Polar Free Infinite Divisibility

• Polar free infinite divisibility is a framework in free probability that organizes infinitely divisible laws into radial (positive) and angular (symmetric or R-diagonal) components using Möbius-conjugated transforms.
• It connects symmetric free infinite divisibility with positive regular measures, utilizing fixed angular laws like the Wigner and arcsine distributions to characterize free multiplicative mixtures.
• The approach extends to non-selfadjoint cases by incorporating R-diagonality, where Haar unitaries and free Poisson laws illustrate the stability of the system under free multiplicative convolution.

Searching arXiv for the cited papers and closely related work on polar formulations of free infinite divisibility. Polar free infinite divisibility is a family of concepts in free probability that organize freely infinitely divisible laws through radial–angular structure. In one explicit formulation, a measure on $\hat{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ is $a$-freely infinitely divisible if it admits fractional roots under Möbius-conjugated polar free powers $F_a^t$ attached to a point $a\in\hat{\mathbb{R}}$ (Perales et al., 26 Aug 2025). In an earlier structural formulation, symmetric $\boxplus$-infinitely divisible laws are encoded by positive regular free infinitely divisible laws through the identity $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$, and multiplicative mixtures with the Wigner or arcsine law provide fixed “angular” components for distinguished polar subclasses (Perez-Abreu et al., 2009). In the non-selfadjoint setting, the analogous polar symmetry is carried by $R$-diagonal distributions, whose $\boxplus$-infinitely divisible subclass is parametrized by positive measures describing the left and right moduli (Bercovici et al., 2016).

1. Analytic framework and scope

Free infinite divisibility is defined with respect to free additive convolution $\boxplus$: a probability measure $\mu$ is freely infinitely divisible if for every $a$0 there exists $a$1 such that

$a$2

Analytically, the basic transforms are the Cauchy transform

$a$3

its reciprocal $a$4, and the free cumulant transform

$a$5

defined on an appropriate cone via the Bercovici–Voiculescu inverse $a$6 (Perez-Abreu et al., 2009). An equivalent formulation uses the Voiculescu transform $a$7, for which free infinite divisibility is characterized by a free Lévy–Khintchine representation (Arizmendi, 2013).

For symmetric $a$8, the free Lévy–Khintchine formula simplifies to

$a$9

where $F_a^t$0 and $F_a^t$1 is a symmetric Lévy measure (Perez-Abreu et al., 2009). On the positive half-line, the relevant subclass is the class $F_a^t$2 of free regular distributions, characterized by a Lévy–Khintchine representation supported on $F_a^t$3 (Perez-Abreu et al., 2009).

Within this analytic setting, “polar” has two distinct but related meanings. The first is structural: symmetric laws are decomposed into a positive radial object and a fixed symmetric angular law such as the Wigner or arcsine distribution (Perez-Abreu et al., 2009). The second is explicit: Möbius transforms convert ordinary fractional free convolution powers into polar powers $F_a^t$4 on $F_a^t$5, leading to the notion of $F_a^t$6-free infinite divisibility (Perales et al., 26 Aug 2025).

2. Symmetric free infinite divisibility as a radial correspondence

A central structural theorem identifies symmetric free infinite divisibility with positive regular free infinite divisibility. If $F_a^t$7 denotes symmetric freely infinitely divisible measures and $F_a^t$8 denotes free regular positive freely infinitely divisible measures, then

$F_a^t$9

The associated Lévy measures satisfy

$a\in\hat{\mathbb{R}}$0

where the superscripts denote the push-forwards under $a\in\hat{\mathbb{R}}$1 and $a\in\hat{\mathbb{R}}$2 (Perez-Abreu et al., 2009).

This correspondence is the basic radial map in the selfadjoint theory. The positive law $a\in\hat{\mathbb{R}}$3 is the regular “radius,” while $a\in\hat{\mathbb{R}}$4 is obtained from it by a symmetric square-root symmetrization. The 2009 paper does not use the word “polar,” but it explicitly describes this mapping as a canonical passage between symmetric and positive regular free infinitely divisible laws, and the construction separates sign from modulus exactly in the way expected from a polar viewpoint (Perez-Abreu et al., 2009).

The theorem is not merely formal. It translates additive free infinite divisibility of a symmetric law into a positivity problem on $a\in\hat{\mathbb{R}}$5, which is then compatible with multiplicative free convolution. This compatibility is what makes fixed angular laws, such as the standard Wigner law or the arcsine law, natural objects in subsequent classifications.

3. Fixed angular laws: type $a\in\hat{\mathbb{R}}$6 and type $a\in\hat{\mathbb{R}}$7

For $a\in\hat{\mathbb{R}}$8, the symmetric measure

$a\in\hat{\mathbb{R}}$9

is called a free multiplicative mixture of the Wigner law $\boxplus$0. If $\boxplus$1, it is called a free type $\boxplus$2 distribution (Perez-Abreu et al., 2009). The key characterization is expressed through multiplicative square roots on the radial side. Writing $\boxplus$3, one has

$\boxplus$4

and in this case

$\boxplus$5

Thus type $\boxplus$6 laws are precisely those symmetric freely infinitely divisible laws whose associated positive regular law is $\boxplus$7-$\boxplus$8 divisible (Perez-Abreu et al., 2009).

A further structural identity is

$\boxplus$9

where $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$0 is the free Poisson law with parameter $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$1. Hence the squared radial law of every multiplicative Wigner mixture is a free compound Poisson distribution in $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$2 (Perez-Abreu et al., 2009).

The arcsine law yields an analogous angular class. A distribution $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$3 is of type $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$4 when it is symmetric and freely infinitely divisible. The characterization is: $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$5 The inclusion relations are strict: $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$6 The paper also shows that type $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$7 is strictly larger than free type $$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$8 by exhibiting the symmetric Beta$$\mathcal{C}_\mu^{\boxplus}(z)=\mathcal{C}_\sigma^{\boxplus}(z^2)$$9 law $R$0 as both type $R$1 and type $R$2, while not being free type $R$3 (Perez-Abreu et al., 2009).

These fixed-angle constructions explain a common misconception. Symmetric free infinite divisibility does not imply realizability as a multiplicative Wigner mixture, and realizability as a Wigner mixture does not exhaust the arcsine-based class. The obstruction on the Wigner side is exactly the failure of $R$4-$R$5 divisibility of the associated positive regular law (Perez-Abreu et al., 2009).

4. Polar powers on $R$6 and $R$7-free infinite divisibility

An explicit notion of polar free infinite divisibility was introduced through repeated polar differentiation of real-rooted polynomials and its asymptotic effect on root measures (Perales et al., 26 Aug 2025). For a polynomial $R$8 of degree $R$9, the affine polar derivative at $\boxplus$0 is

$\boxplus$1

and $\boxplus$2. Polar derivatives commute: $\boxplus$3 At the measure level, ordinary repeated differentiation corresponds to

$\boxplus$4

while the polar version is obtained by Möbius conjugation. If $\boxplus$5 sends $\boxplus$6 to $\boxplus$7, then

$\boxplus$8

The paper defines $\boxplus$9 to be $\boxplus$0-freely infinitely divisible if for every $\boxplus$1 there exists $\boxplus$2 such that

$\boxplus$3

This class is denoted $\boxplus$4 (Perales et al., 26 Aug 2025).

The basic reduction theorem says that if $\boxplus$5, then

$\boxplus$6

and therefore

$\boxplus$7

Except for the trivial case $\boxplus$8, the corresponding roots $\boxplus$9 are unique (Perales et al., 26 Aug 2025).

The same paper derives a nontrivial commutation relation for polar powers. If $\mu$0, $\mu$1, and $\mu$2 satisfy

$\mu$3

then

$\mu$4

This identity is the measure-theoretic shadow of the commutativity of polar derivatives and is the basis for Belinschi–Nica type semigroups

$\mu$5

which form a semigroup on $\mu$6 (Perales et al., 26 Aug 2025).

5. Non-selfadjoint polar symmetry: $\mu$7-diagonal infinite divisibility

In the non-selfadjoint setting, the relevant polar symmetry is $\mu$8-diagonality. An element $\mu$9 is $a$00-diagonal precisely when its polar decomposition

$a$01

has $a$02 a Haar unitary, $a$03, and $a$04 free from $a$05 (Bercovici et al., 2016). This is the noncommutative analogue of rotational invariance, with $a$06 as angular part and $a$07 as radial part.

The Boolean counterpart is the class of eta-diagonal distributions, characterized by an $a$08-series supported only on alternating words in $a$09 and $a$10. The two classes are linked by the Boolean-to-free Bercovici–Pata bijection: $a$11 Restricting this bijection to eta-diagonal distributions gives a bijection between eta-diagonal laws and $a$12-infinitely divisible $a$13-diagonal laws (Bercovici et al., 2016).

A decisive structural theorem parametrizes every compactly supported $a$14-infinitely divisible $a$15-diagonal distribution by a pair of compactly supported Borel probability measures on $a$16: $a$17 The two measures encode the laws of $a$18 and $a$19 in the eta-diagonal model. In the tracial case they coincide, so the parametrization collapses to a single radial measure (Bercovici et al., 2016).

This non-selfadjoint theory extends the polar vocabulary from the real line to rotationally symmetric free laws. It also has a strong stability property: the class $a$20 of $a$21-infinitely divisible $a$22-diagonal distributions is closed under free multiplicative convolution $a$23 (Bercovici et al., 2016).

6. Examples, boundary phenomena, and related viewpoints

Several explicit examples delineate the size of polar free infinitely divisible classes. On the selfadjoint side, the symmetrized free Poisson laws

$a$24

satisfy

$a$25

For $a$26 small, for example $a$27, the candidate multiplicative square root fails to be an $a$28-transform of a positive measure, so $a$29 is symmetric freely infinitely divisible but not type $a$30; the same construction also yields examples not in type $a$31 (Perez-Abreu et al., 2009). By contrast, the symmetric Beta$a$32 law $a$33 satisfies

$a$34

so it lies simultaneously in type $a$35 and type $a$36, while not belonging to free type $a$37 (Perez-Abreu et al., 2009).

In the explicit polar-power theory, Marchenko–Pastur distributions are stable under $a$38-polar powers: $a$39 and the corresponding Belinschi–Nica type semigroup acts by

$a$40

The Cauchy distribution is even more rigid: for every $a$41 and $a$42,

$a$43

so $a$44 is fixed by every polar power and by every $a$45 (Perales et al., 26 Aug 2025).

In the non-selfadjoint theory, the $a$46-circular distribution gives a canonical $a$47-diagonal example. Its determining sequences satisfy $a$48, $a$49, and $a$50 for $a$51, while the radial components $a$52 and $a$53 are free Poisson laws (Bercovici et al., 2016). This parallels the role of Marchenko–Pastur laws on the selfadjoint radial side.

A final terminological point is essential. The 2009 paper on Wigner mixtures and the 2016 paper on $a$54-diagonals do not use the phrase “polar free infinite divisibility”; they supply the radial–angular and rotationally symmetric structures that later literature interprets in polar terms (Perez-Abreu et al., 2009, Bercovici et al., 2016). By contrast, the 2025 work introduces $a$55-free infinite divisibility as an explicit definition on $a$56 (Perales et al., 26 Aug 2025). A related transform-space vocabulary appears in work on Appell polynomials, where the free infinitely divisible law associated with a Laguerre–Pólya function $a$57 has

$a$58

and the finite free $a$59-transforms of the associated Appell polynomials are described as a directional, “polar” approximation scheme for free infinitely divisible distributions (Campbell, 2024). This does not define polar free infinite divisibility, but it places the subject in a broader analytic geometry of free Lévy–Khintchine transforms.