Fractional Free Convolution

• Fractional free convolution is defined as the continuous extension of free convolution powers, enabling interpolation of convolution semigroups in free probability.
• It employs analytic tools such as the R-transform and S-transform, along with subordination and free compression methods, to characterize spectral distributions.
• Recent research links fractional free convolution to repeated differentiation of polynomials and stable limit laws in both real-rooted and rotationally invariant settings.

Searching arXiv for papers on fractional free convolution and related repeated differentiation/free probability connections. Fractional free convolution denotes the extension of free convolution powers from integer iterates to continuous parameters and, in a broader sense, the semigroup structures generated by such powers in free probability. In the additive setting on $\mathbb{R}$, it is the family $\mu^{\boxplus t}$ characterized for $t \ge 1$ by linear scaling of the $R$-transform, $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$, with existence for all $t\ge 0$ precisely in the freely infinitely divisible case (Shlyakhtenko et al., 2020, Campbell, 2024). In the non-normal setting of Brown measures of $R$-diagonal elements, an analogous fractional operation $\mu_a^{\oplus k}$ is defined for $k\ge 1$ through an $S$-transform formula and forms a semigroup interpolating the Brown measures of free sums of $\mu^{\boxplus t}$0-diagonal variables (Campbell et al., 2023). Recent work connects both notions to repeated differentiation of polynomials: on the real line through free additive convolution powers and Appell limits (Campbell, 2024), and in the rotationally invariant complex setting through Brown measures and a quadratic transport map relating zero distributions to $\mu^{\boxplus t}$1-powers (Campbell et al., 2023).

1. Additive fractional powers on the real line

For compactly supported probability measures $\mu^{\boxplus t}$2 on $\mu^{\boxplus t}$3, free additive convolution $\mu^{\boxplus t}$4 is the law of $\mu^{\boxplus t}$5 for freely independent self-adjoint variables $\mu^{\boxplus t}$6, equivalently the limiting empirical spectral distribution of $\mu^{\boxplus t}$7 for independent unitarily invariant Hermitian matrices with limiting measures $\mu^{\boxplus t}$8 (Shlyakhtenko et al., 2020). The analytic description uses the Cauchy transform

$\mu^{\boxplus t}$9

the reciprocal transform $t \ge 1$0, the Voiculescu transform $t \ge 1$1, and the $t \ge 1$2-transform $t \ge 1$3 (Shlyakhtenko et al., 2020, Campbell, 2024). The defining relation for free additive convolution is additivity of the $t \ge 1$4-transform:

$t \ge 1$5

Integer free convolution powers are defined by repeated convolution, and satisfy

$t \ge 1$6

for $t \ge 1$7 (Shlyakhtenko et al., 2020). Fractional free convolution extends this identity to all real $t \ge 1$8: for any compactly supported $t \ge 1$9, there exists a unique compactly supported measure $R$0 such that

$R$1

or equivalently $R$2 for all free cumulants $R$3 (Shlyakhtenko et al., 2020). Existence for arbitrary $R$4 on the full half-line $R$5 is attributed there to Bercovici–Voiculescu and Nica–Speicher, while existence for $R$6 is equivalent to free infinite divisibility (Shlyakhtenko et al., 2020).

This immediately yields two semigroup relations:

$R$7

for $R$8 (Shlyakhtenko et al., 2020). The first is multiplicative in the fractional-power parameter and is structurally analogous to the $R$9-semigroup for Brown measures of $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$0-diagonal elements (Campbell et al., 2023). The second is the ordinary additive convolution semigroup law.

A standard normalization fixes variance by dilation:

$R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$1

where $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$2 is the pushforward under $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$3 (Shlyakhtenko et al., 2020). Under this normalization,

$R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$4

so $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$5 is fixed and higher cumulants decay with $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$6 (Shlyakhtenko et al., 2020). This is the free central-limit scaling, and for mean-zero, variance-one $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$7 one has

$R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$8

as $R_{\mu^{\boxplus t}}(w)=tR_\mu(w)$9, where $t\ge 0$0 is the semicircular law (Shlyakhtenko et al., 2020).

2. Analytic constructions, compression, and variational structure

Fractional free convolution powers admit a subordination characterization. For free additive convolution, there exist analytic maps $t\ge 0$1 such that

$t\ge 0$2

and

$t\ge 0$3

(Shlyakhtenko et al., 2020). Specializing to $t\ge 0$4, there is a single subordination map $t\ge 0$5 with

$t\ge 0$6

for $t\ge 0$7 (Shlyakhtenko et al., 2020). This formulation supports both existence theory and numerical fixed-point iteration.

A complementary interpretation uses free compression, or equivalently principal minors in random matrix theory. If $t\ge 0$8 is a projection with trace $t\ge 0$9 free from a self-adjoint variable $R$0 with law $R$1, and $R$2, then

$R$3

has law $R$4, equivalently

$R$5

(Shlyakhtenko et al., 2020). In cumulant form,

$R$6

For unitarily invariant Hermitian random matrices, the empirical spectral measure of a properly rescaled principal minor converges to $R$7 (Shlyakhtenko et al., 2020). This minor-process realization is one of the most concrete probabilistic models for non-integer free convolution powers.

Shlyakhtenko and Tao further study the normalized process $R$8 through free entropy $R$9 and free Fisher information $\mu_a^{\oplus k}$0 (Shlyakhtenko et al., 2020). Their monotonicity theorem states that

$\mu_a^{\oplus k}$1

is non-decreasing, while

$\mu_a^{\oplus k}$2

is non-increasing for $\mu_a^{\oplus k}$3 (Shlyakhtenko et al., 2020). Equality for some $\mu_a^{\oplus k}$4 occurs if and only if $\mu_a^{\oplus k}$5 is an affine image of the semicircle law (Shlyakhtenko et al., 2020). The paper gives two proofs: a compression-based argument using conjugate variables and a complex-analytic argument based on a Burgers-type PDE for the Cauchy transform (Shlyakhtenko et al., 2020).

The same work also presents a variational description in Gelfand–Tsetlin coordinates. For compactly supported $\mu_a^{\oplus k}$6, one defines a function $\mu_a^{\oplus k}$7 by

$\mu_a^{\oplus k}$8

on the Gelfand–Tsetlin pyramid $\mu_a^{\oplus k}$9 (Shlyakhtenko et al., 2020). Under non-degeneracy assumptions, $k\ge 1$0 is a formal critical point of an action with Lagrangian density

$k\ge 1$1

(Shlyakhtenko et al., 2020). This places fractional free convolution in a geometric framework related to entropy dissipation and minor processes.

3. Fractional convolution of Brown measures of $k\ge 1$2-diagonal elements

In a tracial von Neumann algebra $k\ge 1$3, an element $k\ge 1$4 is $k\ge 1$5-diagonal if in its polar decomposition $k\ge 1$6, the unitary $k\ge 1$7 is Haar and free from $k\ge 1$8; equivalently, the only non-vanishing free cumulants of $k\ge 1$9 are the alternating even cumulants (Campbell et al., 2023). The Brown measure $S$0 of a possibly non-normal operator $S$1 is defined via the Fuglede–Kadison determinant $S$2 and its logarithm $S$3 by

$S$4

as a distribution (Campbell et al., 2023). For $S$5-diagonal $S$6, $S$7 is rotationally invariant (Campbell et al., 2023).

A basic structural result is the Haagerup–Larsen formula. Writing $S$8 for the spectral measure of $S$9 and $\mu^{\boxplus t}$00 for the radial CDF, one has

$\mu^{\boxplus t}$01

where

$\mu^{\boxplus t}$02

and $\mu^{\boxplus t}$03 is the $\mu^{\boxplus t}$04-transform of $\mu^{\boxplus t}$05 (Campbell et al., 2023).

Kösters and Tikhomirov introduced a free convolution of Brown measures for $\mu^{\boxplus t}$06-diagonal elements by transporting additive free convolution on symmetric real measures through a bijection $\mu^{\boxplus t}$07:

$\mu^{\boxplus t}$08

so that for free $\mu^{\boxplus t}$09-diagonal $\mu^{\boxplus t}$10,

$\mu^{\boxplus t}$11

(Campbell et al., 2023). The paper "The fractional free convolution of $\mu^{\boxplus t}$12-diagonal elements and random polynomials under repeated differentiation" extends this operation from integer sums to real powers $\mu^{\boxplus t}$13 (Campbell et al., 2023).

For an $\mu^{\boxplus t}$14-diagonal element $\mu^{\boxplus t}$15 and real $\mu^{\boxplus t}$16, the fractional power $\mu^{\boxplus t}$17 is defined as the rotationally invariant probability measure with radial CDF

$\mu^{\boxplus t}$18

where

$\mu^{\boxplus t}$19

(Campbell et al., 2023). This yields a fractional free convolution semigroup indexed by $\mu^{\boxplus t}$20 (Campbell et al., 2023).

The semigroup property is

$\mu^{\boxplus t}$21

(Campbell et al., 2023). Existence follows from the explicit $\mu^{\boxplus t}$22-transform formula, and uniqueness from Haagerup–Larsen inversion (Campbell et al., 2023). For integer $\mu^{\boxplus t}$23, if $\mu^{\boxplus t}$24 are freely independent copies of $\mu^{\boxplus t}$25, then

$\mu^{\boxplus t}$26

(Campbell et al., 2023). Thus $\mu^{\boxplus t}$27 is an exact interpolation of $\mu^{\boxplus t}$28-fold free summation at the level of Brown measures.

Several basic properties are explicit. If $\mu^{\boxplus t}$29 is the outer radius of $\mu^{\boxplus t}$30, then

$\mu^{\boxplus t}$31

and, on $\mu^{\boxplus t}$32, $\mu^{\boxplus t}$33 has a strictly positive density supported on the closed disk $\mu^{\boxplus t}$34 (Campbell et al., 2023). Writing $\mu^{\boxplus t}$35, the radial density $\mu^{\boxplus t}$36 satisfies

$\mu^{\boxplus t}$37

on the support (Campbell et al., 2023).

4. Stability, explicit examples, and transform formulas

An $\mu^{\boxplus t}$38-diagonal Brown measure $\mu^{\boxplus t}$39 is called $\mu^{\boxplus t}$40-$\mu^{\boxplus t}$41-stable, for $\mu^{\boxplus t}$42, if

$\mu^{\boxplus t}$43

for integers $\mu^{\boxplus t}$44, and hence by the semigroup property for real $\mu^{\boxplus t}$45 (Campbell et al., 2023). The characterization given is

$\mu^{\boxplus t}$46

and this is equivalent to $\mu^{\boxplus t}$47-$\mu^{\boxplus t}$48-stability (Campbell et al., 2023). This parallels the role of free stable laws for additive $\mu^{\boxplus t}$49-powers on $\mu^{\boxplus t}$50, although the operative transform is the $\mu^{\boxplus t}$51-transform of $\mu^{\boxplus t}$52 rather than the $\mu^{\boxplus t}$53-transform of a real measure (Campbell et al., 2023, Campbell, 2024).

The circular element provides the simplest explicit example. If $\mu^{\boxplus t}$54 is standard circular, then

$\mu^{\boxplus t}$55

hence

$\mu^{\boxplus t}$56

and

$\mu^{\boxplus t}$57

(Campbell et al., 2023). Therefore $\mu^{\boxplus t}$58 is uniform on the disk of radius $\mu^{\boxplus t}$59 with density

$\mu^{\boxplus t}$60

(Campbell et al., 2023). In particular, $\mu^{\boxplus t}$61 is again circular (Campbell et al., 2023).

For free Haar unitaries $\mu^{\boxplus t}$62 and $\mu^{\boxplus t}$63, one has

$\mu^{\boxplus t}$64

which matches the general fractional formula because $\mu^{\boxplus t}$65 (Campbell et al., 2023). The resulting radial CDF is

$\mu^{\boxplus t}$66

(Campbell et al., 2023).

The multiplicative behavior of free $\mu^{\boxplus t}$67-diagonal elements is also particularly clean. If $\mu^{\boxplus t}$68 are free $\mu^{\boxplus t}$69-diagonal, then

$\mu^{\boxplus t}$70

(Campbell et al., 2023). Under the polynomial correspondence developed there, this implies multiplicativity of radial quantile functions:

$\mu^{\boxplus t}$71

(Campbell et al., 2023). The same paper also computes Brown measures for commutators and anticommutators of free $\mu^{\boxplus t}$72-diagonal variables. For free $\mu^{\boxplus t}$73-diagonal $\mu^{\boxplus t}$74,

$\mu^{\boxplus t}$75

and the outer radius is $\mu^{\boxplus t}$76 (Campbell et al., 2023). When $\mu^{\boxplus t}$77 are circular, this yields

$\mu^{\boxplus t}$78

and

$\mu^{\boxplus t}$79

(Campbell et al., 2023).

These explicit formulas show that the fractional operation on Brown measures is not merely formal interpolation. It preserves strong structural regularity—rotational invariance, explicit radial inversion, and sharp support control—while matching genuine operator models at integer times (Campbell et al., 2023).

5. Repeated differentiation and polynomial root dynamics

A major recent theme is that repeated differentiation of high-degree polynomials produces free convolution semigroups in the large-degree limit. In the real-rooted setting, the heuristic

$\mu^{\boxplus t}$80

is identified in the literature summarized by "Free infinite divisibility, fractional convolution powers, and Appell polynomials" (Campbell, 2024). There, the exponent $\mu^{\boxplus t}$81 is generally non-integer, so repeated differentiation naturally generates fractional free additive powers (Campbell, 2024). At the level of finite free probability, differentiation rescales the finite free $\mu^{\boxplus t}$82-transform:

$\mu^{\boxplus t}$83

(Campbell, 2024). This exact transform identity explains why the large-degree limit is governed by fractional free convolution.

In the complex rotationally invariant setting, the connection is formulated in terms of empirical root distributions of random polynomials

$\mu^{\boxplus t}$84

where the coefficients $\mu^{\boxplus t}$85 are iid complex random variables satisfying

$\mu^{\boxplus t}$86

and the deterministic profile $\mu^{\boxplus t}$87 satisfies the asymptotic regularity condition

$\mu^{\boxplus t}$88

for a continuous $\mu^{\boxplus t}$89 with $\mu^{\boxplus t}$90 on $\mu^{\boxplus t}$91 and $\mu^{\boxplus t}$92 for $\mu^{\boxplus t}$93 (Campbell et al., 2023). The empirical root measure $\mu^{\boxplus t}$94 then converges in probability to a rotationally invariant measure $\mu^{\boxplus t}$95 determined by the Legendre–Fenchel transform of $\mu^{\boxplus t}$96 (Campbell et al., 2023). Writing

$\mu^{\boxplus t}$97

the radial law is

$\mu^{\boxplus t}$98

(Campbell et al., 2023).

The connection to Brown-measure fractional convolution uses the quadratic map

$\mu^{\boxplus t}$99

on rotationally invariant measures (Campbell et al., 2023). If $t \ge 1$00 is the limiting empirical root distribution of the $t \ge 1$01-th derivative of $t \ge 1$02, then

$t \ge 1$03

(Campbell et al., 2023). Thus the differentiation flow of zeros becomes, after the quadratic pullback $t \ge 1$04, the $t \ge 1$05-semigroup on Brown measures of $t \ge 1$06-diagonal elements (Campbell et al., 2023).

In radial quantile form, if $t \ge 1$07 is the radial CDF of $t \ge 1$08, then

$t \ge 1$09

(Campbell et al., 2023). This identity is the precise differentiation-flow law that matches the compression-induced transform defining $t \ge 1$10 (Campbell et al., 2023). The real-rooted finite-free theory of Appell polynomials supplies an analogous transform-level explanation with $t \ge 1$11-transforms instead of $t \ge 1$12-transforms (Campbell, 2024).

6. Limit laws, Appell structures, and current scope

Repeated differentiation yields central-limit and stable-law phenomena in both the additive and Brown-measure frameworks. In the Brown-measure setting, for $t \ge 1$13, the paper (Campbell et al., 2023) defines an $t \ge 1$14-stable rotationally invariant measure $t \ge 1$15 through the radial quantile

$t \ge 1$16

and shows that $t \ge 1$17 is differentiation-stable, equivalently that $t \ge 1$18 is $t \ge 1$19-$t \ge 1$20-stable (Campbell et al., 2023). Under a tail assumption

$t \ge 1$21

for a slowly varying $t \ge 1$22, there exists a slowly varying $t \ge 1$23 such that suitable rescalings of $t \ge 1$24 converge weakly to $t \ge 1$25 as $t \ge 1$26 (Campbell et al., 2023). In the compact-support case $t \ge 1$27, the limit is the circular law under $t \ge 1$28, with

$t \ge 1$29

(Campbell et al., 2023).

In the real-rooted setting, "Free infinite divisibility, fractional convolution powers, and Appell polynomials" identifies Appell sequences as asymptotic limits of repeated differentiation when root bounds are removed (Campbell, 2024). An Appell sequence $t \ge 1$30 satisfies $t \ge 1$31 (Campbell, 2024). If $t \ge 1$32 belongs to the Laguerre–Pólya class, the associated Appell polynomials are

$t \ge 1$33

and their normalized versions converge to a freely infinitely divisible law $t \ge 1$34 whose $t \ge 1$35-transform is given explicitly by a free Lévy–Khintchine formula (Campbell, 2024). The finite free $t \ge 1$36-transform of $t \ge 1$37 is a truncation of the analytic $t \ge 1$38-transform of $t \ge 1$39:

$t \ge 1$40

and after normalization,

$t \ge 1$41

(Campbell, 2024). This gives a direct analytic bridge between fractional free convolution powers and Appell polynomial asymptotics.

The same paper extends these ideas to rectangular finite free probability. There, the differentiation operator $t \ge 1$42 is replaced by

$t \ge 1$43

and one obtains a rectangular finite free convolution $t \ge 1$44 together with a rectangular finite free $t \ge 1$45-transform (Campbell, 2024). Crucially, the parameter $t \ge 1$46 extends continuously to $t \ge 1$47, giving “rectangular fractional free convolution powers” (Campbell, 2024). In large-degree scaling regimes, repeated application of $t \ge 1$48 produces fractional rectangular powers and corresponding law-of-large-numbers, central-limit, and interpolation regimes (Campbell, 2024). This suggests that fractional free convolution is best viewed not as a single isolated construction but as a family of semigroup mechanisms adapted to different free-probabilistic geometries.

Several limitations are explicit in the current literature. The Brown-measure construction in (Campbell et al., 2023) requires $t \ge 1$49-diagonality and rotational invariance, and its parameter domain is $t \ge 1$50. The Appell and finite-free framework in (Campbell, 2024) highlights open problems involving heavy-tailed limits, analytic finite free $t \ge 1$51-transform theory beyond formal power series, and multiplicative analogues. The additive theory on $t \ge 1$52 is more mature analytically, with subordination, entropy monotonicity, and minor-process interpretations already established (Shlyakhtenko et al., 2020). A plausible implication is that future progress will depend on extending such analytic control—subordination, PDE formulations, and infinite-divisibility criteria—to the non-normal and rectangular settings where fractional convolution now appears naturally but is not yet fully unified.