---
title: Fractional Free Convolution
url: https://www.emergentmind.com/topics/fractional-free-convolution
type: topic
---

# Fractional Free Convolution

Searching arXiv for recent 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 [2009.01882, 2412.20488]. 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 $R$-diagonal variables [2307.11935]. Recent work connects both notions to repeated differentiation of polynomials: on the real line through free additive convolution powers and Appell limits [2412.20488], and in the rotationally invariant complex setting through Brown measures and a quadratic transport map relating zero distributions to $\oplus$-powers [2307.11935].

## 1. Additive fractional powers on the real line

For compactly supported probability measures $\mu,\nu$ on $\mathbb{R}$, free additive convolution $\mu\boxplus\nu$ is the law of $X+Y$ for freely independent self-adjoint variables $X,Y$, equivalently the limiting empirical spectral distribution of $A+B$ for independent unitarily invariant Hermitian matrices with limiting measures $\mu,\nu$ [2009.01882]. The analytic description uses the Cauchy transform
$$
G_\mu(z)=\int_{\mathbb{R}} \frac{1}{z-x}\,d\mu(x),
$$
the reciprocal transform $F_\mu(z)=1/G_\mu(z)$, the Voiculescu transform $\phi_\mu(z)=F_\mu^{-1}(z)-z$, and the $R$-transform $R_\mu(w)=\phi_\mu(1/w)$ [2009.01882, 2412.20488]. The defining relation for free additive convolution is additivity of the $R$-transform:
$$
R_{\mu\boxplus\nu}(w)=R_\mu(w)+R_\nu(w).
$$

Integer free convolution powers are defined by repeated convolution, and satisfy
$$
R_{\mu^{\boxplus k}}(w)=kR_\mu(w)
$$
for $k\in\mathbb{N}$ [2009.01882]. Fractional free convolution extends this identity to all real $t\ge 1$: for any compactly supported $\mu$, there exists a unique compactly supported measure $\mu^{\boxplus t}$ such that
$$
R_{\mu^{\boxplus t}}(w)=tR_\mu(w),
$$
or equivalently $\kappa_n(\mu^{\boxplus t})=t\kappa_n(\mu)$ for all free cumulants $\kappa_n$ [2009.01882]. Existence for arbitrary $\mu$ on the full half-line $t\ge 1$ is attributed there to Bercovici–Voiculescu and Nica–Speicher, while existence for $t\in(0,1)$ is equivalent to free infinite divisibility [2009.01882].

This immediately yields two semigroup relations:
$$
(\mu^{\boxplus k})^{\boxplus \ell}=\mu^{\boxplus k\ell},\qquad
\mu^{\boxplus k}\boxplus \mu^{\boxplus \ell}=\mu^{\boxplus(k+\ell)},
$$
for $k,\ell\ge 1$ [2009.01882]. The first is multiplicative in the fractional-power parameter and is structurally analogous to the $\oplus$-semigroup for Brown measures of $R$-diagonal elements [2307.11935]. The second is the ordinary additive convolution semigroup law.

A standard normalization fixes variance by dilation:
$$
\mu_t:=D_{t^{-1/2}}(\mu^{\boxplus t}),
$$
where $D_\lambda(\mu)$ is the pushforward under $x\mapsto \lambda x$ [2009.01882]. Under this normalization,
$$
\kappa_m(\mu_t)=t^{1-m/2}\kappa_m(\mu),
$$
so $\kappa_2$ is fixed and higher cumulants decay with $t$ [2009.01882]. This is the free central-limit scaling, and for mean-zero, variance-one $\mu$ one has
$$
D_{t^{-1/2}}(\mu^{\boxplus t})\to \mu_{sc}
$$
as $t\to\infty$, where $\mu_{sc}$ is the semicircular law [2009.01882].

## 2. Analytic constructions, compression, and variational structure

Fractional free convolution powers admit a subordination characterization. For free additive convolution, there exist analytic maps $\omega_\mu,\omega_\nu:\mathbb{C}^+\to\mathbb{C}^+$ such that
$$
G_{\mu\boxplus\nu}(z)=G_\mu(\omega_\mu(z))=G_\nu(\omega_\nu(z)),
$$
and
$$
\omega_\mu(z)+\omega_\nu(z)-z=F_{\mu\boxplus\nu}(z)
$$
[2009.01882]. Specializing to $\mu^{\boxplus t}$, there is a single subordination map $\omega_t$ with
$$
G_{\mu^{\boxplus t}}(z)=G_\mu(\omega_t(z)),\qquad
\omega_t(z)=z-(t-1)F_{\mu^{\boxplus t}}(z)
$$
for $t\ge 1$ [2009.01882]. 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 $p$ is a projection with trace $1/k$ free from a self-adjoint variable $X$ with law $\mu$, and $\pi(X)=[pXp]$, then
$$
k\,\pi(X)
$$
has law $\mu^{\boxplus k}$, equivalently
$$
R_{k\pi(X)}(w)=kR_X(w),\qquad R_{\pi(X)}(w)=R_X(w/k)
$$
[2009.01882]. In cumulant form,
$$
\kappa_n(\pi(X))=k^{1-n}\kappa_n(X).
$$
For unitarily invariant Hermitian random matrices, the empirical spectral measure of a properly rescaled principal minor converges to $\mu^{\boxplus k}$ [2009.01882]. 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 $D_{t^{-1/2}}(\mu^{\boxplus t})$ through free entropy $\chi$ and free Fisher information $\Phi$ [2009.01882]. Their monotonicity theorem states that
$$
t\mapsto \chi(D_{t^{-1/2}}(\mu^{\boxplus t}))
$$
is non-decreasing, while
$$
t\mapsto \Phi(D_{t^{-1/2}}(\mu^{\boxplus t}))
$$
is non-increasing for $t\ge 1$ [2009.01882]. Equality for some $t>1$ occurs if and only if $\mu$ is an affine image of the semicircle law [2009.01882]. 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 [2009.01882].

The same work also presents a variational description in Gelfand–Tsetlin coordinates. For compactly supported $\mu$, one defines a function $\lambda(s,y)$ by
$$
\mu^{\boxplus(1/s)}((-\infty,\lambda(s,y)/s])=y/s
$$
on the Gelfand–Tsetlin pyramid $\Delta=\{(s,y):0<s<1,\ 0<y<s\}$ [2009.01882]. Under non-degeneracy assumptions, $\lambda$ is a formal critical point of an action with Lagrangian density
$$
L(\lambda_s,\lambda_y)=\log(\lambda_y)+\log(\sin(\pi\lambda_s))
$$
[2009.01882]. This places fractional free convolution in a geometric framework related to entropy dissipation and minor processes.

## 3. Fractional convolution of Brown measures of $R$-diagonal elements

In a tracial von Neumann algebra $(\mathcal{M},\tau)$, an element $a\in\mathcal{M}$ is $R$-diagonal if in its polar decomposition $a=uh$, the unitary $u$ is Haar and free from $h=|a|$; equivalently, the only non-vanishing free cumulants of $(a,a^*)$ are the alternating even cumulants [2307.11935]. The Brown measure $\mu_a$ of a possibly non-normal operator $a$ is defined via the Fuglede–Kadison determinant $\Delta$ and its logarithm $L$ by
$$
\mu_a:=\frac{1}{2\pi}\Delta^2L(a-\lambda1)
$$
as a distribution [2307.11935]. For $R$-diagonal $a$, $\mu_a$ is rotationally invariant [2307.11935].

A basic structural result is the Haagerup–Larsen formula. Writing $\nu_a:=\mu_{aa^*}$ for the spectral measure of $aa^*$ and $F_a(r)=\mu_a(\overline{\mathbb{D}_r})$ for the radial CDF, one has
$$
F_a(r)=
\begin{cases}
0,& r\in[0,\lambda_1),\\[4pt]
1+\mathscr{S}_{a^*a}^{\langle -1\rangle}(r^{-2}),& r\in[\lambda_1,\lambda_2),\\[4pt]
1,& r\ge \lambda_2,
\end{cases}
$$
where
$$
\lambda_1:=\left(\int_0^\infty x^{-2}\,d\mu_{|a|}(x)\right)^{-1/2},\qquad
\lambda_2:=\left(\int_0^\infty x^2\,d\mu_{|a|}(x)\right)^{1/2},
$$
and $\mathscr{S}_{a^*a}$ is the $S$-transform of $\mu_{a^*a}$ [2307.11935].

Kösters and Tikhomirov introduced a free convolution of Brown measures for $R$-diagonal elements by transporting additive free convolution on symmetric real measures through a bijection $\mathcal{H}$:
$$
\mu_a\oplus\mu_b:=\mathcal{H}\big(\mathcal{H}^{-1}(\mu_a)\boxplus \mathcal{H}^{-1}(\mu_b)\big),
$$
so that for free $R$-diagonal $a,b$,
$$
\mu_{a+b}=\mu_a\oplus\mu_b
$$
[2307.11935]. The paper "The fractional free convolution of $R$-diagonal elements and random polynomials under repeated differentiation" extends this operation from integer sums to real powers $k\ge 1$ [2307.11935].

For an $R$-diagonal element $a$ and real $k>1$, the fractional power $\mu_a^{\oplus k}$ is defined as the rotationally invariant probability measure with radial CDF
$$
\mu_a^{\oplus k}(\overline{\mathbb{D}_r})=
\begin{cases}
1+\mathscr{S}_k^{\langle -1\rangle}(r^{-2}),& r\in(0,\lambda_2^{(k)}),\\[4pt]
1,& r\ge \lambda_2^{(k)},
\end{cases}
$$
where
$$
\mathscr{S}_k(z):=\frac{1+z/k}{k(1+z)}\,\mathscr{S}_{aa^*}(z/k),\qquad
\lambda_2^{(k)}:=\sqrt{k}\,\lambda_2
$$
[2307.11935]. This yields a fractional free convolution semigroup indexed by $k\ge 1$ [2307.11935].

The semigroup property is
$$
(\mu_a^{\oplus j})^{\oplus l}=\mu_a^{\oplus jl},\qquad j,l\ge 1
$$
[2307.11935]. Existence follows from the explicit $S$-transform formula, and uniqueness from Haagerup–Larsen inversion [2307.11935]. For integer $k$, if $a_1,\dots,a_k$ are freely independent copies of $a$, then
$$
\mu_{a_1+\cdots+a_k}=\mu_a^{\oplus k}
$$
[2307.11935]. Thus $\mu_a^{\oplus k}$ is an exact interpolation of $k$-fold free summation at the level of Brown measures.

Several basic properties are explicit. If $R$ is the outer radius of $\operatorname{supp}\mu_a$, then
$$
\mu_a^{\oplus k}(\{0\})=\max\{0,\ 1-k(1-\mu_a(\{0\}))\}
$$
and, on $\mathbb{C}\setminus\{0\}$, $\mu_a^{\oplus k}$ has a strictly positive density supported on the closed disk $|z|\le \sqrt{k}\,R$ [2307.11935]. Writing $F_k(r)=1+\mathscr{S}_k^{\langle -1\rangle}(r^{-2})$, the radial density $\varrho_k(r)$ satisfies
$$
F_k'(r)=2\pi r\,\varrho_k(r),\qquad
\varrho_k(r)=\frac{1}{\pi r^4\big|\mathscr{S}_k'(\mathscr{S}_k^{\langle -1\rangle}(r^{-2}))\big|}
$$
on the support [2307.11935].

## 4. Stability, explicit examples, and transform formulas

An $R$-diagonal Brown measure $\mu_a$ is called $\alpha$-$\oplus$-stable, for $\alpha\in(0,2]$, if
$$
\mu_a^{\oplus m}=\mathcal{D}_{m^{1/\alpha}}\mu_a
$$
for integers $m\ge 1$, and hence by the semigroup property for real $m\ge 1$ [2307.11935]. The characterization given is
$$
\mathscr{S}_{aa^*}(z)=\theta\,\frac{(-z)^{\frac{2}{\alpha}-1}}{1+z},\qquad \theta>0,
$$
and this is equivalent to $\alpha$-$\oplus$-stability [2307.11935]. This parallels the role of free stable laws for additive $\boxplus$-powers on $\mathbb{R}$, although the operative transform is the $S$-transform of $aa^*$ rather than the $R$-transform of a real measure [2307.11935, 2412.20488].

The circular element provides the simplest explicit example. If $c$ is standard circular, then
$$
\mathscr{S}_{cc^*}(z)=\frac{1}{1+z},
$$
hence
$$
\mathscr{S}_k(z)=\frac{1}{k(1+z)}
$$
and
$$
F_{\mu_c^{\oplus k}}(r)=\frac{r^2}{k},\qquad 0\le r\le \sqrt{k}
$$
[2307.11935]. Therefore $\mu_c^{\oplus k}$ is uniform on the disk of radius $\sqrt{k}$ with density
$$
d\mu_c^{(\oplus k)}(z)=\frac{1}{\pi k}\mathbf{1}_{|z|\le \sqrt{k}}\,d^2z
$$
[2307.11935]. In particular, $k^{-1/2}(c_1+\cdots+c_k)$ is again circular [2307.11935].

For free Haar unitaries $u_1,\dots,u_k$ and $u^{(k)}=\sum_{i=1}^k u_i$, one has
$$
\mathscr{S}_{u^{(k)}u^{(k)*}}(z)=\frac{z+k}{k^2(z+1)},
$$
which matches the general fractional formula because $\mathscr{S}_{uu^*}\equiv 1$ [2307.11935]. The resulting radial CDF is
$$
F_{\mu_{u^{(k)}}}(r)=(k-1)\frac{r^2}{k^2-r^2},\qquad 0\le r\le \sqrt{k}
$$
[2307.11935].

The multiplicative behavior of free $R$-diagonal elements is also particularly clean. If $x,y$ are free $R$-diagonal, then
$$
\mathscr{S}_{xyy^*x^*}(z)=\mathscr{S}_{xx^*}(z)\mathscr{S}_{yy^*}(z)
$$
[2307.11935]. Under the polynomial correspondence developed there, this implies multiplicativity of radial quantile functions:
$$
F_{xy}^{\langle -1\rangle}(x)=F_x^{\langle -1\rangle}(x)\,F_y^{\langle -1\rangle}(x)
$$
[2307.11935]. The same paper also computes Brown measures for commutators and anticommutators of free $R$-diagonal variables. For free $R$-diagonal $x,y$,
$$
\mathscr{S}_{xy\pm yx}(z)=\frac{2+z}{4(1+z)}\,\mathscr{S}_{x^*x}(z/2)\,\mathscr{S}_{y^*y}(z/2)
$$
and the outer radius is $\lambda_2=\sqrt{2}\lambda_2^x\lambda_2^y$ [2307.11935]. When $x,y$ are circular, this yields
$$
\mathscr{S}_{xy-yx}(z)=\frac{1}{(1+z)(2+z)}
$$
and
$$
\mu_{xy-yx}(\overline{\mathbb{D}_r})
=\frac{-1+\sqrt{1+4r^2}}{2},\qquad 0<r<\sqrt{2}
$$
[2307.11935].

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 [2307.11935].

## 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
$$
\mu_{p_j}\approx \mu_p^{\boxplus \frac{d}{d-j}},\qquad p_j(x)=p^{(j)}\!\left(\frac{d-j}{d}x\right)
$$
is identified in the literature summarized by "Free infinite divisibility, fractional convolution powers, and Appell polynomials" [2412.20488]. There, the exponent $d/(d-j)$ is generally non-integer, so repeated differentiation naturally generates fractional free additive powers [2412.20488]. At the level of finite free probability, differentiation rescales the finite free $R$-transform:
$$
R_{p_j}^{\,d-j}(s)=R_p^d\!\left(\frac{d-j}{d}s\right)\mod[s^{d-j+1}]
$$
[2412.20488]. 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
$$
P_n(z)=\sum_{k=0}^n \xi_k P_{k,n} z^k,
$$
where the coefficients $\xi_k$ are iid complex random variables satisfying
$$
\mathbb{P}(\xi_0=0)=0,\qquad \mathbb{E}\log(1+|\xi_0|)<\infty,
$$
and the deterministic profile $P_{k,n}$ satisfies the asymptotic regularity condition
$$
\lim_{n\to\infty}\sup_{0\le k\le n}\left||P_{k,n}|^{1/n}-P\!\left(\frac{k}{n}\right)\right|=0
$$
for a continuous $P:[0,1]\to[0,\infty)$ with $P(t)>0$ on $[0,1)$ and $P(t)=0$ for $t>1$ [2307.11935]. The empirical root measure $\mu_n$ then converges in probability to a rotationally invariant measure $\mu$ determined by the Legendre–Fenchel transform of $u(t)=-\log P(t)$ [2307.11935]. Writing
$$
I(s):=\sup_{t\ge 0}(st+\log P(t)),
$$
the radial law is
$$
\mu(\mathbb{D}_r)=I'(\log r),\qquad r>0
$$
[2307.11935].

The connection to Brown-measure fractional convolution uses the quadratic map
$$
(\sq\mu)(\mathbb{D}_r):=\mu(\mathbb{D}_{\sqrt r}),\qquad
\sq^{-1}(\mu)(\mathbb{D}_r)=\mu(\mathbb{D}_{r^2})
$$
on rotationally invariant measures [2307.11935]. If $\mu_t$ is the limiting empirical root distribution of the $\lceil tn\rceil$-th derivative of $P_n((1-t)^2x)$, then
$$
\mu_t=\sq\Big[\big(\sq^{-1}\mu\big)^{\oplus \frac{1}{1-t}}\Big],\qquad 0<t<1
$$
[2307.11935]. Thus the differentiation flow of zeros becomes, after the quadratic pullback $\sq^{-1}$, the $\oplus$-semigroup on Brown measures of $R$-diagonal elements [2307.11935].

In radial quantile form, if $\Phi_t$ is the radial CDF of $\mu_t$, then
$$
\Phi_t^{\langle -1\rangle}(x)=
\frac{x(1-t)\,\Phi_0^{\langle -1\rangle}((1-t)x+t)}{x(1-t)+t},
\qquad x\in(0,1)
$$
[2307.11935]. This identity is the precise differentiation-flow law that matches the compression-induced transform defining $\mu_a^{\oplus k}$ [2307.11935]. The real-rooted finite-free theory of Appell polynomials supplies an analogous transform-level explanation with $R$-transforms instead of $S$-transforms [2412.20488].

## 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 $\alpha\in(0,2]$, the paper [2307.11935] defines an $\alpha$-stable rotationally invariant measure $\mu_\alpha$ through the radial quantile
$$
\Phi_{0,\alpha}^{\langle -1\rangle}(x)=
\theta\,\frac{x}{(1-x)^{\frac{2}{\alpha}-1}},\qquad \theta>0,
$$
and shows that $\mu_\alpha$ is differentiation-stable, equivalently that $\sq^{-1}\mu_\alpha$ is $\alpha$-$\oplus$-stable [2307.11935]. Under a tail assumption
$$
\lim_{r\to\infty}\frac{1-\Phi_0(r)}{L(r)\,r^{-\frac{\alpha}{2-\alpha}}}=1
$$
for a slowly varying $L$, there exists a slowly varying $g$ such that suitable rescalings of $\mu_t$ converge weakly to $\mu_\alpha$ as $t\to 1^-$ [2307.11935]. In the compact-support case $\alpha=2$, the limit is the circular law under $\sq$, with
$$
\lim_{t\to1^-}\Phi_t((1-t)r)=r,\qquad r\in(0,1)
$$
[2307.11935].

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 [2412.20488]. An Appell sequence $\{A_d\}_{d\ge 1}$ satisfies $A_d'(x)=dA_{d-1}(x)$ [2412.20488]. If $f$ belongs to the Laguerre–Pólya class, the associated Appell polynomials are
$$
A_{d,f}(z)=f(D)z^d
$$
and their normalized versions converge to a freely infinitely divisible law $\mu_f$ whose $R$-transform is given explicitly by a free Lévy–Khintchine formula [2412.20488]. The finite free $R$-transform of $A_{d,f}$ is a truncation of the analytic $R$-transform of $\mu_f$:
$$
R_{A_{d,f}}^d(s)=-\frac{f'(ds)}{f(ds)}\mod[s^{d+1}],
$$
and after normalization,
$$
R_{\hat A_d}^d(s)=-\frac{f'(s)}{f(s)}\mod[s^{d+1}]
$$
[2412.20488]. 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 $D$ is replaced by
$$
M_n=xD^2+(n+1)D,
$$
and one obtains a rectangular finite free convolution $\#_d^{\,n}$ together with a rectangular finite free $R$-transform [2412.20488]. Crucially, the parameter $n$ extends continuously to $n>-1$, giving “rectangular fractional free convolution powers” [2412.20488]. In large-degree scaling regimes, repeated application of $M_n$ produces fractional rectangular powers and corresponding law-of-large-numbers, central-limit, and interpolation regimes [2412.20488]. 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 [2307.11935] requires $R$-diagonality and rotational invariance, and its parameter domain is $k\ge 1$. The Appell and finite-free framework in [2412.20488] highlights open problems involving heavy-tailed limits, analytic finite free $R$-transform theory beyond formal power series, and multiplicative analogues. The additive theory on $\mathbb{R}$ is more mature analytically, with subordination, entropy monotonicity, and minor-process interpretations already established [2009.01882]. 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.

Source: https://www.emergentmind.com/topics/fractional-free-convolution