Infinitesimal Freeness in Noncommutative Probability

• Infinitesimal freeness is an extension of Voiculescu’s freeness that incorporates O(1/N) corrections in noncommutative probability spaces.
• It refines the algebraic structure by introducing extra linear functionals to model small deviations and organize higher-order independence relations.
• This concept is pivotal in random matrix theory, aiding in the analysis of spectral distributions and eigenvector behaviors.

Infinitesimal freeness is an extension of Voiculescu’s freeness in non-commutative probability theory, capturing leading-order ($O(1/N)$) corrections in the asymptotic behavior of random matrices and refining the algebraic structure by introducing additional linear functionals to model infinitesimal deviations from classical freeness. It plays a central role in organizing higher-order independence relations, analyzing spectral properties and eigenvectors in random matrix models, and providing a combinatorial and analytic bridge between free and conditional independence frameworks.

1. Fundamental Definitions and Formalism

Let $(\mathcal{A},\tau)$ be a non-commutative probability space with unital algebra $\mathcal{A}$ and unital linear functional $\tau:\mathcal{A} \to \mathbb{C}$. Two unital subalgebras $\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}$ are free if, for any $a_i \in \mathcal{A}_{j_i}$ ($j_1 \ne j_2 \ne \cdots$) with $\tau(a_i) = 0$, $\tau(a_1 \cdots a_n) = 0$.

Infinitesimal non-commutative probability space: A triple $(\mathcal{A},\tau,\tau')$ where $\tau'$ is a linear map with $\tau'(1) = 0$. Subalgebras $\mathcal{A}_1, \mathcal{A}_2$ are infinitesimally free (Belinschi–Shlyakhtenko) if they are free w.r.t.\ $\tau$ and for all centered alternating tuples,

$$\tau'(a_1\cdots a_n) = \sum_{k=1}^n \tau(a_1 \cdots a_{k-1} \, \tau'(a_k) \, a_{k+1} \cdots a_n).$$

This is a first-order correction formula: $\tau'$ encodes the infinitesimal deviation from classical freeness.

In the conditional setting, a triple $(\mathcal{A}, \tau, \varphi)$ is a conditional non-commutative probability space with two unital linear functionals; conditionally free ("c-free") subalgebras satisfy:

$$\tau(a_1\cdots a_n) = 0, \quad \varphi(a_1\cdots a_n) = \varphi(a_1)\cdots\varphi(a_n)$$

for alternating $\tau$-centered tuples.

2. Hierarchy and Relations to Other Notions

Higher-order independence concepts fit into a strict hierarchy captured by the following implications (Cébron et al., 2022):

• Freeness of type B $\implies$ Cyclic monotone independence $\implies$ Monotone independence
• Freeness of type B $\to$ Conditional freeness under specific secondary state constructions

Freeness of type B, as developed in the algebraic context, is an extension acting on an ideal $V \subset \mathcal{A}$ with a second state $\varphi$. Infinitesimal freeness is recovered by projection onto the kernel of the trace. Conditional freeness arises as a specialization of type B freeness, and cyclic monotone and monotone independence are further derived by restriction and state selection (Cébron et al., 2022).

The following table summarizes the algebraic structures:

Concept	Functionals Involved	Key Structural Rule
Freeness	$\tau$	$\tau(a_1\cdots a_n) = 0$
Infinitesimal Freeness	$\tau, \tau'$	$\tau'(a_1\cdots a_n) = \sum_{k=1}^n\ldots$
Type B Freeness	$\tau, V, \tau'$	Enhancement; ideal-based, relates to infinitesimal freeness
Conditional Freeness (c-free)	$\tau, \varphi$	Multi-state factorization for alternating, centered tuples
Cyclic-Monotone	Restriction of type B	Mixed-state cyclic conditions

This hierarchy is realized concretely in random matrix models via $1/N^2$ expansions, with each level corresponding to finer asymptotic or algebraic detail.

3. Random Matrices and Asymptotic Expansions

For deterministic families $(\mathbf{A}_N,\mathbf{B}_N)$ of $N \times N$ matrices with bounded *-distributions, and $U_N$ a Haar unitary, Voiculescu’s asymptotic freeness theorem admits a precise extension (Cébron et al., 2022):

$$\mathbb{E} \Bigl[ \frac{1}{N} \operatorname{Tr} \bigl(P(\mathbf{A}_N, U_N \mathbf{B}_N U_N^*) \bigr) \Bigr] = (\tau_{\mathbf{A}_N} \star \tau_{\mathbf{B}_N})(P) + O(N^{-2})$$

If $\tau_{\mathbf{A}_N}(P)$ and $\tau_{\mathbf{B}_N}(P)$ possess $1/N$ expansions, the mixed expectation admits a first-order correction, and one obtains infinitesimal freeness in the joint limit:

$$\tau_{\mathbf{A}_N}(P) = \tau(P) + N^{-1}\tau'(P) + o(N^{-1})\quad\Rightarrow\quad \mathcal{A}_1,\mathcal{A}_2 \text{ are infinitesimally free w.r.t.\ } (\tau, \tau').$$

Beyond expectation, using concentration arguments, “type B” freeness holds almost surely for the corresponding ideals.

These results underpin major limits in random matrix theory, such as the empirical spectral distribution (ESD) and eigenvector spectral distribution (“eigenvector-ESD”), where the latter requires conditional or infinitesimal conditional freeness for asymptotics (Cébron et al., 2022, Cébron et al., 2022).

4. Cumulant Framework and Combinatorial Structure

Infinitesimal freeness and its extensions are organized using free, Boolean, c-free, and infinitesimal cumulants, all indexed by non-crossing partitions and subject to Möbius inversion formulas.

In the c-free/infinitesimal relation, the Δ*-map construction provides a systematic transformation: the infinitesimal cumulants are derived from c-free cumulants by adding an extra argument and symmetrizing:

$$\kappa_n^{\varphi'}(a_1,\dots,a_n) = \sum_{m=1}^n \kappa_{n+1}^{(c)}(a_m, \dots, a_n, a_1, \dots, a_m)$$

This connects infinitesimal and c-free laws at the level of distributions (Fevrier et al., 2018).

The combinatorial structure is highly transparent in the Motzkin-path approach: infinitesimal corrections to moments are isolated to specific Motzkin (or Dyck) paths—pyramid paths for pure infinitesimal free moments, pyramid+flat for infinitesimal c-freeness, and flat paths for infinitesimal Boolean independence. The key outcome is the Leibniz-type rule for infinitesimal c-free moments (Lenczewski, 27 Dec 2025):

$$\varphi'(a_1 \cdots a_n) = \varphi'(a_1)\varphi(a_2 \cdots a_n) + \sum_{m=2}^n \psi'(a_m)\varphi(a_1\cdots a_{m-1} a_{m+1} \cdots a_n)$$

This condition is precisely equivalent to the path classification: only concatenations of a pyramid path and a flat path contribute to the first-order infinitesimal terms.

5. Infinitesimal Conditional Freeness and Dualities

The infinitesimal conditional freeness framework further generalizes infinitesimal freeness by working with triples $(\psi, \varphi, \omega)$: two unital states and a tracial, possibly non-unital, infinitesimal functional with $\omega(1) = 0$ (Cébron et al., 2022). For alternating tuples,

• Ordinary c-freeness: $$\psi(a_1\cdots a_n) = 0,\; \varphi(a_1\cdots a_n) = \prod \varphi(a_j)$$
• Infinitesimal correction: $$\omega(a_1\cdots a_n) = \sum_{j=1}^n \left(\prod_{i<j}\varphi(a_i)\right)\omega(a_j)\left(\prod_{i>j}\varphi(a_i)\right)$$

This structure interpolates between scalar infinitesimal freeness (when $\psi = \varphi$) and ordinary c-freeness, and can be realized as an $O(1/N)$ correction in the asymptotics of specific random matrix models (“Vortex model”), where the infinitesimal functional captures corrections to both trace and vector states. This framework unifies cyclic-monotone, cyclic-Boolean, and other cyclic types of independence (Cébron et al., 2022).

6. Operator-Valued and Higher-Order Extensions

Infinitesimal freeness has been extended to the operator-valued (amalgamated) setting, where the functionals take values in a subalgebra $\mathcal{B}$ (Tseng, 2019). The central result is the equivalence between operator-valued infinitesimal (OVI) freeness over $(\mathcal{A},\mathcal{B}, E, E')$ and ordinary operator-valued freeness with respect to an upper-triangular $2 \times 2$ matrix extension:

• The “upper-triangularization” method reduces combinatorial and analytic questions about infinitesimal freeness to standard operator-valued freeness.
• Cumulants and analytic transforms (Cauchy, $R$-, $S$-transforms) admit infinitesimal companions (e.g., $r_x(b)$, $\partial S_x(b)$) encoding derivatives in the extended probability space.
• The method generalizes to “higher-order” or “higher-rank” infinitesimal structures by considering larger upper-triangular matrix algebras.

This operator-valued machinery leads to systematic derivations of moment-cumulant formulas and subordination equations for infinitesimal laws and reveals that all features of infinitesimal freeness can be read from the off-diagonal entries of these matrix-valued transforms.

7. Applications and Impact in Random Matrix Theory

Infinitesimal freeness, its conditional and operator-valued variants, and related higher-order independence types underpin the $1/N$ expansion phenomena in unitarily invariant random matrix models. These concepts are directly responsible for:

• Explaining the convergence of empirical spectral distributions (free convolution) and eigenvector spectral distributions (conditional free convolution) in sums of unitarily invariant ensembles (Cébron et al., 2022).
• Quantifying and predicting the location and behavior of outlier eigenvalues, such as the BBP phase transition for low-rank perturbations of GUE matrices. The emergence and precise location of an outlier are consequences of asymptotic conditional/infinitesimal freeness and can be calculated via associated convolution operations (monotone, conditional free) (Cébron et al., 2022).
• Organizing the combinatorial content of $1/N^2$ expansions via the Weingarten calculus, non-crossing partitions, and Motzkin-path combinatorics.

A plausible implication is that infinitesimal freeness provides the universal language and technical infrastructure for understanding all first-order corrections to free probability in random matrix asymptotics, and acts as a mediating concept for connecting free, Boolean, monotone, and c-free independence via deformation and cyclic constructions.

• "Freeness of type $B$ and conditional freeness for random matrices" (Cébron et al., 2022)
• "A construction which relates c-freeness to infinitesimal freeness" (Fevrier et al., 2018)
• "Asymptotic cyclic-conditional freeness of random matrices" (Cébron et al., 2022)
• "A Unified Approach to Infinitesimal Freeness with Amalgamation" (Tseng, 2019)
• "Infinitesimal moments in free and c-free probability and Motzkin paths" (Lenczewski, 27 Dec 2025)