilde{\sigma}_{\alpha}$ and $\sigma_{\beta}$ are free semicircular
random variables with variances $\alpha$ and $\beta$ respectively. Given any
self-adjoint $a\in\tilde{\mathscr{A}}$, let

$\Lambda_{a,s}=\\{\left.u+iv\in\mathbb{C}\right||v|<v_{a,s}(u)\\},$

where $v_{a,s}$ is defined in Definition 2.3(2). We note that this definition
of $\Lambda_{a,s}$ is consistent with the notation in the previous section by
Proposition 3.12.

In this section, we study the Brown measure of $x_{0}+c_{\alpha,\beta}$, where
$x_{0}\in\mathscr{A}^{\Delta}$ is self-adjoint, and $\alpha\geq 0$ (the case
$\alpha=0$ agrees with the one computed in Theorem 4.5). The computation in
[15] relies only on the computations of holomorphic functions; the compactness
of the support of $\mu_{x_{0}}$ in [15] does not play a role (since the
computation of the holomorphic maps can be restricted to a Stolz angle). The
Brown measure computation of [15] can be automatically carried over to the
unbounded self-adjoint $x_{0}$.

We start by a few notations; these notations are well-defined for
$a\in\tilde{\mathscr{A}}$. Later we will apply these notations to
$x_{0}\in\mathscr{A}^{\Delta}$ since the Brown measure of
$x_{0}+c_{\alpha,\beta}$ is defined only for $x_{0}\in\mathscr{A}^{\Delta}$
but not for $x_{0}\in\tilde{\mathscr{A}}$. More discussions can be found in
Proposition 3.6 of [15].

######  Definition 5.1.

  1. 1.

Given any self-adjoint $a\in\tilde{\mathscr{A}}$ and $r\in\mathbb{R}$. Define

$H_{a,r}(z)=z+rG_{a}(z),\quad z\not\in\sigma(a).$

This definition is consistent with (2.1), when $r>0$, and consistent with
(4.1), when $r<0$. Furthermore, $H_{a,r}$ is an injective conformal map on
$\Lambda_{a,|r|}^{c}$.

  2. 2.

Let $a\in\tilde{\mathscr{A}}$ be self-adjoint. Also let $\alpha\geq 0$ and
$\beta>0$ and write $s=\alpha+\beta$. Define
$f_{a,\alpha,\beta}:\mathbb{R}\to\mathbb{R}$ by

$f_{a,\alpha,\beta}(u)=\mathrm{Re}[H_{\alpha-\beta}(u+iv_{x_{0},s}(u))].$

This function is strictly increasing and is a homeomorphism onto $\mathbb{R}$.
Furthermore, $f_{a,\alpha,\beta}^{\prime}(u)>0$ for all
$u\in\Lambda_{a,s}\cap\mathbb{R}$.

  3. 3.

Let $\alpha>0$ and $\beta>0$ and write $s=\alpha+\beta$. Define
$\varphi_{x_{0},\alpha,\beta}(u)=2\beta
v_{a,s}(f_{x_{0},\alpha,\beta}^{-1}(u))/s$ and

$\Omega_{x_{0},\alpha,\beta}=\left.\\{u+iv\in\mathbb{C}\right||v|<\varphi_{x_{0},\alpha,\beta}(u)\\}.$

By Definition 3.1 and Proposition 3.6 of [15],
$\Omega_{x_{0},\alpha,\beta}=[H_{x_{0},\alpha-\beta}(\Lambda_{x_{0},s}^{c})]^{c}$.
Remark that $\varphi_{x_{0},0,t}=\varphi_{x_{0},t}$, which is defined in
Theorem 4.1(4).

We will see that $\Omega_{x_{0},\alpha,\beta}$ is an open set of full measure
with respect to the Brown measure of $x_{0}+c_{\alpha,\beta}$.

######  Remark 5.2.

The author uses a different parametrization for the elliptic variable in [15].
In [15] the elliptic variable is parametrized using $s$ and $t$ by
$\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$. The parametrization in [15] is
motivated by [14]. In this paper, the parametrization is a linear
transformation of the one in [15], by $\alpha=s-t/2$ and $\beta=t/2$.

If we apply Theorem 4.5 to compute the Brown measure of
$x_{0}+c_{\alpha,\beta}=x_{0}+\tilde{\sigma}_{\alpha}+i\sigma_{\beta}$, the
Brown measure is written in terms of the law of
$x_{0}+\tilde{\sigma}_{\alpha}$. The key to write the Brown measure in terms
of $x_{0}$ is the following proposition in [15].

######  Proposition 5.3 (Theorem 3.4 of [15]).

Let $a=x_{0}+\tilde{\sigma}_{\alpha}$. Then, by writing
$f_{x_{0},\alpha,\beta}(u_{0})=u$, we have

$f_{a,\beta}^{-1}(u)=u+\beta\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}},\quad
u\in\mathbb{R}.$

We are ready to state the main result in this case.

######  Theorem 5.4.

The Brown measure of $x_{0}+c_{\alpha,\beta}$ is absolutely continuous with
respect to the Lebesgue measure on the plane. The open set
$\Omega_{\alpha,\beta}$ is a set of full measure of the Brown measure. The
density of the Brown measure on $\Omega_{\alpha,\beta}$ is given by

$w_{\alpha,\beta}(u+iv)=\frac{1}{4\pi\beta}\left(1+2\beta\frac{d}{du}\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}}\right),\quad
u+iv\in\Omega_{\alpha,\beta},$

where $f_{x_{0},\alpha,\beta}(u_{0})=u$. The density is strictly positive and
constant along the vertical segments in $\Omega_{x_{0},\alpha,\beta}$.

######  Remark 5.5.

If $\alpha=0$ and $\beta=t$, then Proposition 5.3 implies that

$w_{\alpha,\beta}(u+iv)=\frac{1}{4\pi
t}\left(1+2\frac{d}{du}(f_{x_{0},t}^{-1}(u)-u)\right)=\frac{1}{2\pi
t}\left(\frac{d}{du}f_{x_{0},\beta}^{-1}(u)-\frac{1}{2}\right),$

giving the density of the Brown measure of $x_{0}+i\sigma_{t}$.

When $x_{0}\in\mathscr{A}$ is a bounded self-adjoint random variable, the
Brown measure of $x_{0}+c_{t}$ is computed in [16]. The following corollary
computes the Brown measure of $x_{0}+c_{t}$ for possibly-unbounded self-
adjoint $x_{0}\in\mathscr{A}^{\Delta}$. The highlights are, as in [16], the
Brown measure has full measure on the open set $\Lambda_{x_{0},t}$, and the
density can be written as terms of the derivative of the function
$\psi_{x_{0},t}$ in Theorem 2.4(3).

######  Corollary 5.6.

The Brown measure of $x_{0}+c_{t}$ is absolutely continuous with respect to
the Lebesgue measure on the plane. The open set $\Lambda_{x_{0},t}$ is a set
of full measure of the Brown measure. The density on $\Lambda_{x_{0},t}$ has
the form

$w_{t/2,t/2}(u+iv)=\frac{1}{2\pi t}\frac{d\psi_{x_{0},t}}{du}=\frac{1}{\pi
t}\left(1-\frac{t}{2}\frac{d}{du}\int\frac{x\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}\right),\quad
u+iv\in\Lambda_{x_{0},t},$

where $\psi_{x_{0},t}$ is defined in Theorem 2.4. The density is strictly
positive and constant along the vertical segments in $\Lambda_{x_{0},t}$.

###### Proof.

Take $\alpha=\beta=t/2$ in Theorem 5.4. Then, $\alpha+\beta=t$ and
$2\beta/(\alpha+\beta)=1$, so that $f_{x_{0},\alpha,\beta}$ is the identity
map, and

$\Omega_{x_{0},t/2,t/2}=\left.\\{u+iv\in\mathbb{C}\right||v|<v_{x_{0},t}(u)\\}=\Lambda_{x_{0},t}.$

Then we can compute the density using the formula in Theorem 5.4

$\displaystyle w_{t/2,t/2}(u+iv)$ $\displaystyle=\frac{1}{2\pi
t}\left(1+t\frac{d}{du}\int\frac{(u-x)\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}\right)$
$\displaystyle=\frac{1}{2\pi t}\frac{d\psi_{x_{0},t}}{du},\quad
u+iv\in\Lambda_{x_{0},t};$

the last equality follows from the definition of
$\psi_{x_{0},t}(u)=H_{x_{0},t}(u+iv_{x_{0},t}(u))$. By Definiton 2.3(2) of
$v_{x_{0},t}$, we have

$\int\frac{(u-x)\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}=\frac{u}{t}-\int\frac{x\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}.$

The other claimed formula of $w_{t}$ follows from a straightforward algebraic
computation. ∎

The following corollary establishes the push-forward properties between the
Brown measure of $x_{0}+c_{\alpha,\beta}$, the Brown measure of $x_{0}+c_{s}$,
and the law of $x_{0}+\sigma_{t}$, where $s=\alpha+\beta$ is the total
variance of the elliptic variable $c_{\alpha,\beta}$.

######  Corollary 5.7.

Write $s=\alpha+\beta$. Let
$U_{\alpha,\beta}:\Lambda_{x_{0},s}\to\Omega_{x_{0},\alpha,\beta}$ be defined
by

$U_{\alpha,\beta}(u+iv)=f_{x_{0},\alpha,\beta}(u)+i\frac{2\beta v}{s}.$ (5.1)

Then $U_{\alpha,\beta}$ extends to a homeomorphism from
$\overline{\Lambda}_{x_{0},s}$ to $\overline{\Omega}_{\alpha,\beta}$ and
agrees with $H_{x_{0},\alpha-\beta}$ on the boundary of $\Lambda_{x_{0},s}$.
Furthermore, the following push-forward properties hold.

  1. 1.

The push-forward of the Brown measure of $x_{0}+c_{s}$ under the map
$U_{\alpha,\beta}$ is the Brown measure of $x_{0}+c_{\alpha,\beta}$.

  2. 2.

The push-forward of the Brown measure of $x_{0}+c_{\alpha,\beta}$ by the map

$Q_{\alpha,\beta}(u+iv)=\begin{cases}\frac{1}{\alpha-\beta}[su-2\beta
f_{x_{0},\alpha,\beta}^{-1}(u)],\quad&\textrm{if $\alpha\neq\beta$}\\\
H_{x_{0},s}(u+iv_{x_{0},s}(u)),\quad&\textrm{if
$\alpha=\beta=s/2$}.\end{cases}$

is the law of $x_{0}+\sigma_{s}$.

######  Remark 5.8.

Write $u=f_{x_{0},\alpha,\beta}(u_{0})$. When $\alpha\neq\beta$, we can
compute that

$Q_{\alpha,\beta}(u+iv)=u_{0}+s\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}}=H_{x_{0},s}(u_{0}+iv_{x_{0},s}(u)).$

If $\alpha=\beta=s/2$, then $u=u_{0}$ and the above equation reduces to
$H_{x_{0},s}(u+iv_{x_{0},s}(u))$, which is the definition of
$Q_{\alpha,\beta}$ when $\alpha=\beta=s/2$.

###### Proof.

That $U_{\alpha,\beta}$ extends to a homeomorphism from
$\overline{\Lambda}_{x_{0},s}$ to $\overline{\Omega_{\alpha,\beta}}$ and
agrees with $H_{x_{0},\alpha-\beta}$ on the boundary of $\Lambda_{x_{0},s}$
follows from the proof of Proposition 4.2 of [15].

For $\alpha\neq\beta$, the proof follows from the one given for Theorem 4.1 of
[15]. For the case $\alpha=\beta=s/2$, the proof follows from the one given
for Theorem 3.13 of [16]. ∎

Using the push-forward property in Point 1 of Corollary 5.7, we can express
the density of the Brown measure of $x_{0}+c_{\alpha,\beta}$ in terms of the
density of the Brown measure of $x_{0}+c_{s}$, if $\alpha+\beta=s$.

######  Corollary 5.9.

For any $r=2\beta/s$, by writing $u+iv=U_{\alpha,\beta}(u_{0}+iv_{0})$ for all
$u_{0}+iv_{0}\in\Lambda_{x_{0},s}$, we have

$w_{\alpha,\beta}(u+iv)=\frac{1}{r}\frac{w_{s/2,s/2}(u_{0}+iv_{0})}{r+2\pi(1-r)s\cdot
w_{s/2,s/2}(u_{0}+iv_{0})}.$

###### Proof.

This follows from a direct computation of integration of substitution using
Corollary 5.7. Interested readers are referred to the proof of Corollary 4.3
of [15]. ∎

Figure 2: Visualization of the map $U_{\alpha,\beta}$ with $\alpha=1/8$ and
$\beta=7/8$.

Figure 2 shows a visualization of the map $U_{\alpha,\beta}$, with
$\mu_{x_{0}}=\frac{1}{3}\delta_{-1}+\frac{2}{3}\delta_{1}$, $\alpha=1/8$ and
$\beta=7/8$. The blue equally-spaced vertical lines on the left hand side of
the figure are mapped to the corresponding vertical lines on the right hand
side of the figure by $U_{\alpha,\beta}$. The blue vertical lines on the right
hand side are no longer equally-spaced. In the next proposition, we
investigate the spacing of the vertical lines on the right hand side by
looking at the second derivative of $f_{x_{0},\alpha,\beta}$. If
$f_{x_{0},\alpha,\beta}^{\prime\prime}>0$ on an interval $I$, then the spacing
between the image of the vertical lines in $\Lambda_{x_{0},s}$ is increasing
on $I$. Since the push-forward of the Brown measure of $x_{0}+c_{s}$ by
$U_{\alpha,\beta}$ is the Brown measure of $x_{0}+c_{\alpha,\beta}$, this
proposition describes how mass is transferred under this push-forward map —
whether the mass is “squeezed” or “stretched” (relative to the total mass on
$\Omega_{x_{0},\alpha,\beta}$). More precisely, the mass of

$\\{\left.u_{0}+iv_{0}\in\Lambda_{x_{0},s}\right|a\leq u_{0}\leq b\\}$

with respect to the Brown measure $x_{0}+c_{s}$ is transferred to the set

$\\{\left.u+iv\in\Omega_{x_{0},\alpha,\beta}\right|f_{x_{0},\alpha,\beta}(a)\leq
u\leq f_{x_{0},\alpha,\beta}(b)\\}.$

If $f_{x_{0},\alpha,\beta}^{\prime\prime}<0$ on $(a,b)$ and
$f_{x_{0},\alpha,\beta}^{\prime\prime}>0$ on $(b,c)$, then the push-forward
map $U_{\alpha,\beta}$ “squeezes” the mass towards the vertical line
intersecting $b$. Similarly, if $f_{x_{0},\alpha,\beta}^{\prime\prime}>0$ on
$(a,b)$ and $f_{x_{0},\alpha,\beta}^{\prime\prime}<0$ on $(b,c)$, then the
push-forward map $U_{\alpha,\beta}$ “stretches” the mass away from the
vertical line intersecting $b$.

######  Proposition 5.10.

Writing $s=\alpha+\beta$, we have

$f_{x_{0},\alpha,\beta}^{\prime\prime}(u_{0})=2\pi(\alpha-\beta)w_{s/2,s/2}^{\prime}(u_{0})=\frac{\alpha-\beta}{s}\psi_{x_{0},s}^{\prime\prime}(u_{0})$

for all $u_{0}\in\Lambda_{x_{0},s}\cap\mathbb{R}$, where
$\psi_{x_{0},s}(u_{0})=H_{x_{0},s}(u_{0}+iv_{x_{0},s}(u_{0}))$ is defined in
Theorem 2.4(3).

###### Proof.

Using the formula

$f_{x_{0},\alpha,\beta}(u_{0})=u_{0}+(\alpha-\beta)\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}}=u_{0}+\frac{\alpha-\beta}{s}(\psi_{x_{0},s}(u_{0})-u_{0}),$

we have

$f_{x_{0},\alpha,\beta}^{\prime\prime}(u_{0})=\frac{\alpha-\beta}{s}\psi_{x_{0},s}^{\prime\prime}(u_{0}).$

The other formula for $f_{x_{0},\alpha,\beta}^{\prime\prime}$ follows from
$\psi_{x_{0},s}^{\prime}(u_{0})=2\pi sw_{s/2,s/2}(u_{0})$ by Corollary 5.6. ∎

##  6 Examples: Cauchy case

In this section, we compute the Brown measures of $x_{0}+c_{\alpha,\beta}$
where $x_{0}$ has the Cauchy distribution

$d\mu_{x_{0}}(x)=\frac{1}{\pi}\frac{dx}{1+x^{2}}.$

Since the density of $\mu_{x_{0}}$ has polynomial decay at $\pm\infty$,
$x_{0}\in\mathscr{A}^{\Delta}$, so does
$x_{0}+c_{\alpha,\beta}\in\mathscr{A}^{\Delta}$. We first compute the Brown
measure of $x_{0}+c_{t}$; the computation in the process is also useful for
the computation of the Brown measure of $x_{0}+c_{\alpha,\beta}$. Lastly, we
compute the Brown measure of $x_{0}+i\sigma_{t}$ by putting $\alpha=0$ and
$\beta=t$ to the Brown measure of $x_{0}+c_{\alpha,\beta}$.

###  6.1 Adding a circular variable

In this section, we compute the Brown measure of $x_{0}+c_{t}$ as in the
following theorem. Some of the computations will be used again when we compute
the Brown measure of $x_{0}+c_{\alpha,\beta}$.

######  Theorem 6.1.

When $x_{0}$ is Cauchy distributed, the boundary of the domain
$\Lambda_{x_{0},t}$ has the form

$\partial\Lambda_{x_{0},t}=\left\\{u\pm
iv\left|u^{2}=\frac{1}{v}(1+v)(t-v-v^{2}),v>0\right.\right\\}.$ (6.1)

The upper boundary
$(\partial\Lambda_{x_{0},t})\cap(\mathbb{C}^{+}\cup\mathbb{R})$ of
$\Lambda_{x_{0},t}$ is the graph of a positive unimodal function with peak
$\frac{-1+\sqrt{1+4t}}{2}$ at $0$. The Brown measure of $x_{0}+c_{t}$ has full
measure on $\Lambda_{x_{0},t}$, with density

$w_{t/2,t/2}(\lambda)=\frac{1}{2\pi
t}\frac{t+4v^{2}(1+v)^{2}}{(1+v)(t+2v^{2}(1+v))}$

where $v=v_{x_{0},t}(u)$.

######  Remark 6.2.

As $|u|\to\infty$, the density $w_{t}(u)$ does not approach $0$. In fact, as
$|u|\to\infty$, $v_{x_{0},t}(u)\to 0$, so $w_{t}(u)$ approaches $1/(2\pi t)$.
The density $w_{t}(\lambda)$ on $\Lambda_{x_{0},t}$ still defines a
probability measure because the function $v_{x_{0},t}(u)\approx t/u^{2}$ as
$|u|\to\infty$.

Figure 3 plots the eigenvalue simulation of $x_{0}+c_{t}$, the density of the
Brown measure of $x_{0}+c_{t}$, as well as the function $w_{t}(u)$ for
$u\in\mathbb{R}$ at $t=1$.

Figure 3: $5000\times 5000$ matrix simulation of eigenvalues of $x_{0}+c_{t}$
(top), density plot of the Brown measure of $x_{0}+c_{t}$ (middle), and
$w_{t/2,t/2}(u)$ for $u\in\mathbb{R}$ (bottom), at $t=1$. Top two plotted with
boundary of $\Lambda_{x_{0},t}$.

We start by computing the function $v_{x_{0},t}(u)$, then we compute the
derivative of $\psi_{x_{0},t}$ (See Theorem 2.4 for definition of
$\psi_{x_{0},t}$).

######  Lemma 6.3.

For each $u\in\mathbb{R}$, $v_{x_{0},t}(u)$ is the unique positive number $v$
satisfying

$u^{2}=\frac{1}{v}(1+v)(t-v-v^{2}).$ (6.2)

Thus, $u$ and $dv_{x_{0},t}/du$ have opposite sign; in particular,
$v_{x_{0},t}$ is unimodal with peak at $0$, and
$v_{x_{0},t}(0)=\frac{-1+\sqrt{1+4t}}{2}$.

###### Proof.

Let $u\in\mathbb{R}$. We can compute (computer software such as Mathematica
could be helpful) that

$\frac{1}{t}=\int\frac{d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}=\frac{1+v_{x_{0},t}(u)}{v_{x_{0},t}(u)(u^{2}+(1+v_{x_{0},t}(u))^{2})},$

so that

$u^{2}=\frac{1}{v_{x_{0},t}(u)}[1+v_{x_{0},t}(u)][t-v_{x_{0},t}(u)-v_{x_{0},t}(u)^{2}].$

This proves (6.2).

Differentiating both sides of (6.2) with respect to $u$, we have

$2u=\left(-\frac{t}{v_{x_{0},t}(u)^{2}}-2(1+v_{x_{0},t}(u))\right)\frac{dv_{x_{0},t}}{du}$
(6.3)

which shows $u$ and $dv_{x_{0},t}/du$ have oppposite sign since we must have
$-t/v_{x_{0},t}(u)-2(1+v_{x_{0},t}(u))<0$. It follows that $dv_{x_{0},t}/du<0$
for $u>0$ and $dv_{x_{0},t}/du>0$ for $u<0$, proving $v_{x_{0},t}$ is unimodal
with peak at $0$. The value $v_{x_{0},t}(0)$ comes from solving the equation
(6.2) at $u=0$. ∎

######  Proposition 6.4.

We have

$\frac{d\psi_{x_{0},t}}{du}=\frac{t+4v^{2}(1+v)^{2}}{(1+v)(t+2v^{2}(1+v))},$

where $v=v_{x_{0},t}(u)$.

###### Proof.

Since

$\frac{1+v}{v(u^{2}+(1+v)^{2})}=\int\frac{d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}=\frac{1}{t},$

we can compute (again, computer software could be useful)

$\psi_{x_{0},t}(u)=u+t\int\frac{(u-x)\,d\mu_{x_{0}}(x)}{(u-x)^{2}+v_{x_{0},t}(u)^{2}}=u+t\frac{u}{u^{2}+(1+v)^{2}}=u+\frac{uv}{1+v}.$
(6.4)

Using (6.2) and (6.3), we can compute the derivative

$\frac{d}{du}\frac{uv}{1+v}=\frac{v}{1+v}+\frac{u}{(1+v)^{2}}\frac{dv}{du}=\frac{v(-t+2v(1+v)^{2})}{(1+v)(t+2v^{2}(1+v))}$
(6.5)

and so

$\frac{d\psi_{x_{0},t}}{du}=\frac{t+4v^{2}(1+v)^{2}}{(1+v)(t+2v^{2}(1+v))},$

completing the proof. ∎

###### Proof of Theorem 6.1.

By Proposition 3.12,
$\Lambda_{x_{0},t}=\\{\left.u+iv\in\mathbb{C}\right||v|<v_{x_{0},t}(u)\\}$;
thus, (6.1) and the description of the boundary of $\Lambda_{x_{0},t}$ follow
from (6.2).

The formula of the density of the Brown measure follows from Corollary 5.6 and
Proposition 6.4. ∎

###  6.2 Adding an elliptic variable

The main result about the Brown measure of $x_{0}+c_{\alpha,\beta}$ is as
follows.

######  Theorem 6.5.

When $x_{0}$ is Cauchy distributed, the function
$\varphi_{x_{0},\alpha,\beta}$ in Definition 5.1(3) is unimodal with peak
$\frac{\beta(-1+\sqrt{1+4s})}{s}$ at $u=0$. The boundary of the corresponding
$\Omega_{x_{0},\alpha,\beta}$ (See Definition 5.1(3)) has the form

$\partial\Omega_{x_{0},\alpha,\beta}=\left\\{u\pm
ib\in\mathbb{C}\left|u^{2}=\frac{(b\alpha+\beta)^{2}(4\beta^{2}-2b\beta-b^{2}s)}{b\beta^{2}(bs+2\beta)}\right.\right\\}.$
(6.6)

The function $b=\varphi_{x_{0},\alpha,\beta}(u)$ has a decay of order
$1/u^{2}$ as $|u|\to\infty$.

The density of the Brown measure has the form

$w_{\alpha,\beta}(\lambda)=\frac{1}{4\pi\beta}\frac{b^{4}s^{3}+4b^{3}s^{2}\beta+4b^{2}s\beta^{2}+4\beta^{4}}{b^{4}s^{2}\alpha+4b^{3}s\alpha\beta+4b^{2}\alpha\beta^{2}+4b\beta^{4}+4\beta^{4}},\quad\lambda\in\Omega_{x_{0},\alpha,\beta}$

where $b=\varphi_{x_{0},\alpha,\beta}(u)$.

######  Remark 6.6.

The density $w_{\alpha,\beta}$ does not approach $0$ as
$u=\mathrm{Re}(\lambda)$ approaches infinity. In fact, as $|u|\to\infty$,
$\varphi_{x_{0},\alpha,\beta}(u)\to 0$ and so

$\lim_{|u|\to\infty}w_{\alpha,\beta}(u)=\frac{1}{4\pi\beta}.$

Figure 4 plots an eigenvalue simulation of $x_{0}+c_{\alpha,\beta}$, the
density of the Brown measure of $x_{0}+c_{\alpha,\beta}$, and the graph of the
function $w_{\alpha,\beta}(u)$ for $u\in\mathbb{R}$ at $\alpha=1/8$ and
$\beta=7/8$.

Figure 4: $5000\times 5000$ matrix simulation of eigenvalues of
$x_{0}+c_{\alpha,\beta}$ (top), density plot of the Brown measure of
$x_{0}+c_{\alpha,\beta}$ (middle), and $w_{\alpha,\beta}(u)$ for
$u\in\mathbb{R}$ (bottom), at $\alpha=1/8$ and $\beta=7/8$. Top two plotted
with boundary of $\Omega_{x_{0},\alpha,\beta}$.

We start proving Theorem 6.5 by the following lemma which concerns the
derivative of $f_{x_{0},\alpha,\beta}$.

######  Lemma 6.7.

Let $u=f_{x_{0},\alpha,\beta}(u_{0})$ for $u_{0}\in\mathbb{R}$. Then, by
writing $s=\alpha+\beta$ and $v=v_{x_{0},s}(u_{0})$,

$\frac{du}{du_{0}}=\frac{(1+v)(s^{2}+4\alpha
v^{2}(1+v))-(\alpha-\beta)sv}{s(1+v)(s+2v^{2}(1+v))}.$

###### Proof.

Using a computation similar to (6.4), we have

$\begin{split}u=f_{x_{0},\alpha,\beta}(u_{0})&=u_{0}+(\alpha-\beta)\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v^{2}}\\\
&=u_{0}+\frac{\alpha-\beta}{s}\frac{u_{0}v}{1+v}.\end{split}$ (6.7)

Then using (6.5) with $s$ in place of $t$ and $u_{0}$ in place of $u$, we have

$\frac{du}{du_{0}}=1+\frac{\alpha-\beta}{s}\frac{v(-s+2v(1+v)^{2})}{(1+v)(s+2v^{2}(1+v))}.$

The conclusion now follows from an algebraic computation. ∎

######  Lemma 6.8.

Let $u=f_{x_{0},\alpha,\beta}(u_{0})$ for $u_{0}\in\mathbb{R}$. Then, by
writing $s=\alpha+\beta$ and $v=v_{x_{0},s}(u_{0})$, we have

$\frac{d}{du}\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v}=\frac{v(-s+2v(1+v)^{2})}{(1+v)(s^{2}+4\alpha
v^{2}(1+v))-(\alpha-\beta)sv}.$

###### Proof.

Using a computation similar to (6.4), we have

$\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v}=\frac{1}{s}\frac{u_{0}v}{1+v}.$

Thus, by a computation similar to (6.5) and Lemma 6.7,

$\displaystyle\frac{d}{du}\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v}$
$\displaystyle=\frac{1}{s}\frac{v(-s+2v(1+v)^{2})}{(1+v)(s+2v^{2}(1+v))}\frac{du_{0}}{du}$
$\displaystyle=\frac{v(-s+2v(1+v)^{2})}{(1+v)(s^{2}+4\alpha
v^{2}(1+v))-(\alpha-\beta)sv},$

completing the proof. ∎

###### Proof of Theorem 6.5.

First, the boundary of $\Omega_{x_{0},\alpha,\beta}$ is the image of the graph
$\\{(u_{0},v_{x_{0},t}(u_{0}))|\,u_{0}\in\mathbb{R}\\}$ under the map
$U_{\alpha,\beta}$ (See (5.1)). By (6.2) and (6.7), if we write
$u=f_{x_{0},\alpha\beta}(u_{0})$, then

$u^{2}=\frac{(1+v)(s-v-v^{2})}{v}\left(1+\frac{\alpha-\beta}{s}\frac{v}{1+v}\right)^{2}=\frac{1}{s^{2}}\frac{s-v-v^{2}}{v}\frac{(s+2\alpha
v)^{2}}{1+v}$

where $v=v_{x_{0},s}(u_{0})$. The claimed formula of
$\partial\Omega_{x_{0},\alpha,\beta}$ follows from the fact that
$b=\varphi_{x_{0},\alpha,\beta}(u)=2\beta v/s$. It is clear that
$\varphi_{x_{0},\alpha,\beta}(u)$ has a decay of order $1/u^{2}$ as
$|u|\to\infty$, because $v\to 0$ as $|u|\to\infty$.

Recall from Theorem 6.1 that $v_{x_{0},s}(0)=\frac{-1+\sqrt{1+4s}}{2}$. Since
$f_{x_{0},\alpha,\beta}(0)=0$ by (6.7) and $f_{x_{0},\alpha,\beta}$ is
strictly increasing, the function

$\varphi_{x_{0},\alpha,\beta}(u)=\frac{2\beta v_{x_{0},s}(u_{0})}{s}$

is unimodal with peak $\frac{\beta(-1+\sqrt{1+4s})}{s}$ at $0$.

By Theorem 5.4 and Lemma 6.8, we have

$\displaystyle w_{\alpha,\beta}(u+iv)$
$\displaystyle=\frac{1}{4\pi\beta}\left(1+2\beta\frac{d}{du}\int\frac{(u_{0}-x)\,d\mu_{x_{0}}(x)}{(u_{0}-x)^{2}+v_{x_{0},s}(u_{0})^{2}}\right)$
$\displaystyle=\frac{1}{4\pi\beta}\left(1+2\beta\frac{v(-s+2v(1+v)^{2})}{(1+v)(s^{2}+4\alpha
v^{2}(1+v))-(\alpha-\beta)sv}\right)$
$\displaystyle=\frac{1}{4\pi\beta}\frac{b^{4}s^{3}+4b^{3}s^{2}\beta+4b^{2}s\beta^{2}+4\beta^{4}}{b^{4}s^{2}\alpha+4b^{3}s\alpha\beta+4b^{2}\alpha\beta^{2}+4b\beta^{4}+4\beta^{4}}$

where we have used $b=\varphi_{x_{0},\alpha,\beta}(u)=(2\beta/s)v$. ∎

We close this section by an example illustrating how mass is transformed by
the push-forward under $U_{\alpha,\beta}$ (See (5.1)), as proved in
Proposition 5.10. By Proposition 5.10 and Theorem 6.1,

$f_{x_{0},\alpha,\beta}^{\prime\prime}(u_{0})=2\pi(\alpha-\beta)w_{s/2,s/2}^{\prime}(u_{0})=(\alpha-\beta)\frac{4v(1+v)(1+3v(1+v))-s}{s(1+v)^{2}(1+2v^{2}(1+v))^{2}}\frac{dv}{du_{0}}$

where $v=v_{x_{0},s}(u_{0})$. Recall that $v_{x_{0},s}(u_{0})$ is unimodal,
$dv/du_{0}=0$ if and only if $u_{0}=0$. As an example, when $s=1$, we can
solve, by the relation of $u_{0}$ and $v_{x_{0},s}(u_{0})$ in Lemma 6.2, that
$f_{x_{0},\alpha,\beta}^{\prime\prime}(u_{0})=0$ if and only if $u_{0}=0$ or
$u_{0}=\pm\frac{-3+\sqrt{15}}{6}\approx\pm 2.56141$.

The top diagram of Figure 5 shows vertical blue line segments inside
$\Lambda_{x_{0},1}$. The spacing between the blue line segments is $0.25$. The
middle diagram of Figure 5 shows the corresponding vertical blue line segments
after being mapped by $U_{1/8,7/8}$: each blue line segment intersecting
$u_{0}\in\mathbb{R}$ in the top diagram is mapped to a vertical line segment
intersecting $f_{x_{0},1/8,7/8}(u_{0})$ in the middle diagram. The bottom
diagram plots the differences

$[f_{x_{0},1/8,7/8}(u_{0}+0.05)-f_{x_{0},1/8,7/8}(u_{0})]-[f_{x_{0},1/8,7/8}(u_{0})-f_{x_{0},1/8,7/8}(u_{0}-0.05)]$

for $u_{0}=0.05k$ for $k=0,1,\ldots,120$. Figure 5 agrees with the theoretical
computation in the preceding paragraph. The spacings between the image of the
vertical line segments with real part less than or equal $2.5$ in the top
diagram are increasing, as shown in the middle diagram, whereas the spacings
between the image of the vertical line segments with real part greater than
$2.5$ in the top diagram are decreasing, as shown in the middle diagram. The
bottom diagram also shows a sign change at a value slightly greater than
$2.5$.

Figure 5: Equally-spaced blue lines inside $\Lambda_{x_{0},1}$ (Top). The
corresponding blue lines under the map $U_{1/8,7/8}$ (Middle). Differences
between the lines: $f_{x_{0},1/8,7/8}(u_{0}+0.05)-f_{x_{0},7/8,7/8}(u_{0})$
and $f_{x_{0},1/8,7/8}(u_{0})-f_{x_{0},1/8,7/8}(u_{0}-0.05)$ (Bottom).

###  6.3 Adding an imaginary multiple of semicircular variable

In this section, we take $\alpha=0$ and $\beta=t$ in Theorem 6.5 to obtain the
Brown measure of $x_{0}+i\sigma_{t}$. One significant simplification over the
previous cases is that the function $\varphi_{x_{0},t}$, whose graph is the
(upper) boundary of $\Omega_{x_{0},t}$, and the density $w_{t}(u+iv)$ can be
written explicitly in terms of $u$, instead of $v_{x_{0},t}(u)$.

######  Theorem 6.9.

When $x_{0}$ has the Cauchy distribution, the domain $\Omega_{x_{0},t}$ has
the form

$\Omega_{x_{0},t}=\left\\{u+iv\in\mathbb{C}\left||v|\leq\frac{4t}{\sqrt{u^{2}+1}(\sqrt{u^{2}+1}+\sqrt{u^{2}+1+4t})}\right.\right\\}.$

The density of the Brown measure of $x_{0}+i\sigma_{t}$ is given by

$w_{t}(\lambda)=\frac{1}{4\pi
t}\frac{4t+(1+u^{2})^{2}}{(1+u^{2})^{3/2}\sqrt{u^{2}+1+4t}},\quad\lambda=u+iv\in\Omega_{x_{0},t}.$

######  Lemma 6.10.

When $x_{0}$ has the Cauchy distribution, the function $\varphi_{x_{0},t}$
defined in Theorem 4.1(4) can be computed as

$\varphi_{x_{0},t}(u)=\frac{4t}{\sqrt{u^{2}+1}(\sqrt{u^{2}+1}+\sqrt{u^{2}+1+4t})},\quad
u\in\mathbb{R}.$

###### Proof.

By putting $\alpha=0$ and $\beta=t$ in (6.6), we get

$u^{2}=\frac{t^{2}(4t^{2}-2bt-b^{2}t)}{bt^{2}(bt+2t)}=\frac{(4t-2b-b^{2})}{b(b+2)}$

where $b=\varphi_{x_{0},0,t}(u)=\varphi_{x_{0},t}$, as indicated in Definition
5.1. Thus, since $\varphi_{x_{0},t}(u)>0$, we can solve the above equation and
get

$\varphi_{x_{0},t}(u)=\frac{-\sqrt{u^{2}+1}+\sqrt{u^{2}+1+4t}}{\sqrt{u^{2}+1}}=\frac{4t}{\sqrt{u^{2}+1}(\sqrt{u^{2}+1}+\sqrt{u^{2}+1+4t})},$

completing the proof. ∎

###### Proof of Theorem 6.9.

The formula for $\Omega_{x_{0},t}$ follows directly from Lemma 6.10. To
compute the density of the Brown measure, by Remark 5.5, we again put
$\alpha=0$ and $\beta=t$ in Theorem 6.5, and get

$w_{t}(\lambda)=\frac{1}{4\pi
t}\frac{4t^{4}+tb^{2}(bt+2t)^{2}}{4t^{3}(bt+t)}=\frac{1}{4\pi
t}\frac{4t+b^{2}(b+2)^{2}}{4t(b+1)},\quad\lambda\in\Omega_{x_{0},t}.$

where $b=\varphi_{x_{0},t}(u)$. By putting in the formula of
$\varphi_{x_{0},t}(u)$ in Lemma 6.10, we have

$w_{t}(\lambda)=\frac{1}{4\pi
t}\frac{4t+(1+u^{2})^{2}}{(1+u^{2})^{3/2}\sqrt{u^{2}+1+4t}},\quad\lambda=u+iv\in\Omega_{x_{0},t}.$

The theorem is established. ∎

In the case $x_{0}+i\sigma_{t}$, the density $w_{t}(\lambda)$ is very
explicit, in terms of $u=\mathrm{Re}(\lambda)$. It is not hard to see that

$\lim_{|u|\to\infty}w_{t}(u)=\frac{1}{4\pi t}.$

Recall that Figure 1 plots an eigenvalue simulation of $x_{0}+i\sigma_{t}$,
the density of the Brown measure of $x_{0}+i\sigma_{t}$ and the function
$w_{t}(u)$ for $u\in\mathbb{R}$ at $t=1$. Figure 6 shows the plots of the
Brown measure densities of $x_{0}+c_{t}$ ($t=1$), $x_{0}+c_{\alpha,\beta}$
($\alpha=1/8$, $\beta=7/8$), and $x_{0}++i\sigma_{t}$ ($t=1$). It also shows
the graphs of $w_{\alpha,\beta}(u)$, $u\in\mathbb{R}$, for comparison. Note
that although we observe the trend from Figure 1 that $w_{\alpha,\beta}(0)$
decreases as $\beta$ increases while keeping $\alpha+\beta=1$, we do not lose
mass around vertical strips around the imaginary axis. The map
$U_{\alpha,\beta}$ indeed pushes mass towards the imaginary axis in a vertical
strip including the origin, by the discussion in the last paragraph of Section
6.2.

Figure 6: Left: Brown measure densities of $x_{0}+c_{1}$ (top),
$x_{0}+c_{1/8,7/8}$ (middle), and $x_{0}+i\sigma_{1}$ (bottom). Right: Graphs
of $w_{1/2,1/2}$ (top), $w_{1/8,7/8}$ (middle), and $w_{1}$ (bottom).

##  7 Acknowledgments

The author would like to thank Hari Bercovici and Roland Speicher who asked
the author in two different seminars whether one can extend the results in
[13, 15] to unbounded random variables. Their question was the starting point
of this paper, and the author had useful discussions with them. The author
would also like to thank Marek Bożejko, Brian Hall, Eugene Lytvynov for useful
conversations. The computer simulation and the Brown measure plot in Figure 1
was plotted by Hall.

## References

  * [1] Belinschi, S. T., and Bercovici, H. A new approach to subordination results in free probability.  J. Anal. Math. 101 (2007), 357–365.
  * [2] Bercovici, H., and Voiculescu, D. Free convolution of measures with unbounded support.  Indiana Univ. Math. J. 42, 3 (1993), 733–773.
  * [3] Biane, P. On the free convolution with a semi-circular distribution.  Indiana Univ. Math. J. 46, 3 (1997), 705–718.
  * [4] Biane, P. Processes with free increments.  Math. Z. 227, 1 (1998), 143–174.
  * [5] Biane, P., and Lehner, F. Computation of some examples of Brown’s spectral measure in free probability.  Colloq. Math. 90, 2 (2001), 181–211.
  * [6] Biane, P., and Speicher, R. Stochastic calculus with respect to free Brownian motion and analysis on Wigner space.  Probab. Theory Related Fields 112, 3 (1998), 373–409.
  * [7] Brown, L. G. Lidskii’s theorem in the type II case.  In Geometric methods in operator algebras (Kyoto, 1983), vol. 123 of Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow, 1986, pp. 1–35.
  * [8] Driver, B. K., Hall, B. C., and Kemp, T. The Brown measure of the free multiplicative Brownian motion.  preprint arXiv:1903.11015 (2019).
  * [9] Evans, L. C. Partial Differential Equations.  Vol 19, Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
  * [10] Fuglede, B., and Kadison, R. V. Determinant theory in finite factors.  Ann. of Math. (2) 55 (1952), 520–530.
  * [11] Girko, V. L. The elliptic law.  Teor. Veroyatnost. i Primenen. 30, 4 (1985), 640–651.
  * [12] Haagerup, U., and Schultz, H. Brown measures of unbounded operators affiliated with a finite von Neumann algebra.  Math. Scand. 100, 2 (2007), 209–263.
  * [13] Hall, B. C., and Ho, C.-W. The Brown measure of the sum of self-adjoint element and an imaginary multiple of a semicircular element.  arXiv:2006.07168 (2020).
  * [14] Ho, C.-W. The two-parameter free unitary Segal-Bargmann transform and its Biane-Gross-Malliavin identification.  J. Funct. Anal. 271, 12 (2016), 3765–3817.
  * [15] Ho, C.-W. The Brown measure of the sum of a self-adjoint element and an elliptic element.  arXiv:2007.06100 (2020).
  * [16] Ho, C.-W., and Zhong, P. Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions.  arXiv:1908.08150 (2019).
  * [17] Jajte, R. Strong limit theorems in noncommutative probability, vol. 1110 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1985.
  * [18] Maassen, H. Addition of freely independent random variables.  J. Funct. Anal. 106, 2 (1992), 409–438.
  * [19] Nica, A., and Speicher, R. Lectures on the combinatorics of free probability, vol. 335 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2006.
  * [20] Rudin, W. Principles of mathematical analysis, third ed.  McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.  International Series in Pure and Applied Mathematics.
  * [21] Saitô, K. Non-commutative extension of Lusin’s theorem.  Tohoku Math. J. (2) 19 (1967), 332–340.
  * [22] Segal, I. E. A non-commutative extension of abstract integration.  Ann. of Math. (2) 57 (1953), 401–457.
  * [23] Śniady, P. Random regularization of Brown spectral measure.  J. Funct. Anal. 193, 2 (2002), 291–313.
  * [24] Takesaki, M. Theory of operator algebras. II, vol. 125 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2003.  Operator Algebras and Non-commutative Geometry, 6.
  * [25] Voiculescu, D. Addition of certain non-commuting random variables.  Journal of Functional Analysis 66, 3 (1986), 323 – 346.
  * [26] Voiculescu, D. Limit laws for random matrices and free products.  Invent. Math. 104, 1 (1991), 201–220.
  * [27] Voiculescu, D. The analogues of entropy and of Fisher’s information measure in free probability theory. I.  Comm. Math. Phys. 155, 1 (1993), 71–92.
  * [28] Voiculescu, D. The coalgebra of the free difference quotient and free probability.  Internat. Math. Res. Notices, 2 (2000), 79–106.