††thanks: This work is currently under review.

# A Theoretical Paradigm for Thermal Rectification via Phonon Filtering and
Energy Carrier Confinement

Brian F. Donovan<EMAIL_ADDRESS>Physics Department
United States Naval Academy
Annapolis, MD 21402  Ronald J. Warzoha  Department of Mechanical Engineering
United States Naval Academy
Annapolis, MD 21402

###### Abstract

We provide a theoretical framework for the development of a solid-state
thermal rectifier through a confinement in the available population of phonons
on one side of an asymmetrically graded film stack. Using a modification of
the phonon gas model to account for phonon filtering and population
confinement, we demonstrate that for an ideal material, with low phonon
anharmonicity, significant thermal rectification can be achieved even in the
absence of ballistic phonon transport. This formalism is used to illustrate
thermal rectification in a thin-film of diamond (1-5 nm) graded to dimensions
$>$ 1 $\mu$m exhibiting theoretical values of thermal rectification ratios
between 0.75 and 6. Our theoretical formulation for thermal rectification is
therefore expected to produce opportunities to design advanced solid-state
devices that enable a variety of critical technologies.

††preprint: APS/123-QED

Introduction.–The development of solid-state thermal architectures is expected
to result in transformative technological breakthroughs similar to those
realized in the information technologies sector. For instance, thin-film
thermal rectifiers have the capacity to revolutionize phonons as information
carriers, allowing for the materialization of phononic computing [1].
Similarly, thermal biasing is pivotal for improvements in thermal barrier
coating effectiveness and heat mitigation in electronic devices [2].
Problematically, such biasing has traditionally been achieved either when
thermal gradients are sufficiently large to produce corresponding gradations
in temperature-dependent thermal conductivity across a set of dissimilar
materials [2, 3, 4, 5, 6, 7] or when there exists mass gradation in the
direction of heat flow [8].

Recent work by Chang et al. [9] describes a process to produce thermal
rectification via the formation of an asymmetric structure that induces
asymmetric boundary scattering of phonons with respect to the direction of
heat flow. In their work, the authors find that a partial mass loading of
asymmetrically deposited, amorphous C9H16Pt particles on the outer surface of
a boron nitride nanotube results in a measured thermal rectification of $\sim$
7$\%$. Others have also found asymmetry to be an effective mechanism to
achieve thermal rectification [10, 11, 12, 13, 14, 4, 15]. Still, the
magnitude of thermal rectification achieved through asymmetric structuring
remains relatively low (i.e. $<$ 10-50$\%$) in the absence of significant
thermal gradients in the direction of heat flow [10]. In fact, the authors are
aware of only one study that experimentally demonstrates a thermal
rectification ratio well above this [16] by tunnel-coupling metals to
superconducting elements. We note, however, the difficulty associated with the
integration of such a device into practical thermal applications. In this
work, we provide a physical construct that can be used to design thermal
rectifiers that do not rely on thermal and/or mass gradients to produce an
observable thermal rectification effect.

To this end, we focus on the conditions necessary for thermal rectification
from the perspective of available wavelengths in the phonon density of states
as thermal energy traverses thin-film material stacks in opposing directions.

Modeling thermal rectification via phonon confinement.– To demonstrate thermal
rectification in thin-film, asymmetric material stacks, we employ the
analytical treatment of phonon transport constructed by Callaway [17] and
represented by,

$\kappa=\frac{1}{3}\sum_{j}\int_{k}C_{v}\nu ldk$ (1)

and,

$C_{v}=\frac{1}{2\pi^{2}}\sum_{j}\int_{k}\hbar\omega\frac{\partial
f_{BE}}{\partial T}k^{2}dk$ (2)

where $\kappa$ and Cv are thermal conductivity and volumetric heat capacity,
respectively. In the above expressions, j represents the polarization index,
$\nu$ the phonon group velocity, $\omega$ the angular frequency, k the
wavevector and $\frac{\partial f_{BE}}{\partial T}$ the temperature derivative
of the Bose-Einstein distribution. As well, the phonon mean free path, $l$ is
incorporated into the physical representation of thermal conductivity to
account for energy carrier scattering.

The impact of extrinsic scattering mechanisms on the spectrum of phonon
wavelengths that contribute to the thermal conductivity is analyzed using
Matthiessen’s rule, which allows us to isolate the intrinsic phonon mean free
path, $l_{in}$, and the mean free path considering the sample boundaries,
$l_{bound}$. This results in a representative thermal conductivity for a given
phonon polarization as described by,

$\kappa=\frac{1}{3}\int
C_{v}\nu\big{[}l_{in}^{-1}+l_{bound}^{-1}\big{]}^{-1}dk$ (3)

In this work, we limit our analysis to one-dimensional heat flow across thin
film configurations. Thus, lbound is treated as the characteristic length of
an individual film.

Within the framework of spectral contributions to thermal conductivity, we
deduce that thermal rectification occurs when long wavelength phonons are
unable to carry thermal energy across the film in the presence of a very thin,
specular material boundary on one side of the film. Thus we rewrite the
equation for the thermal conductivity (Eqn. 4) to highlight its wavelength
($\lambda$) dependence, and explicitly identify the components that are a
function of phonon wave vector ($k=2\pi/\lambda$). Moreover, it will become
informative to discuss impacts to the thermal conductivity within the context
of the phonon density of states, DOS$(k)$, thus the integral that determines
the thermal conductivity for a given phonon polarization as a function of wave
vector can be written as,

$\kappa(k)=\int_{k_{min}}^{k_{max}}\hbar\omega(k)\mathrm{DOS}(k)\frac{\partial
f_{BE}}{\partial T}\nu(k)\big{[}l_{in}(k)^{-1}+l_{bound}^{-1}\big{]}^{-1}dk$
(4)

This integral is taken over the available wave vector space from
$k_{min}=\pi/d$ to $k_{max}=\pi/a$, where $d$ is the physical dimension of the
entire crystal and $a$ is the lattice constant of the crystal. In the limit of
most bulk formulations of thermal conductivity, the minimum wave vector is
assumed to be zero (i.e. an infinite crystal).

Typically, the impact of nanostructuring on thermal conductivity is assumed to
be accounted for entirely by modifications to the phonon mean free path. In
this case, phonons that have wavelengths greater than the dimensions of a
given nanostructure (a material layer having nanometer thickness) are assumed
to be present when calculating thermal conductivity using the standard model,
but are thought to heavily scatter at any material boundary (i.e. grain
boundaries, film boundaries, or other interfaces). It is useful in this
context to consider spectral contributions to the thermal conductivity. In
Fig. 1, we observe the spectral thermal conductivity of the longitudinal
acoustic phonon branch of diamond, which is calculated from the integrand of
Eq. 4, plotted against phonon wavelength for an infinite crystal
($d\rightarrow\infty$) with finite scattering dimensions of 5 mm. Further, we
include the spectral thermal conductivity for 5 $\mu$m, 500 nm, 50 nm, and 5
nm films of diamond modeled using this same full integral for an infinite
crystal, and account for the impacts of nanostructuring on thermal
conductivity using a mean free path that is limited by boundary scattering.

Figure 1:  Spectral thermal conductivity as a function of phonon wavelength
from a typical scattering limited thin film model. This function is integrated
to obtain the thermal conductivity for a given material system. Curves are
included for diamond films with thicknesses of 5 mm (black), 5$\mu$m (red),
500 nm (blue), 50 nm (green) and 5 nm (purple). Also included are dashed lines
indicating where the wavelength of phonons corresponding to the film thickness
lies. Phonons with wavelengths greater than the film thickness contribute far
less (orders of magnitude) than the primary heat carrying phonons in each
system.

Figure 1 provides the relative contribution to the total thermal conductivity
of single-layer diamond films with varying thickness at any given phonon
wavelength. The Callaway model can be used to determine the effect that a
given population of phonons has on total thermal conductivity by integrating
over all possible phonon wavelengths. This has been addressed within the wider
literature, most notably through the concept of mean free path accumulation
[18, 19, 20, 21, 22, 23]. One immediate conclusion that can be drawn from this
analysis is that the peak phonon wavelength is significantly smaller than the
film’s characteristic dimension, and is further reduced as the sample
dimension shrinks.

We note now that this peak phonon wavelength contributes orders of magnitude
more to the thermal conductivity than wavelengths that are on the order of the
film thickness. To illustrate this point, the characteristic dimension used to
determine the phonon mean free path by boundary scattering is plotted in Fig.
1 with a dashed, vertical line for each film thickness (or the bounds of the
figure for the 5 mm film). When examining phonon wavelengths that extend
beyond the limit of the physical boundary, their inclusion in the computation
of film thermal conductivity can be omitted due to their relatively low
contributions to thermal conductivity when compared to those wavelengths that
span the peak. In the context of phonon filtering, boundary scattering
preferentially eliminates (or filters out) contributions to the thermal
conductivity from phonons that are greater than the film thickness.

As a consequence of the relatively low contribution to the thermal
conductivity made by phonons having wavelengths greater than the film
thickness, we assert that these dimensions can be considered as the
predominant confining features of the material. In fact, in a free standing
nanostructured system (a suspended nanoparticle or unsupported membrane), the
confinement of available phonons is clearly reduced to the nanostructured
dimension and incorporation of longer wavelength phonons that are even
available to undergo scattering would be non-physical [24, 25].

In this work, we examine the impact of treating those boundaries, which have
previously been considered only as scattering sites, as sites that confine the
population of phonons available to transfer thermal energy across a layered
material system. This implies that the lower bound of our integration in wave
vector space is governed by the characteristic dimension of the nanostructured
component. In the case of a spherical nanoparticle, for example, the relevant
dimension is the diameter ($d$) of the nanoparticle. Likewise, for a free
standing film with one dimensional heat flow, the film thickness ($t$) serves
as the relevant dimension for phonon confinement.

In order to account for possible phonon confinement effects, we modify the
bounds of our integration such that we disregard wave vectors below
$k_{min}=\pi/t$. Thus, every component of the thermal conductivity in Eq. 4
that depends on the wave vector will assume the effects of phonon filtering or
dimensional phonon confinement. This simple modification provides for a more
rigorous treatment of the physics that govern thermal transport in multi-layer
nanoscale films, and allows for an additional mechanism to achieve thermal
rectification in multi-layer material systems.

We validate our confined phonon model by first comparing computations of
temperature-dependent thermal conductivity for single-layer diamond films to
those obtained by integrating across all phonon wavelengths, as shown in Fig.
2. In this case we use the acoustic branches of a real diamond phonon
dispersion to determine the phonon frequency and group velocity [26, 27] and
assume a spherical density of states. The intrinsic mean free path is
determined by accounting for phonon-phonon scattering and phonon-impurity
scattering when fit to literature data for bulk diamond systems. Values for
bulk diamond thermal conductivity computed with the confined phonon integral
are found to be consistent with those values available in the wider literature
[28, 18, 29, 30].

Figure 2:  Thermal conductivity of diamond modeled over temperature using both
a standard scattering limited Callaway model as well as a confined phonon
population model. Models are shown for samples with bulk dimensions (black,
solid line) as well as films of 5 $\mu$m, 50 nm, and 5 nm. In the thin film
models, the standard “Full Integral” scattering limited approach (blue line)
matches the confined phonon population approach (red dashes) very well. The
only deviation comes at low temperatures with the single-nanometer-scale film.

In Fig. LABEL:fig:KvsT, the confined phonon integration matches the full
integral almost identically. The only deviation occurs at low temperature with
single-digit nanometer sized films, which is physically appropriate
considering the phonon population is not yet fully occupied in such thin films
and long wavelength phonons begin to govern thermal transport. The comparison
in Fig. 2 lends confidence that a confined phonon model captures phonon
filtering identically to the Callaway model integrated over a full spectrum of
all possible wavelengths.

Thermal rectification in nanostructured film stack.–In order to use a phonon
confinement effect to achieve thermal rectification, layered (or graded)
structures must be fabricated in such a way that a confined phonon population
can be injected from one layer into an otherwise non-confined layer. Within a
practical context, an ultra-thin film can be deposited above a thick film
(whose phonon population is unconfined) to achieve this effect. Here, the
ultra-thin film acts to filter out long wavelength phonons, where the
available phonon population across the film boundary remains limited to the
spectrum of incoming phonons.

For the available phonon population to remain consistent on opposing sides of
the film boundary, the interface between the phonon filter layer and adjacent
thick layer(s) must be lattice matched and coherent. If the morphology of the
interface results in diffuse and anharmonic phonon scattering, then the
transport of thermal energy from one film to the other would result in a
redistribution of phonons that assumes the thick film’s enlarged density of
states, rendering this formulation invalid.

Likewise, the manifestation of thermal rectification requires a lack of
redistribution of the phonon population from anharmonic phonon-phonon
scattering within the thick layer itself. While we account for these effects
to determine the intrinsic mean free path within our model, we assume that the
phonons injected into the thick film from the filter layer do not scatter
anharmonically into other wavelengths that may not have been available to
begin with. Each of these requirements are fulfilled in several existing
material systems, including grain-graded, nanocrystalline diamond membranes
(whose grain boundaries are twinned) [31] and Si-based superlattices [32].

Provided the above requirements are met, this treatment can be used to design
a solid-state thermal rectifier using multilayer thin-films. The most palpable
system to imagine is a bilayer stack having one ultra-thin layer deposited
above a thick layer. In Fig. 3a we provide an example of thermal rectification
in a bilayer diamond film system that consists of a 1 $\mu$m film above a 1 nm
film. Fig. 3 provides for an interesting result. In the typical “full
integral” model, we see that the thermal conductivity is not directionally
dependent when calculated using mean free paths that are limited by boundary
scattering (as expected). Conversely, the phonon confinement analysis does
yield a directionally-dependent thermal conductivity. When we apply these
physics to the case when heat flows from the thick film to the thin film, the
confined thermal model results in a thermal conductivity distribution
identical to that obtained from the “full integral” model. However, if heat
flows in the other direction, we obtain a significantly reduced thermal
conductivity in the thick layer as we are not fully populating all available
phonon modes. The directional-dependence of thermal conductivity results in
thermal rectification, and is extremely significant in the limit of full
phonon confinement.

Figure 3:  Thermal conductivity of diamond through the thickness of a bilayer
(a) and multilayer (b) film going from the bottom side of the total film stack
up (thin to thick) and the topside down (thick to thin). Models are shown
using only scattering limited modeling (blue solid line and dashes, no
directional difference) and the confined phonon population model (red dashed
line from bottom up and solid line from top down). In the top-down direction
both modeling approaches match, however in the bottom up, the confined phonon
population model results in a limited thick-film thermal conductivity due to a
lack of long wavelength phonons present in the incoming phonon population.
This results in a significant difference in thermal conductivity and a large
thermal rectification effect.

Mathematically, the resulting thermal rectification originates from
modifications to the limits of integration in Eq. 4. Considering the case when
heat emanates from the thick-film side, $k_{min}={\pi}~{}{\mu\mathrm{m}^{-1}}$
(as well as $l_{bound}=1~{}\mu\mathrm{m}$), which then become limited in the
thin film portion of the stack. In the opposing direction, however,
$k_{min}=\pi~{}\mathrm{nm}^{-1}$ (and $l_{bound}=1~{}\mathrm{nm}$). Even
though the mean free path is relaxed as the heat moves into the thick layer
($l_{bound}=1~{}\mu\mathrm{m}$), $k_{min}$ does not change and so the
confinement results in diminished thermal transport.

In Fig. 3b we also examine these physics for a 20 layer material stack with
logarithmically increasing thickness (in this case, ranging from 1 nm to 1
$\mu$m). This is the type of dimensional evolution that one might see from
nucleation and grain growth in a typical top-down film deposition technique
[31]. Here, our confinement still originates from the initial filtering layer
(1 nm). We again note that this model assumes that no anharmonic interactions
occur, however, traversing this many interfaces in a real material system
would inevitably lead to some redistribution of phonon populations.
Nevertheless, this provides an upper limit to thermal rectification (in the
absence of any external temperature gradient or material asymmetry).

A thermal rectification ratio, $TR$, is used to calculate the degree of
thermal rectification achieved relative to other works [12, 8] and is
represented by,

$TR=\frac{\kappa_{TD}-\kappa_{BU}}{\kappa_{BU}}$ (5)

where $\kappa_{TD}$ is the thermal conductivity with heat moving from the top
down and $\kappa_{BU}$ is the thermal conductivity from the bottom up.

The thermal rectification ratio is plotted for the bilayer system over various
phonon filtering layer thicknesses in Fig. 4. In this formulation we compare
the directional thermal conductivities in the thick layer as it represents the
largest thermal resistor in the bilayer stack. We also include distributions
for a number of different thick layer length scales, which reveal that despite
primary limitations of the phonon filter layer, boundary scattering will still
play a role in the thick layer’s available phonon population. Effectively,
thick layers with less boundary scattering are at greater risk for diminished
thermal conductivity due to phonon confinement. However, even in a bilayer
that consists of a 1 $\mu$m film with a 1 nm filter layer, the thermal
rectification ratio is $>$ 75$\%$. We note that this is considerably higher
than thermal rectification ratios that have been reported in literature (for
cases where a large temperature gradient across the structure is absent). In
the case where there is significant dimensional mismatch, extremely large
thermal rectification ratios can be achieved ($>$ 600%). The upper range of
these thermal rectification ratios is expected to revolutionize a wide range
of thermal devices and facilitate the development of phononic computing.

Figure 4:  Thermal rectification over various phonon filter dimensions (i.e.
the thin layer in the bilayer film stack) computed for diamond films using the
confined phonon population model. This is shown for multiple dimensions of the
thick film, where thermal rectification effects become more extreme with a
greater disparity between the filter layer and the top layer. Rectification is
computed using the directional difference in the thermal conductivities in the
top layer.

Figure 4 suggests that when the filter layer thickness is greater than
$\sim$10 nm, thermal rectification is negligible. This can be readily
understood via the spectral contributions to thermal conductivity shown in
Fig. 1. Here, the peak contributors to the thermal conductivity exist in the
sub-10 nm range (particularly in the thin film regime). We should note that
while the bilayer system has a rectification factor that is governed by the
thicker component, the multilayer system does not have the same spatial
response. When multiple layers are present, the thermal rectification ratio is
computed using average directional thermal conductivities in Eq. 5. This
results in a thermal rectification ratio that is slightly reduced (63% in the
20 layer system that goes from 1 nm to 1 $\mu$m compared to 75% in the same
extremes for a bilayer).

Discussion.– The use of a phonon filter layer to confine a population of
phonons across a thin-film boundary is extremely promising for the realization
of effective thermal rectifiers. These physical dynamics enable thermal
rectification at realistic device scales with sample configurations that
readily lend themselves to microelectronics processing techniques. Critically,
extreme thermal rectification ratios of several hundred percent are predicted
for these systems, which will finally allow for robust experimental
demonstrations of thermal rectification.

We have enabled this physical understanding by modifying the well-known
Callaway formulation for the thermal conductivity of a material based on heat
capacity and thermal carrier dynamics. Critical to realizing thermal
rectification, we have extended the model by considering the confinement of
phonons when film thickness is sufficiently small. By restricting the
integration limits to the characteristic dimension(s) of a film, we can
directly observe the impact of phonon confinement on the density of phonon
states available for thermal transport while simultaneously capturing the
phonon filtering effects that are well-known within the scientific community.
This can be extended to multilayer, graded materials, which are an important
class of materials for integration into larger system platforms.

Though this analysis relies on a lack of anharmonic interactions across
interfaces and within each material layer, it does provide an upper limit to
thermal rectification that can be achieved in an ideal system. In diamond (a
material that does, in fact, demonstrate negligible anharmonic scattering
[26]), this upper limit is computed to be $\sim$600%, which is orders of
magnitude larger than any experiments yet to be reported in literature. This
is particularly significant given the additional potential to combine this
effect with other mechanisms that are known to produce thermal rectification,
such as geometric asymmetry and an imposed thermal gradient. Even if a
fraction of this limit should be realized experimentally, it would allow for
the production of practical thermal devices and represent a major advancement
in the thermal sciences.

Acknowledgements.– BFD and RJW would like to acknowledge the financial support
of Mr. Peter Morrison and the Office of Naval Research under Contract No.
N0001419WX00501. The authors are also grateful for critical insight provided
by Dr. Andrew Smith.

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# Superconvergence in free probability limit theorems for arbitrary triangular
arrays

Hari Bercovici, Ching-Wei Ho, Jiun-Chau Wang, and Ping Zhong  Hari Bercovici:
Department of Mathematics, Indiana University, Bloomington, IN 47405, United
States<EMAIL_ADDRESS>Ching-Wei Ho: Institute of Mathematics, Academia
Sinica, Taipei 10617, Taiwan; Department of Mathematics, University of Notre
Dame, Notre Dame, IN 46556, United States<EMAIL_ADDRESS>Jiun-Chau Wang:
Department of Mathematics and Statistics, University of Saskatchewan,
Saskatoon, S7N 5E6, Canada<EMAIL_ADDRESS>Ping Zhong: Department of
Mathematics and Statistics, University of Wyoming, Laramie, WY 82071, United
States<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

It is known that limit theorems for triangular arrays with identically
distributed rows yields convergence of densities rather than just convergence
in distribution. We show that this superconvergence result holds—at least at
points at which the limit density is nonzero—even if the rows of the array are
not identically distributed.

###### Key words and phrases:

free convolution, superconvergence

###### 2010 Mathematics Subject Classification:

46L54

##  1\. Introduction

The central limit theorem of free probability [15] asserts that for a sequence
of freely independent, identically distributed random variables
$X_{1},X_{2},\dots$ with zero expected value and unit variance, the variables

$Y_{n}=\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$

converge in distribution to the standard semicircular law with density

$\frac{1}{2\pi}\sqrt{4-t^{2}}$

for $t\in[-2,2]$. In case the variables $X_{n}$ are also bounded, it was
observed in [6] that the distribution of $Y_{n}$ is necessarily absolutely
continuous for large $n$, and that the densities of these distributions (along
with their derivatives of all orders) converge uniformly to the semicircular
density on every interval $[a,b]\subset(-2,2)$. Moreover, the distribution of
$Y_{n}$ is supported on an interval $[c_{n},d_{n}]$ such that
$\lim_{n\to\infty}c_{n}=-2$ and $\lim_{n\to\infty}d_{n}=2$. This phenomenon
was called _superconvergence_ in [6]. Chistyakov and Götze computed an
asymptotic expansion of the free central limit theorem [10] and the rate of
convergence of the density [9]. Various aspects of superconvergence were
extended to more general limit processes for sums as well as for products of
_infinitesimal arrays_ of random variables, generally under the assumption
that the rows of the arrays are formed of freely independent, identically
distributed variables (see [1, 7, 8, 13, 14]). The purpose of this note is to
remove the identical distribution hypothesis on the arrays. The technique is
closer to the one employed in [6] than to the more recent developments and it
essentially uses only the fact that the $R$-transforms of the variables in an
infinitesimal array are defined on arbitrarily large Stolz angles. In
particular, the subordination property of free convolution is not needed in
this paper. The results do not even require that we work with a convolution of
infinitesimal variables, but they only apply locally to the points that have a
neighborhood where the limit distribution is absolutely continuous. We provide
complete arguments for the additive case. The arguments for products are very
similar and only the relevant differences are pointed out.

##  2\. Additive convolution and superconvergence

The distribution of a sum of freely independent selfadjoint random variables
is the _free additive convolution_ of the distributions of the summands.
Because of this fact, we can forget about the variables themselves and focus
instead on free convolutions. Suppose that $\mu$ is a Borel probability
measure on $\mathbb{R}$, and define two analytic functions on
$\mathbb{H}=\\{z\in\mathbb{C}:\Im z>0\\}$ by

$G_{\mu}(z)=\int_{\mathbb{R}}\frac{d\mu(t)}{z-t},\quad
F_{\mu}(z)=\frac{1}{G_{\mu}(z)},\quad z\in\mathbb{H}.$

The measure $\mu$ is uniquely determined by either of these functions, and the
Stieltjes inversion formula shows that the value of the density of $\mu$ at
almost every $x\in\mathbb{R}$ is equal to the boundary limit
$-\pi^{-1}\lim_{y\downarrow 0}\Im G_{\mu}(x+iy)$. The function $F_{\mu}$
satisfies

(2.1) $\lim_{z\to\infty}\frac{F_{\mu}(z)}{z}=1,$

where the limit is taken as $z=x+iy\to\infty$ such that $|x|/y$ remains
bounded. Given $\alpha,\beta>0$, define the Stolz angle at infinity by

$\Gamma_{\alpha,\beta}=\\{x+iy\in\mathbb{C}:y\geq\max\\{\beta,\alpha|x|\\}\\}.$

Then (2.1) implies that for every $\alpha>0$, there exist $\beta>0$ and an
analytic function $H_{\mu}:\Gamma_{\alpha,\beta}\to\mathbb{H}$ such that

$F_{\mu}(H_{\mu}(z))=z,\quad z\in\Gamma_{\alpha,\beta};$

see [5]. Since $\Im F_{\mu}(z)\geq\Im z$ for $z\in\mathbb{H},$ it follows that
the function

$\varphi_{\mu}(z)=H_{\mu}(z)-z$

satisfies

(2.2) $\Im\varphi_{\mu}(z)\leq 0,\quad z\in\Gamma_{\alpha,\beta}.$

The function $\varphi_{\mu}$ may have analytic continuations to other regions
of the form $\Gamma_{\alpha^{\prime},\beta^{\prime}}$ that still satisfy (2.2)
but possibly $\Im(z+\varphi_{\mu}(z))\leq 0$ for some
$z\in\Gamma_{\alpha^{\prime},\beta^{\prime}}$, so the quantity
$F_{\mu}(z+\varphi_{\mu}(z))$ is not defined. Any two such analytic
continuations coincide on their common domain of definition, and we continue
denoting by $\varphi_{\mu}$ the function obtained by assembling all of these
possible analytic continuations to domains of the form
$\Gamma_{\alpha^{\prime},\beta^{\prime}}$. Thus, if $\varphi_{\mu}$ is defined
at a point $z=x+iy\in\mathbb{H}$, then it is also defined at all points
$x^{\prime}+iy^{\prime}$ such that $y^{\prime}\geq y$ and
$|x^{\prime}|/y^{\prime}\leq|x|/y$. The equation

$F_{\mu}(z+\varphi_{\mu}(z))=z$

persists for $z$ in every connected open set $U$ such that
$U\subset\Gamma_{\alpha^{\prime},\beta^{\prime}}$ for some
$\alpha^{\prime},\beta^{\prime}>0$ and $z+\varphi_{\mu}(z)\in\mathbb{H}$ for
every $z\in U$. The related identity

(2.3) $F_{\mu}(z)+\varphi_{\mu}(F_{\mu}(z))=z$

holds in every connected open set $V\subset\mathbb{H}$ with the property that
$F_{\mu}(z)$ is in the domain of $\varphi_{\mu}$ for $z\in V$.

The function $\varphi_{\mu}$ defined above is known as the _Voiculescu
transform_ of $\mu$ and it serves to linearize the free additive convolution
of probability measures on $\mathbb{R}$; that is

$\varphi_{\mu_{1}\boxplus\mu_{2}}=\varphi_{\mu_{1}}+\varphi_{\mu_{2}}$

on the common domain of $\varphi_{\mu_{1}}$ and $\varphi_{\mu_{2}}$. We also
recall that a measure $\mu$ is $\boxplus$-_infinitely divisible_ if and only
if $\varphi_{\mu}$ is defined on the entire $\mathbb{H}$ and
$\varphi_{\mu}(\mathbb{H})\subset\mathbb{C}\backslash\mathbb{H}$ [5]. If $\mu$
is $\boxplus$-infinitely divisible, the identity (2.3) holds for for every
$z\in\mathbb{H}$ and thus $F_{\mu}$ maps $\mathbb{H}$ conformally onto a
domain $\Omega_{\mu}\subset\mathbb{H}.$ It was shown in [5] that

$\Omega_{\mu}=\\{z\in\mathbb{H}:z+\varphi_{\mu}(z)\in\mathbb{H}\\}.$

In fact, there exists a continuous function $f_{\mu}:\mathbb{R}\to[0,+\infty)$
such that

$\Omega_{\mu}=\\{x+iy:x\in\mathbb{R},y>f_{\mu}(x)\\},$

and the map $x\mapsto H_{\mu}(x+if_{\mu}(x))$ (extended by continuity when
$f_{\mu}(x)=0$) is a homeomorphism of $\mathbb{R}$ onto $\mathbb{R}$, see [11]
or [7, Section 2]. Moreover, the function $f_{\mu}$ is real-analytic wherever
it is nonzero. If $f_{\mu}>0$ on an interval $[A,B]$ then $\mu$ has a real-
analytic density on the interval $[a,b]$, where

$a=H_{\mu}(A+if_{\mu}(A)),\quad b=H_{\mu}(B+if_{\mu}(B)),$

and the density at a point $t=H_{\mu}(s+if_{\mu}(s))$, $s\in[A,B]$, is given
by

$-\frac{1}{\pi}\Im
G_{\mu}(t)=-\frac{1}{\pi}\Im\frac{1}{s+if_{\mu}(s)}=\frac{1}{\pi}\frac{f_{\mu}(s)}{s^{2}+f_{\mu}(s)^{2}}.$

Every interval $[a,b]$ on which $\mu$ has a positive density is of the form
described above. The following fact is used in the proof of the main result.

######  Lemma 2.1.

Suppose that $\mu$ is a $\boxplus$-infinitely divisible measure on
$\mathbb{R},$ and let $s\in\mathbb{R}$ be such that $f_{\mu}(s)>0$. Then

$\frac{\partial}{\partial t}\Im H_{\mu}(s+it)>0\text{ for }t\geq f_{\mu}(s).$

###### Proof.

When $\mu$ is a point mass, we have $\Im H_{\mu}(s+it)=t$, so the result is
immediate. Suppose therefore that $\mu$ is not a point mass and write the
function $\varphi_{\mu}$ in its Nevanlina representation

$\varphi_{\mu}(z)=c+\int_{\mathbb{R}}\frac{1+zx}{z-x}\,d\sigma(x),$

where $c$ is a real constant and $\sigma$ is a nonzero, finite, positive Borel
measure on $\mathbb{R}$. By the Cauchy-Riemann equations, it suffices to show
that

$\Re H^{\prime}_{\mu}(z)>0\text{ for }z=s+it,\,t\geq f_{\mu}(s).$

Easy calculations show that

$H^{\prime}_{\mu}(s+it)=1-\int_{\mathbb{R}}\frac{1+x^{2}}{(s+it-x)^{2}}d\sigma(x),$

and

$\Im
H_{\mu}(s+it)=t\left[1-\int_{\mathbb{R}}\frac{1+x^{2}}{|s+it-x|^{2}}d\sigma(x)\right].$

The fact that $f_{\mu}(s)>0$ means that

$\int_{\mathbb{R}}\frac{1+x^{2}}{|s+it-x|^{2}}d\sigma(x)=1,\quad
t=f_{\mu}(s),$

and thus

$\int_{\mathbb{R}}\frac{1+x^{2}}{|s+it-x|^{2}}d\sigma(x)\leq 1,\quad t\geq
f_{\mu}(s).$

On the other hand, one has

$\Re\int_{\mathbb{R}}\frac{1+x^{2}}{(s+it-x)^{2}}d\sigma(x)<\left|\int_{\mathbb{R}}\frac{1+x^{2}}{(s+it-x)^{2}}d\sigma(x)\right|\leq\int_{\mathbb{R}}\frac{1+x^{2}}{|s+it-x|^{2}}d\sigma(x)\leq
1$

for $t\geq f_{\mu}(s)$, and this implies the desired inequality $\Re
H^{\prime}_{\mu}(z)>0$. ∎

######  Remark 2.2.

In the above proof, we were able to use an explicit formula for the
derivative. Calculations are more cumbersome for free multiplicative
convolutions and therefore the following general fact will be useful. Suppose
that $\alpha\in\mathbb{R}$ and $\varepsilon,\delta>0$, and that
$f:(\alpha-\varepsilon,\alpha+\varepsilon)\to\mathbb{R}$ is a differentiable
function. Set

$D=\\{x+iy:x\in(\alpha-\varepsilon,\alpha+\varepsilon)\text{ and
}|y-f(x)|<\delta\\},$

and let $H:D\to\mathbb{C}$ be an analytic function such that

$\Im H(x+iy)\begin{cases}>0,&y>f(x),\\\ <0,&y<f(x).\end{cases}$

Then

(2.4) $\frac{\partial}{\partial y}\Im H(x+iy)>0\text{ for
}y=f(x),\>x\in(\alpha-\varepsilon,\alpha+\varepsilon).$

To see this, we observe that the function $H(x+if(x))$ is necessarily real and
increasing on $(\alpha-\varepsilon,\alpha+\varepsilon)$, and thus

$0\leq\frac{d}{dx}H(x+if(x))=H^{\prime}(x+if(x))(1+if^{\prime}(x)).$

Writing $H^{\prime}(x+if(x))=a+ib$ with $a,b\in\mathbb{R}$, we see that
$b=-af^{\prime}(x)$ and

$\frac{d}{dx}H(x+if(x))=a-bf^{\prime}(x)=a(1+f^{\prime}(x)^{2}).$

Since it is easily seen that $H^{\prime}(x+if(x))\neq 0$ (otherwise, the
function $H(x+if(x))$ behaves locally as a power function, and $H(x+if(x))$
cannot be real on $(\alpha-\varepsilon,\alpha+\varepsilon)$), we conclude that
$a>0$, and (2.4) follows from the Cauchy–Riemann equations.

Weak convergence is easily described in terms of the functions $F_{\mu}$ or
$\varphi_{\mu}$. If a sequence $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ of Borel
probability measures is tight, then there exist $\alpha,\beta>0$ such that
$\Gamma_{\alpha,\beta}$ is contained in the domain of $\varphi_{\mu_{n}}$ for
every $n\in\mathbb{N}$. Moreover, the weak convergence of
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ to $\mu$ is equivalent to the local uniform
convergence of $\varphi_{\mu_{n}}$ to $\varphi_{\mu}$ on
$\Gamma_{\alpha,\beta}$, as well as to the local uniform convergence of
$F_{\mu_{n}}$ to $F_{\mu}$ on $\mathbb{H}$. See [5] for the proofs of these
results.

Suppose now that $k_{1},k_{2},\dots\in\mathbb{N}$, and that
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an array of Borel
probability measures on $\mathbb{R}$. This array is said to be _infinitesimal_
if

$\lim_{n\to\infty}\min_{1\leq i\leq
k_{n}}\mu_{n,i}\left((-\varepsilon,\varepsilon)\right)=1$

for every $\varepsilon>0$. It was observed in [4] that, given an infinitesimal
array as above and arbitrary $\alpha,\beta>0$, there exists $N\in\mathbb{N}$
such that $\varphi_{\mu_{n,i}}$ is defined on $\Gamma_{\alpha,\beta}$ for
every $n\geq N$ and $i=1,\dots,k_{n}$. In particular, if we set

$\nu_{n}=\mu_{n,1}\boxplus\cdots\boxplus\mu_{n,k_{n}},$

$\varphi_{\nu_{n}}$ is defined on $\Gamma_{\alpha,\beta}$ for $n\geq N$.
Therefore the following result applies in particular to free additive
convolutions of measures in an infinitesimal array and provides our extension
of superconvergence to such an array.

######  Theorem 2.3.

Let $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures on $\mathbb{R}$ that converges weakly to an $\boxplus$-infinitely
divisible measure $\nu.$ Suppose that for every $\alpha,\beta>0$ there exists
$N\in\mathbb{N}$ such that $\varphi_{\nu_{n}}$ is defined in
$\Gamma_{\alpha,\beta}$ for every $n\geq N$. Let $J\subset\mathbb{R}$ be a
compact interval such that $\nu$ is absolutely continuous and $d\nu/dx>0$ in a
neighborhood of $J$. Then $\nu_{n}$ is absolutely continuous on a neighborhood
of $J$ with a real-analytic density for sufficiently large $n$, and the
densities $d\nu_{n}/dx$ converge uniformly on $J$, along with all their
derivatives, to $d\nu/dx$ as $n\to\infty$.

###### Proof.

Since unit point masses are purely singular we may, and shall, assume that the
support of $\nu$ contains more than one point. It suffices to show that
$d\nu_{n}/dx$ converges locally uniformly to $d\nu/dx$ on a set on which the
latter density is strictly positive. Fix $\alpha_{0},\beta_{0}>0$ such that
$\Gamma_{\alpha_{0},\beta_{0}}$ is contained in the domain of
$\varphi_{\nu_{n}}$ for every $n\in\mathbb{N}$ and $\varphi_{\nu_{n}}$
converges locally uniformly to $\varphi_{\nu}$ on
$\Gamma_{\alpha_{0},\beta_{0}}$. As noted above, the function $\varphi_{\nu}$
is defined on $\mathbb{H}$. Fix a point $x_{0}\in\mathbb{R}$ such that $\nu$
is absolutely continuous in a neighborhood of $x_{0}$ and
$(d\nu/dx)(x_{0})>0$. It follows that there exists $s_{0}\in\mathbb{R}$ such
that $f_{\nu}(s_{0})>0$ and $x_{0}=H_{\nu}(s_{0}+if_{\nu}(s_{0}))$. Given an
arbitrary number $\varepsilon>0$ with $\varepsilon<f_{\nu}(s_{0})$, Lemma 2.1
allows us to choose an interval $[A,B]$ containing $s_{0}$ such that

$Q=\\{s+it:s\in[A,B],|t-f_{\nu}(s_{0})|\leq\varepsilon\\}$

is contained in $\mathbb{H}$, $|f_{\nu}(s)-f_{\nu}(s_{0})|<\varepsilon$ for
$s\in[A,B]$, and

$\frac{\partial}{\partial t}\Im H_{\nu}(s+it)>0,\quad s+it\in Q.$

Choose also $\delta>0$ such that the interior of the rectangle

$K=\left\\{s+it:s\in[A,B],\delta\leq t\leq\frac{1}{\delta}\right\\}$

has nonempty intersection with $\Gamma_{\alpha_{0},\beta_{0}}$ and $Q\subset
K$. Finally, choose $\alpha,\beta>0$ such that

$K\cup\Gamma_{\alpha_{0},\beta_{0}}\subset\Gamma_{\alpha,\beta}.$

By hypothesis, there exists $N\in\mathbb{N}$ be such that $\varphi_{\nu_{n}}$
is defined on $\Gamma_{\alpha,\beta}$ for every $n\geq N$. The sequence of
restrictions $\\{\varphi_{\nu_{n}}|_{\Gamma_{\alpha_{0},\beta_{0}}}\\}_{n\geq
N}$ converges to $\varphi_{\nu}$ uniformly in some closed disk contained in
$\Gamma_{\alpha_{0},\beta_{0}}$, and since these functions take values in
$\mathbb{-H}$, the Vitali-Montel theorem implies that $\varphi_{\nu_{n}}$
actually converges locally uniformly to $\varphi_{\nu}$ on
$\Gamma_{\alpha,\beta}$, and thus $H_{\nu_{n}}$ also converges locally
uniformly to $H_{\nu}$ on $\Gamma_{\alpha,\beta}$. In particular,
$H_{\nu_{n}}$ converges uniformly to $H_{\nu}$ on a neighborhood of $K$ so,
after replacing $N$ by a larger value, we may assume that

(2.5) $\frac{\partial}{\partial t}\Im H_{\nu_{n}}(s+it)>0,\quad s+it\in
Q,\,n\geq N.$

Note also that the set

$K_{+}=\\{s+it\in K:t\geq f_{\nu}(s_{0})+\varepsilon\\}$

is contained in $\Omega_{\nu}$, while

$K_{-}=\\{s+it\in K:t\leq f_{\nu}(s_{0})-\varepsilon\\}$

is disjoint from $\Omega_{\nu}$. Thus, possibly making $N$ even larger, we may
assume that

$\Im H_{\nu_{n}}(z)>0\text{ for }z\in K_{+}\text{ and }\Im
H_{\nu_{n}}(z)<0\text{ for }z\in K_{-},$

provided that $n\geq N$. Combining this with (2.5), we see that for every
$n\geq N$ and for every $s\in[A,B]$ there exists a unique $f_{n}(s)>0$ such
that

$\Im H_{\nu_{n}}(s+it)>0\text{ for }t>f_{n}(s)\text{ and }\Im
H_{\nu_{n}}(s+it)<0\text{ for }\delta<t<f_{n}(s).$

Of course, we have $|f_{n}(s)-f_{\nu}(s)|<\varepsilon$. Since $\varepsilon$
can be made arbitrarily small, we conclude that $f_{n}$ converges to $f_{\nu}$
uniformly on $[A,B]$.

We observe next that $F_{\nu_{n}}(H_{\nu_{n}}(z))=z$ for
$z\in\Gamma_{\alpha_{0},\beta_{0}},$ and our choice of $K$, along with
analytic continuation, show that

$F_{\nu_{n}}(H_{\nu_{n}}(s+it))=s+it,\quad s+it\in Q,\,t>f_{n}(s),\,n\geq N.$

Now, the set $Q$ is convex, and thus (2.5) implies that $H_{\nu_{n}}$ is
injective with an analytic inverse on a neighborhood of $Q$, and that
$F_{\nu_{n}}$ coincides with the inverse of $H_{\nu_{n}}$ on the set
$\\{s+it\in Q:t>f_{n}(s)\\}$. It follows that $F_{\nu_{n}}$ has an analytic
continuation to a neighborhood of $H_{\nu_{n}}(Q).$ In particular,
$F_{\nu_{n}}$ has an analytic continuation across the segment

$\\{H_{\nu_{n}}(s+if_{n}(s)):s\in[A,B]\\},$

and this segment tends to

$\\{H_{\nu}(s+if_{\nu}(s)):s\in[A,B]\\}$

which is a neighborhood of $x_{0}$. This shows that $\nu_{n}$ is absolutely
continuous with a real-analytic density $d\nu_{n}/dx$ in a neighborhood of
$x_{0}$, and this density satisfies the formula

$\frac{d\nu_{n}}{dx}\left(H_{\nu_{n}}(s+if_{n}(s)\right)=\frac{1}{\pi}\frac{f_{n}(s)}{s^{2}+f_{n}(s)^{2}},\quad
s\in[A,B].$

Finally, to show that the densities $d\nu_{n}/dx$ converge uniformly to
$d\nu/dx$ in a neighborhood of $x_{0}$, it suffices to show that $F_{\nu_{n}}$
converges uniformly to $F_{\nu}$ in a neighborhood of $x_{0}$. This, as well
as the convergence of the derivatives, follows from the formula

$F_{\nu_{n}}(z)=H_{\nu_{n}}^{-1}(z)=\frac{1}{2\pi i}\int_{\partial
Q}\frac{\zeta H_{\nu_{n}}(\zeta)}{H_{\nu_{n}}(\zeta)-z}\,d\zeta$

that holds for $z\in\mathbb{H}$ close to $x_{0}$ and for sufficiently large
$n$. ∎

The following corollary formalizes the discussion preceding Theorem 2.3.

######  Corollary 2.4.

Suppose that $k_{1},k_{2},\dots\in\mathbb{N}$, and that
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an infinitesimal array of
Borel probability measures on $\mathbb{R}$. Set

$\nu_{n}=\mu_{n,1}\boxplus\cdots\boxplus\nu_{n,k_{n}}.$

Assume that $\nu_{n}$ converges to a $\boxplus$-infinite divisible measure
$\nu$. Then for any compact interval $J\subset\mathbb{R}$ such that $\nu$ is
absolutely continuous and $d\nu/dx>0$ in a neighborhood of $J$, $\nu_{n}$ is
absolutely continuous on a neighborhood of $J$ with a real-analytic density
for sufficiently large $n$, and the densities $d\nu_{n}/dx$ converge uniformly
on $J$ to $d\nu/dx$ as $n\to\infty$.

##  3\. Multiplicative convolution on $\mathbb{R}_{+}$

Suppose that $\mu$ is a Borel probability measure on
$\mathbb{R}_{+}=[0,+\infty)$, other than the point mass $\delta_{0}$ at the
origin. The analytic functions

$\psi_{\mu}(z)=\int_{\mathbb{R}_{+}}\frac{zt}{1-zt}\,d\mu(t),\quad\eta_{\mu}(z)=\frac{\psi_{\mu}(z)}{1+\psi_{\mu}(z)},\quad
z\in\mathbb{C}\backslash\mathbb{R}_{+},$

are real-valued on $(-\infty,0)$ and the measure $\mu$ is uniquely determined
by either $\psi_{\mu}$ or $\eta_{\mu}$. Indeed, we have

(3.1) $zG_{\mu}(z)=\frac{1}{1-\eta_{\mu}\left(\frac{1}{z}\right)},\quad
z\in\mathbb{C}\setminus\mathbb{R}_{+},$

to which the Stieltjes inversion can be applied to recover $\mu$. Moreover,
$\psi_{\mu}$ and $\eta_{\mu}$ map $\mathbb{H}$ to itself and, in addition,

$\arg(\eta_{\mu}(z))\geq\arg z,\quad z\in\mathbb{H},$

where ‘$\arg$’ stands for the principal value of the argument, so $\arg
z\in(0,\pi)$ for $z\in\mathbb{H}$, see [2]. It was shown in [5] that there
exist an open set $V$, containing some interval of the form $(-a,0)$, and an
analytic function $\Sigma_{\mu}$ defined on $V$ such that

$\eta_{\mu}(z\Sigma_{\mu}(z))=z,\quad z\in V.$

The related equation $\eta_{\mu}(z)\Sigma_{\mu}(\eta_{\mu}(z))=z$ holds for
$z$ in every connected open set $U\subset\mathbb{C}\setminus\mathbb{R}_{+}$
that intersects ${\mathbb{R}}_{-}$ and such that $\eta_{\mu}(z)$ belongs to
the domain of $\Sigma_{\mu}$. The function $\Sigma_{\mu}$ serves an analogous
role relative to multiplicative free convolution to that of $\varphi_{\mu}$
relative to additive free convolution, namely

$\Sigma_{\mu\boxtimes\nu}(z)=\Sigma_{\mu}(z)\Sigma_{\nu}(z)$

for $z$ in every domain on which the three functions are defined. The measure
$\mu$ is $\boxtimes$-infinitely divisible precisely when the function
$\Sigma_{\mu}$ continues analytically to the entire domain
$\mathbb{C}\backslash\mathbb{R}_{+}$ and this analytic continuation can be
written as

$\Sigma_{\mu}(z)=\gamma\exp\left[\int_{[0,+\infty]}\frac{1+tz}{z-t}\,d\sigma(t)\right],\quad
z\in\mathbb{C}\backslash\mathbb{R}_{+},$

where $\gamma>0$ and $\sigma$ is a finite, positive Borel measure on the one-
point compatification $[0,+\infty]$ of ${\mathbb{R}}_{+}$ [5].

Suppose now that $\mu$ is $\boxtimes$-infinitely divisible. The equation

$\eta_{\mu}(z)\Sigma_{\mu}(\eta_{\mu}(z))=z$

extends by analytic continuation to the entire slit plane
$\mathbb{C}\backslash\mathbb{R}_{+}.$ In particular, it shows that
$\eta_{\mu}$ maps $\mathbb{C}\backslash\mathbb{R}_{+}$ conformally onto a
domain $\Omega_{\mu}$ that is symmetric relative to the real line. In addition
to the interval $(-\infty,0]$, the boundary of $\Omega_{\mu}\cap\mathbb{H}$
consists of a curve of the form

$C=\\{re^{ih_{\mu}(r)}:r\in(0,+\infty)\\},$

where $h_{\mu}:(0,+\infty)\to[0,\pi)$ is a continuous function, real analytic
wherever it is strictly positive. In other words,

$\Omega_{\mu}\cap\mathbb{H}=\\{re^{i\theta}:r>0,h_{\mu}(r)<\theta<\pi\\}.$

Moreover, the function $\eta_{\mu}$ extends continuously and injectively to
$\mathbb{H}\cup(0,+\infty)$ such that the range
$\eta_{\mu}\left((0,+\infty)\right)$ is exactly the curve $C$ and that the
inversion relationships between $\eta_{\mu}$ and the map
$\Phi(z)=z\Sigma_{\mu}(z)$ now extend to the boundary of the relevant domains.
(See [8, Section2] for a review of these results concerning boundary behavior,
and the references therein for their origin.) Thus, there is at most one value
$t_{0}>0$ such that $\eta_{\mu}(t_{0})=1$. Such a point exists precisely when
$h_{\mu}(1)=0$. Outside possibly the point $1/t_{0}$, the measure $\mu$ is
absolutely continuous and its density is locally analytic wherever it is
strictly positive. We give a short proof of this analyticity for future
reference. Suppose that $x_{0}>0$ is a point where the density of $\mu$ is
positive. We write
$1/x_{0}=r_{0}e^{ih_{\mu}(r_{0})}\Sigma_{\mu}(re^{ih_{\mu}(r_{0})})$ for some
$r_{0}>0$ such that $h_{\mu}(r_{0})>0$. The continuity of $h_{\mu}$ yields
$h_{\mu}(r)>0$ for $r$ near $r_{0}$. For such $r$, the function
$\Phi(z)=z\Sigma_{\mu}(z)$ has a non-zero complex derivative at
$z=re^{ih_{\mu}(r)}$. This is because $\Phi^{\prime}(re^{ih_{\mu}(r)})=0$
would imply that the positive number $\Phi(re^{ih_{\mu}(r)})$ has multiple
preimages located on the curve $C$, contradicting the fact that $\Phi$ is
injective on $C$. Since the inversion equation

$\Phi(\eta_{\mu}(z))=z$

holds for $z\in\mathbb{H}$ close to $1/x_{0}$, we conclude that $\eta_{\mu}$
can be analytically continued to a neighborhood of $1/x_{0}$ as the inverse of
the function $\Phi$. The desired analyticity now follows from (3.1) and the
Stieltjes inversion formula. (One can also write an implicit formula for the
density, just as in the additive case, to see its analyticity directly; see
for instance [8, (3.1)].)

Suppose now that $k_{1}<k_{2}<\cdots$ is a sequence of natural numbers and
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an infinitesimal array of
probability measures on $\mathbb{R}_{+}$; that is,

$\lim_{n\to\infty}\min_{1\leq i\leq
k_{n}}\mu_{n,i}((1-\varepsilon,1+\varepsilon))=1$

for every $\varepsilon\in(0,1)$. The analog of the domain
$\Gamma_{\alpha,\beta}$ in the context of multiplicative free convolution is
the domain

$\Omega_{\rho,\theta}=\left\\{re^{it}:\rho<r<\frac{1}{\rho},t\in(\theta,2\pi-\theta)\right\\},$

where $\rho>0$ and $\theta\in(0,\pi)$. By the results in [3], given
arbitrarily small $\rho>0$ and $\theta\in(0,\pi)$, all the functions
$\Sigma_{\mu_{n,i}}$ are defined in $\Omega_{\rho,\theta}$ if $n$ is
sufficiently large. As shown in [5], the weak convergence of probability
measures on $\mathbb{R}_{+}$ can be translated into local uniform convergence
of the corresponding $\eta$-functions, or into local uniform convergence of
the corresponding $\Sigma$-functions on some $\Omega_{\rho,\theta}$. The
following result is analogous to Theorem 2.3 and therefore it provides a
superconvergence result for infinitesimal arrays.

######  Theorem 3.1.

Let $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures on $\mathbb{R_{+}}$ that converges weakly to an
$\boxtimes$-infinitely divisible measure $\nu.$ Suppose that for every
$\rho>0$ and $\theta\in(0,\pi)$ there exists $N\in\mathbb{N}$ such that
$\Sigma_{\nu_{n}}$ is defined in $\Omega_{\rho,\theta}$ for every $n\geq N$.
Let $J\subset(0,+\infty)$ be a compact interval such that $\nu$ is absolutely
continuous and $d\nu/dx>0$ in a neighborhood of $J$. Then $\nu_{n}$ is
absolutely continuous with an analytic density on a neighborhood of $J$ for
$n$ sufficiently large, and the densities $d\nu_{n}/dx$ converge uniformly on
$J$, along with all their derivatives, to $d\nu/dx$ as $n\to\infty$.

###### Sketch of proof.

To simplify notation, we set

$\Phi(z)=z\Sigma_{\nu}(z)\text{ and }\Phi_{n}(z)=z\Sigma_{\nu_{n}}(z),\quad
n\in\mathbb{N}.$

Suppose that $x_{0}>0$ is a point where the density of $\nu$ is positive. As
seen above, we have

$\frac{1}{x_{0}}=\Phi(r_{0}e^{ih_{\nu}(r_{0})})$

for some $r_{0}>0$ such that $h_{\nu}(r_{0})>0$. The point
$r_{0}e^{ih_{\nu}(r_{0})}$ belongs to the domain $\Omega_{\rho,\theta}$
provided that $\rho$ and $\theta$ are sufficiently small. Fix such values
$\rho$ and $\theta$, and note that $\Phi_{n}$ is defined on
$\Omega_{\rho,\theta}$ provided that $n$ is sufficiently large. As seen in the
earlier discussions, the essential point is to verify that, for
$z\in\mathbb{H}$ sufficiently close to $1/x_{0}$, we have
$\eta_{\nu_{n}}(z)\in\Omega_{\rho,\theta}$ and the identity

(3.2) $\Phi_{n}(\eta_{\nu_{n}}(z))=z$

holds for $n$ sufficiently large. The absolute continuity of $\nu_{n}$ near
$x_{0}$ would follow as a byproduct. Note that the identity (3.2) holds for
$z\in(-T,-1/T)$ if $T$ is large enough, and it extends by analytic
continuation to the largest interval of this form with the property that
$\eta_{\nu_{n}}((-T,-1/T))\subset\Omega_{\rho,\theta}$.

We observe now that, by using the polar coordinates, the argument of Remark
2.2 (or alternatively, more explicit calculations from [12, Lemma 4.2]) yields

$\frac{\partial}{\partial\theta}\arg\left(\Phi(re^{i\theta})\right)>0\text{
for }r=r_{0}\text{ and }\theta=h_{\nu}(r_{0}).$

So we may choose a neighborhood $W$ of $r_{0}e^{ih_{\nu}(r_{0})}$ and an
integer $N$ such that $\Phi_{n}$ is defined on $\Omega_{\rho,\theta}$ for
$n\geq N$, $W\subset\Omega_{\rho,\theta}\cap\mathbb{H}$, and

$\frac{\partial}{\partial\theta}\arg\left(\Phi_{n}(re^{i\theta})\right)>0\text{
for $re^{i\theta}\in W\text{ and }n\geq N.$}$

In particular, the complex derivative $\Phi_{n}^{\prime}$ does not vanish on
$W$, so that $\Phi_{n}|_{W}$ has an analytic inverse. Then we choose
$\varepsilon>0$ so small that

  1. (1)

$\left|h_{\nu}(r)-h_{\nu}(r_{0})\right|<\varepsilon$ for
$|r-r_{0}|\leq\varepsilon$,

  2. (2)

the compact set

$K=\\{re^{i\theta}:|r-r_{0}|\leq\varepsilon,|\theta-
h_{\nu}(r_{0})|\leq\varepsilon\\}$

is contained in $W$,

  3. (3)

$\Phi(re^{ih_{\nu}(r_{0})+i\varepsilon})\in\mathbb{H}$ for
$|r-r_{0}|\leq\varepsilon$, and

  4. (4)

$\Phi(re^{ih_{\nu}(r_{0})-i\varepsilon})\in-\mathbb{H}$ for
$|r-r_{0}|\leq\varepsilon.$

Since $\Phi_{n}$ converges to $\Phi$ uniformly on $K$, we can assume that
properties (2) and (3) also hold for $\Phi_{n}$ after making $N$ bigger. It
follows that there exists a unique function
$h_{n}:[r_{0}-\varepsilon,r_{0}+\varepsilon]\to(h_{\nu}(r_{0})-\varepsilon,h_{\nu}(r_{0})+\varepsilon)$
such that

$\Phi_{n}(re^{ih_{n}(r)})\in(0,+\infty),\quad|r-r_{0}|\leq\varepsilon.$

The function $\Phi_{n}$ is one-to-one on $K$, and the interval

$\\{\Phi_{n}(re^{ih_{n}(r)}):|r-r_{0}|\leq\varepsilon\\}$

is a neighborhood of $1/x_{0}$ for sufficiently large $n$. We need to show
that (3.2) holds in the set

$\\{\Phi_{n}(re^{i\theta}):|r-r_{0}|\leq\varepsilon,h_{n}(r)<\theta\leq
h_{\nu}(r)+\varepsilon\\},$

provided that $n$ is sufficiently large. This will follow by analytic
continuation once we show that $\Phi_{n}(K^{\prime})\subset\mathbb{H}$, where

$K^{\prime}=\\{re^{i\theta}:|r-r_{0}|\leq\varepsilon,\theta\in[h_{\nu}(r)+\varepsilon,\pi]\\}.$

This last fact follows immediately because $\Phi(K^{\prime})\subset\mathbb{H}$
and $\Phi_{n}$ converges uniformly on $K^{\prime}$ to $\Phi$. ∎

The following corollary formalizes the discussion preceding Theorem 3.1.

######  Corollary 3.2.

Suppose that $k_{1},k_{2},\dots\in\mathbb{N}$, and that
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an infinitesimal array of
Borel probability measures on $\mathbb{R}_{+}$. Set

$\nu_{n}=\mu_{n,1}\boxtimes\cdots\boxtimes\nu_{n,k_{n}}.$

Assume that $\nu_{n}$ converges to a $\boxtimes$-infinite divisible measure
$\nu$. Then for any compact interval $J\subset(0,\infty)$ such that $\nu$ is
absolutely continuous and $d\nu/dx>0$ in a neighborhood of $J$, $\nu_{n}$ is
absolutely continuous on a neighborhood of $J$ with a real-analytic density
for sufficiently large $n$, and the densities $d\nu_{n}/dx$ converge uniformly
on $J$ to $d\nu/dx$ as $n\to\infty$.

##  4\. Multiplicative convolution on $\mathbb{T}$

Finally, we consider the superconvergence phenomenon on the unit circle
$\mathbb{T}=\\{e^{i\theta}:\theta\in[0,2\pi)\\}$. If $\mu$ is a Borel
probability measure on $\mathbb{T}$, one sets again

$\psi_{\mu}(z)=\int_{\mathbb{T}}\frac{z\zeta}{1-z\zeta}\,d\mu(\zeta),\quad\eta_{\mu}(z)=\frac{\psi_{\mu}(z)}{1+\psi_{\mu}(z)},$

but these functions are now defined on the unit disk
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$. The density of $\mu$ relative to the
normalized arclength measure $m=d\theta/2\pi$ is given almost everywhere by

$\frac{d\mu}{dm}(\xi)=\lim_{r\uparrow
1}\Re\frac{1+\eta_{\mu}(r\,\overline{\xi})}{1-\eta_{\mu}(r\,\overline{\xi})},\quad\xi\in\mathbb{T}.$

We restrict our considerations to the case in which
$\int_{\mathbb{T}}\zeta\,d\mu(\zeta)\neq 0$ and we observe that this condition
is satisfied for (all but finitely many) measures in an infinitesimal array.
In terms of $\psi$ and $\eta$, this condition amounts to the requirement that
$\psi_{\mu}^{\prime}(0)=\eta_{\mu}^{\prime}(0)\neq 0$. This implies the
existence of an analytic function $\Sigma_{\mu}$, defined in a disk
$\rho\mathbb{D}=\left\\{z\in\mathbb{C}:\left|z\right|<\rho\right\\}$ with
$\rho\in(0,1)$ and with values in $\mathbb{D},$ such that

$\eta_{\mu}(z\Sigma_{\mu}(z))=z,\quad z\in\rho\mathbb{D}.$

We denote by $\rho_{\mu}$ the radius of convergence of the Taylor series of
$\Sigma_{\mu}$, so $\Sigma_{\mu}$ is defined in $\rho_{\mu}\mathbb{D}$. The
free multiplicative convolution of measures on $\mathbb{T}$ satisfies the
identity

$\Sigma_{\mu\boxtimes\nu}(z)=\Sigma_{\mu}(z)\Sigma_{\nu}(z),\quad|z|<\min\\{\rho_{\mu},\rho_{\nu}\\}.$

Among the $\boxtimes$-infinitely divisible measures on $\mathbb{T}$, the
normalized arclength measure $m$ is the only one with zero first moment. The
other $\boxtimes$-infinitely divisible measures $\mu$ on $\mathbb{T}$ are
characterized by the fact that $\rho_{\mu}\geq 1$ and
$\Phi_{\mu}(\mathbb{D})\subset\mathbb{D},$ where
$\Phi_{\mu}(z)=z\Sigma_{\mu}(z)$. Clearly, if $\mu$ is $\boxtimes$-infinitely
divisible, the equation

(4.1) $\Phi_{\mu}(\eta_{\mu}(z))=z$

extends by analytic continuation to the entire disk $\mathbb{D}$, showing that
$\eta_{\mu}$ maps $\mathbb{D}$ conformally onto a domain
$\Omega_{\mu}\subset\mathbb{D}.$ In fact, by the results in [12] (see also [8,
Section 5] for a review), $\eta_{\mu}$ extends to a homeomorphism of
$\overline{\mathbb{D}}$ onto $\overline{\Omega_{\mu}}$. In particular,
$\eta_{\mu}$ maps $\mathbb{T}$ homeomorphically onto $\partial\Omega_{\mu}$.
The domain $\Omega_{\mu}$ is starlike, in particular its boundary is a closed
curve of the form

$\\{R_{\mu}(\zeta)\zeta:\zeta\in\mathbb{T}\\},$

where $R_{\mu}:\mathbb{T}\to(0,1]$ is a continuous function that is real-
analytic at all points $\zeta$ satisfying $R_{\mu}(\zeta)<1$. Thus, one has

$\Omega_{\mu}=\\{r\zeta:\zeta\in\mathbb{T},0\leq r<R_{\mu}(\zeta)\\}.$

For a $\boxtimes$-infinitely divisible measure $\mu$, there is at most one
point $\zeta_{0}$ such that $\eta_{\mu}(\zeta_{0})=1$, and this happens
precisely when $R_{\mu}(1)=1$. Outside the point $\overline{\zeta_{0}}$, the
measure $\mu$ is absolutely continuous with a locally analytic density
wherever this density is positive. This is a consequence of the identity (4.1)
that holds for $z\in\mathbb{D}$ close to a point $\xi_{0}\in\mathbb{T}$ such
that $|\eta_{\mu}(\xi_{0})|<1$ because (as we also discuss in the proof below)
$\Phi^{\prime}(\eta_{\mu}(\xi_{0}))\neq 0$.

Suppose that $k_{1}<k_{2}<\cdots$ is a sequence of natural numbers and
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an infinitesimal array of
Borel probability measures on $\mathbb{T}$, that is,

$\lim_{n\to\infty}\min_{1\leq i\leq
k_{n}}\mu_{n,i}(\\{\zeta\in\mathbb{T}:|\zeta-1|<\varepsilon\\})=1$

for every $\varepsilon>0$. According to [3], given an arbitrary
$\rho\in(0,1)$, we have $\rho_{\mu_{n,i}}>\rho$ for all $1\leq i\leq k_{n}$
provided that $n$ is sufficiently large.

The weak convergence of probability measures on $\mathbb{T}$ can be translated
into local uniform convergence of the $\eta$-functions. If all the measures
under consideration satisfy $\int_{\mathbb{T}}\zeta\,d\mu(\zeta)\neq 0$ it
also translates into uniform convergence of $\Phi$-functions on
$\rho\mathbb{D}$ for some $\rho>0$.

We are now ready to state the circle version of the superconvergence result.

######  Theorem 4.1.

Let $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures on $\mathbb{T}$ that converges weakly to an $\boxtimes$-infinitely
divisible measure $\nu$, where $\int_{\mathbb{T}}\zeta\,d\nu(\zeta)\neq 0$.
Suppose that $\lim_{n\to\infty}\rho_{\nu_{n}}=1$. Let $J\subset\mathbb{T}$ be
a compact arc such that $\nu$ is absolutely continuous and its density is
positive in a neighborhood of $J$. Then $\nu_{n}$ is absolutely continuous
with a real-analytic density on a neighborhood of $J$ for $n$ sufficiently
large, and the densities of $\nu_{n}$ converge uniformly on $J$, along with
all their derivatives, to that of $\nu$ as $n\to\infty$.

###### Sketch of proof.

We use the notation $\Phi=\Phi_{\nu}$ and $\Phi_{n}=\Phi_{\nu_{n}}$,
$n\in\mathbb{N}.$ Suppose that $\xi_{0}\in\mathbb{T}$ is such that
$\eta_{\nu}(\xi_{0})\in\mathbb{D}$. As in the proof of Theorem 3.1, we need to
show that, for sufficiently large $n$, we have
$\eta_{\nu_{n}}(z)\in\rho_{\nu_{n}}\mathbb{D}$ for $z\in\mathbb{D}$ close to
$\xi_{0}$, and that $\Phi_{n}(\eta_{\nu_{n}}(z))=z$ for such $z$. By
hypothesis, we may assume that $\rho_{\nu_{n}}>\rho>|\eta_{\nu}(\xi_{0})|$ for
some $\rho\in(0,1),$ and that $|\Phi_{n}(z)|<1$ for $|z|<\rho$. It is easily
seen, by using logrithmic polar coordinate, from Remark 2.2 (see also [12,
Section 3] for a proof by explcit formulas) that

$\frac{\partial}{\partial r}\log\left|\Phi(re^{i\theta})\right|>0$

at the point $r_{0}e^{i\theta_{0}}=\eta_{\nu}(\xi_{0})$. Thus, we may also
assume that

$\frac{\partial}{\partial r}\log\left|\Phi_{n}(re^{i\theta})\right|>0$

for $re^{i\theta}$ in a fixed neighborhood of $\eta_{\nu}(\xi_{0})$ and for
large $n$. It follows that there is a sequence $\\{R_{n}\\}_{n\in\mathbb{N}}$
of functions defined in a neighborhood of $e^{i\theta_{0}}$, with values in
$(0,1)$, such that

$|\Phi_{n}(r\zeta)|\begin{cases}<1,&r<R_{n}(\zeta),\\\ =1,&r=R_{n}(\zeta),\\\
>1,&r>R_{n}(\zeta).\end{cases}$

Moreover, the functions $R_{n}$ converge uniformly to $R_{\nu}$ in a
neighborhood of $e^{i\theta_{0}}$. Then, we need to verify that

$\eta_{\nu_{n}}(\Phi_{n}(z))=z$

for $z=r\zeta$ provided that $r<R_{n}(\zeta)$ and $\zeta$ is close to
$e^{i\theta_{0}}$, say $|\zeta-e^{i\theta_{0}}|\leq\varepsilon$. This
statement follows by analytic continuation because $\Phi_{n}$ converges
uniformly to $\Phi$ in a compact set of the form

$\\{r\zeta:|\zeta-e^{i\theta_{0}}|\leq\varepsilon,r\leq
R_{\nu}(e^{i\theta_{0}})-\delta\\},$

where $\varepsilon,\delta>0$. Absolute continuity of $\nu_{n}$ near
$1/\xi_{0}$, and the uniform convergence of the densities, follow as in the
free additive case. ∎

As in the additive case and the multiplicative case on $\mathbb{R}_{+}$, we
have the following corollary that formalizes the discussion preceding Theorem
4.1.

######  Corollary 4.2.

Suppose that $k_{1},k_{2},\dots\in\mathbb{N}$, and that
$\\{\mu_{n,i}:n,i\in\mathbb{N},i\leq k_{n}\\}$ is an infinitesimal array of
Borel probability measures on $\mathbb{T}$. Set

$\nu_{n}=\mu_{n,1}\boxtimes\cdots\boxtimes\nu_{n,k_{n}}.$

Assume that $\nu_{n}$ converges to a $\boxtimes$-infinite divisible measure
$\nu$ such that $\nu\neq m$. Then for any compact arc $J\subset\mathbb{T}$
such that $\nu$ is absolutely continuous and $d\nu/dx>0$ in a neighborhood of
$J$, $\nu_{n}$ is absolutely continuous on a neighborhood of $J$ with a real-
analytic density for sufficiently large $n$, and the densities $d\nu_{n}/dx$
converge uniformly on $J$ to $d\nu/dx$ as $n\to\infty$.

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