Free Additive Convolution

• Free additive convolution is a fundamental operation in free probability that describes the spectral law of sums of freely independent noncommutative random variables.
• Its analysis leverages the additivity of R-transforms, subordination techniques, and Stieltjes transforms to derive fine spectral properties and support structures.
• The operation underpins applications in random matrix theory, including spectral regularity, dynamic eigenvalue flows, and high-accuracy numerical computations.

Free additive convolution is a central operation in free probability theory, describing the spectral law of the sum of freely independent noncommutative random variables and, in particular, the empirical eigenvalue distribution of sums of large, asymptotically free random matrices. Its analytic description, fine spectral and regularity properties, and computational methods connect noncommutative probability, random matrix theory, operator algebras, and approximation theory.

1. Definition and Fundamental Properties

Let $\mu$ and $\nu$ be Borel probability measures on $\mathbb{R}$. The free additive convolution $\mu \boxplus \nu$ is defined as the limiting spectral law of the sum $A+B$ of large Hermitian (or symmetric) random matrices $A,B$ whose empirical eigenvalue laws converge to $\mu$ and $\nu$ and which are asymptotically free (e.g., $B$ is conjugated by a Haar-distributed unitary/orthogonal matrix independent from $A$) (Bousseyroux et al., 4 Dec 2024). In the probabilistic framework, for freely independent selfadjoint operators $X,Y$ of laws $\mu, \nu$, the law of $X+Y$ is $\mu\boxplus\nu$.

The defining analytic property is the addition of R-transforms. If $R_{\mu}(w)$ and $R_{\nu}(w)$ are the R-transforms of $\mu$ and $\nu$, then

$R_{\mu \boxplus \nu}(w) = R_{\mu}(w) + R_{\nu}(w),$

with $R_{\mu}(w) = K_{\mu}(w) - 1/w$ where $K_{\mu}$ is the functional inverse of the Stieltjes (Cauchy) transform $G_\mu(z) = \int \frac{d\mu(x)}{z-x}$, i.e., $K_{\mu}(w): G_\mu(K_\mu(w)) = w$.

This algebraic additivity is the cornerstone of both theoretical development and explicit calculation in free convolution (Bousseyroux et al., 4 Dec 2024, Campbell, 29 Dec 2024).

2. Analytic Apparatus: Transforms and Subordination

Cauchy and R-Transforms

For a probability measure $\mu$ on $\mathbb{R}$,

• Stieltjes (Cauchy) transform:

$$G_\mu(z) = \int_{\mathbb{R}} \frac{d\mu(x)}{z-x}, \quad z \notin \operatorname{supp}(\mu)$$

• R-transform: For $w$ in a neighborhood of $0$ (in $\mathbb{C}$),

$R_\mu(w) = K_\mu(w) - \frac{1}{w},$

where $K_\mu$ is the inverse of $w = G_\mu(z)$ with respect to $z$.

Free additive convolution is accessed through explicit evaluation or manipulation of R-transforms, leveraging the additivity property.

Subordination

Free additive convolution has an analytic subordination structure: there exist unique analytic subordination functions $$\omega_\mu, \omega_\nu: \mathbb{C}^+ \to \mathbb{C}^+$$ such that

$$G_{\mu\boxplus\nu}(z) = G_\mu(\omega_\nu(z)) = G_\nu(\omega_\mu(z)), \qquad \omega_\mu(z) + \omega_\nu(z) - z = 0,$$

and equivalently,

$$F_{\mu\boxplus\nu}(z) = F_\mu(\omega_\nu(z)) = F_\nu(\omega_\mu(z)),$$

where $F_\mu(z) = -1/G_\mu(z)$ is the negative reciprocal Cauchy transform (Belinschi et al., 2013, Bao et al., 2015, Bao et al., 2018).

The subordination system provides both existence and uniqueness and a practical pathway to recovery of the density and support of $\mu\boxplus\nu$.

3. Spectral Regularity, Edge Behavior, and Support Structure

When both $\mu$ and $\nu$ are absolutely continuous and supported on a single interval with Jacobi-type power-law edge behavior (i.e., density $\sim (x-a)^\alpha$ with $\alpha \in (-1,1)$ near endpoints), the free additive convolution $\mu\boxplus\nu$ is absolutely continuous, supported on a single interval, and has density vanishing as a square root at the endpoints:

$$c^{-1} \sqrt{x-A} \sqrt{B-x} \leq \rho_{\mu\boxplus\nu}(x) \leq C \sqrt{x-A} \sqrt{B-x}$$

for $x \in (A,B)$ (Bao et al., 2018).

For multi-interval ("multi-cut") input measures, the support of $\mu\boxplus\nu$ may have several components. Explicit upper and lower bounds on the number of intervals are given in terms of the number of intervals in the input supports and the location of real off-support zeros of the Cauchy transforms (Moreillon et al., 2022):

$$1 + p_\alpha + p_\beta \leq I_{\alpha\beta} + C_{\alpha\beta} \leq (p_\beta + 1)(n_\alpha - 1) + (p_\alpha + 1)(n_\beta - 1) + 1,$$

where $I_{\alpha\beta}$ is the number of support intervals and $C_{\alpha\beta}$ the number of interior zeros of the convolution density, $n_\alpha$ and $n_\beta$ the respective numbers of intervals in the initial supports, and $p_\alpha, p_\beta$ the numbers of real off-support zeros.

In the case of the free convolution semigroup $\mu^{\boxplus t}$ (the $t$-fold free additive convolution power), the number of support components $n(t)$ is a non-increasing function of $t > 1$ and eventually stabilizes to $1$ under mild conditions on $\mu$ (Huang, 2012). The support varies continuously in the Hausdorff metric as $t$ changes (Williams, 2015).

Atoms in the output are characterized precisely: $\mu\boxplus\nu$ has an atom at $a+b$ if and only if $\mu$ has an atom at $a$ and $\nu$ an atom at $b$ with masses summing to more than $1$, and the mass is $p+q-1$ (Belinschi et al., 2019).

4. Dynamical and Operator Interpretation

The additivity of R-transforms admits a dynamical explanation via generalized Dyson Brownian motion (Bousseyroux et al., 4 Dec 2024). For $M(t) = A + \sqrt{t} B$ with $A$ deterministic and $B$ random and rotationally invariant, the large-$N$ limit of the eigenvalue trajectories $\lambda_i(t)$ obeys

$d\lambda_i(t) = F(t, \lambda_i) dt + o(1),$

where $F$ encodes a sum over $n$-body interactions determined by the free cumulants of $B$. For $n=2$ (Wigner case) this recovers Dyson Brownian motion with logarithmic repulsion; higher cumulants yield non-potential higher-body forces.

The Stieltjes transform $g(z,t)$ of the empirical measure evolves according to a nonlinear Burgers-type PDE

$$\partial_t g = -\left( \partial_t R_{B(t)}(g) \right) \partial_z g,$$

with long-time fixed point characterized by the R-transform addition law.

In the operator-valued setting, the analytic theory extends with subordination maps in the full operator upper half-plane, not just power series around the origin. The analytic dependence of the spectral law of polynomials in freely independent variables on the laws of the summands, via subordination and linearization, provides a general algorithm for the asymptotic eigenvalue distribution of matrix polynomials (Belinschi et al., 2013, Belinschi et al., 2019).

5. Numerical Computation and Algorithms

Efficient, high-accuracy numerical computation of free additive convolutions exploits the regularity of “admissible” and “invertible” measures. For compactly supported input measures with either square-root edge decay or fast decay at infinity, the Cauchy and R-transforms can be represented via Chebyshev (for compact support) or Fourier (for Schwartz-class) series (Olver et al., 2012).

The workflow involves:

• Expansion of input densities (e.g., in Chebyshev polynomials for square-root decaying laws).
• Accurate evaluation and numerical inversion of the Cauchy transform (using companion-matrix techniques).
• Spectrally accurate recovery of the output density via Vandermonde-structured least squares.
• Guaranteed convergence under broad conditions on the input measures.

Alternative algorithms apply the Cauchy integral formula and discretization by the trapezoidal rule, leveraging exponential convergence for analytic densities with square-root edge singularity. This is especially effective for classical laws such as the semicircle and Marchenko–Pastur, and more generally for any measure with the requisite regularity (Cortinovis et al., 2023). See the table below for input/output measure classes in these schemes:

Input class	Output (μ⊞ν) regularity	Series basis
Admissible (Jacobi, Schwartz,...)	Invertible (√-edge, Schwartz)	Chebyshev or Fourier

6. Variational Principles, Infinite Divisibility, and Further Extensions

The logarithmic potential of $\mu\boxplus\nu$ admits a variational characterization in terms of the R-transform of $\mu$ and the logarithmic potential of $\nu$. For $x < \inf\operatorname{supp}(\mu \boxplus \nu)$,

$$U_{\mu\boxplus\nu}(x) = \inf_{g>0} \left\{\int_0^g s R_\mu'(-s)ds + \int \log(\lambda - x + R_\mu(-g))d\nu(\lambda) \right\},$$

where the unique minimizing $g$ is $G_{\mu\boxplus\nu}(x)$ (Concetti et al., 23 Jun 2025). This formula simplifies in the case where $\mu$ is the semicircle or Marchenko–Pastur law and is relevant in the calculation of large deviations for determinants of random matrices.

Free infinite divisibility is characterized by analytic properties of the R-transform (Lévy–Khintchine formula), with fractional convolution powers given by scalar multiplication in $R_\mu$:

$R_{\mu^{\boxplus t}}(s) = t R_\mu(s).$

Connections between finite free convolution on polynomials, asymptotic spectral distributions, empirical root measures, and free infinitely divisible distributions (with the central limit behavior in repeated differentiation of polynomials) are established (Campbell, 29 Dec 2024, Arizmendi et al., 3 May 2025). Corrections at the $1/d$ level reveal linkages between infinitesimal distributions, moment-cumulant expansions, and the subordination structure.

7. Applications and Implications

Free additive convolution underpins the description of spectra for sums of large random matrices, both in global and local regimes. Local laws for random matrix ensembles $A + U B U^*$ assert that the empirical spectral distribution concentrates on $\mu_A \boxplus \mu_B$ down to scales $N^{-2/3}$ in the regular bulk, with explicit error rates (Bao et al., 2015).

The analysis of convolution semigroups provides spectral regularization results, quantifies the merging of support intervals, the fate of atoms, and the evolution of density singularities (Huang, 2012, Williams, 2015). Recent investigations of polynomials in free variables via linearization and operator-valued subordination yield criteria for the location and multiplicity of eigenvalues, with sharp results on the occurrence of atoms (Belinschi et al., 2019).

Specific dynamical interpretations (e.g., generalized Dyson Brownian motion) connect the algebraic structure to flows of eigenvalues governed by cumulant-determined repulsion and higher-order interactions, providing a bridge between algebraic free probability and integrable/stochastic dynamics (Bousseyroux et al., 4 Dec 2024).

The theory has further implications for random matrix large deviations, spin glass complexity via Kac–Rice formulae, and the spectral analysis of high-dimensional random systems (Concetti et al., 23 Jun 2025). The clear structural results on regularity, continuity, and computational tractability make free additive convolution a fundamental tool across several fields.

---

References:

• (Bousseyroux et al., 4 Dec 2024) Free Convolution and Generalized Dyson Brownian Motion
• (Olver et al., 2012) Numerical computation of convolutions in free probability theory
• (Bao et al., 2015) Local Stability of the Free Additive Convolution
• (Bao et al., 2018) On the support of the free additive convolution
• (Moreillon et al., 2022) The support of the free additive convolution of multi-cut measures
• (Huang, 2012) Supports of Measures in a free additive convolution semigroup
• (Williams, 2015) On the Hausdorff Continuity of Free Lèvy Processes and Free Convolution Semigroups
• (Concetti et al., 23 Jun 2025) Variational formula for the logarithmic potential of free additive convolutions
• (Campbell, 29 Dec 2024) Free infinite divisibility, fractional convolution powers, and Appell polynomials
• (Arizmendi et al., 3 May 2025) Finite Free Convolution: Infinitesimal Distributions
• (Belinschi et al., 2019) The atoms of the free additive convolution of two operator-valued distributions
• (Belinschi et al., 2013) Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem
• (Cortinovis et al., 2023) Computing Free Convolutions via Contour Integrals
• (Popa et al., 17 Nov 2025) On some properties of free commutators with semicircular variables