Infinitesimal Conditional Freeness

• Infinitesimal conditional freeness is a noncommutative probability framework that unifies c-freeness and infinitesimal freeness, establishing novel cumulant relations and independence structures.
• It employs combinatorial tools like Motzkin paths and cyclic-conditional cumulants to derive second-order asymptotics in random matrix models, with applications in eigenvector statistics and BBP transitions.
• The framework bridges various noncommutative independence notions by providing explicit moment-cumulant formulas and transformation rules, enhancing the analysis of spectral fluctuations.

Infinitesimal conditional freeness is a generalization within noncommutative probability theory that unifies c-freeness (conditional freeness) and infinitesimal freeness, and is critical for the study of random matrices subject to fine asymptotic expansions. It provides a framework for formulating second-order asymptotics and describes independence structures that interpolate between free, Boolean, monotone, cyclic-monotone, and conditional types of noncommutative independence. The notion emerges naturally in contexts such as the Vortex model, describing matrix ensembles whose symmetry is reduced by conditioning on a preserved vector, and is characterized by new moment-cumulant relations and combinatorial rules (Cébron et al., 2022, Lenczewski, 27 Dec 2025, Fevrier et al., 2018, Cébron et al., 2022).

1. Fundamental Definitions and Conceptual Framework

Infinitesimal conditional freeness involves tuples of unital functionals on a unital algebra $\mathcal{A}$ over $\mathbb{C}$. Given functionals $\psi,\,\varphi: \mathcal{A} \to \mathbb{C}$ and a tracial $\varphi'$, a family of unital subalgebras $\{\mathcal{A}_i\}$ is cyclically conditionally free (cyclic c-free) with respect to $(\psi, \varphi, \varphi')$ if:

• Ordinary c-freeness: For alternating sequences $a_1\cdots a_n$ with $a_j \in \mathcal{A}_{i_j}$, adjacent $a_j$ from different $\mathcal{A}_i$, and each $\psi(a_j)=0$: $\psi(a_1\cdots a_n)=0$ and $$\varphi(a_1\cdots a_n)=\varphi(a_1)\cdots\varphi(a_n)$$.
• Infinitesimal/cyclic condition: For cyclically alternating sequences (additionally, $a_n,a_1$ are from different algebras), $\psi$-centered: $$\varphi'(a_1\cdots a_n) = \varphi(a_1)\cdots\varphi(a_n)$$.

This structure admits a unique free-product operation on $(\psi,\varphi,\varphi')$-triples, characterizable via mixed-moment vanishing and multiplicativity conditions (Cébron et al., 2022).

2. Cumulant and Moment Relations

Infinitesimal conditional freeness requires two cumulant families: the usual c-free cumulants $\kappa^0_n$ for $(\psi,\varphi)$, and cyclic-conditional cumulants $\kappa'_n$ for the infinitesimal part. Their relations are:

• c-free cumulants:

$$\varphi(a_1\cdots a_n) = \sum_{\pi \in NC(n)} \prod_{V \in \pi} \kappa^0_{|V|}(a_V)$$

• Infinitesimal conditional cumulants:

$$\varphi'(a_1\cdots a_n) = \sum_{\pi\in NC(n)} \prod_{V\in\pi} \begin{cases} \kappa'_{|V|}(a_V) & \text{if } V \text{ meets exterior cycle} \ \kappa^0_{|V|}(a_V) & \text{otherwise} \end{cases}$$

A block $V$ meets the exterior cycle if it contains both some index $i$ and its cyclic successor $i+1$ modulo $n$ (Cébron et al., 2022). For $n=2$, $$\kappa'_2(a,b) = \varphi'(ab) - \varphi(a)\varphi(b)$$. For higher $n$, only "cyclically crossing" blocks (that interact with the exterior cycle) contribute nontrivially.

3. Motzkin Path Decompositions and Leibniz-Type Rules

A combinatorial approach utilizes Motzkin paths to give explicit moment decompositions. In this context (Lenczewski, 27 Dec 2025):

• Pyramid paths (one local maximum) correspond to alternating moments in infinitesimal freeness: the moment derivative vanishes unless the path is a pyramid.
• Flat paths correspond to infinitesimal Boolean independence.
• Infinitesimal c-free moments: Only concatenations of a pyramid (for the first variables) and a flat path (for the remainder) have nonzero first-order derivatives.

For subalgebras $\{\mathcal{A}_i\}$ and functionals $(\varphi,\varphi';\,\psi,\psi')$, infinitesimal c-freeness holds if: $$\varphi'(a_1\cdots a_n) = \varphi'(a_1)\varphi(a_2\cdots a_n) + \sum_{m=2}^n \psi'(a_m)\varphi(a_1\cdots a_{m-1}a_{m+1}\cdots a_n)$$ with $a_1\in\ker\varphi$, $a_{2\ldots n}\in\ker\psi$ and indices alternating (Lenczewski, 27 Dec 2025). This interprets infinitesimal c-free moments as a Leibniz sum over derivatives of subalgebra states.

4. Random Matrix Ensembles and Asymptotic Conditional Freeness

The operational context for infinitesimal conditional freeness is provided by large $N$ random matrix models, particularly the Vortex model (Cébron et al., 2022, Cébron et al., 2022):

• Consider deterministic $N\times N$ matrices $A_N,\,B_N$ and a unit vector $v_N$. Let $U_N$ be Haar-distributed on the subgroup fixing $v_N$.
• As $N\to\infty$, $A_N$ and $U_N B_N U_N^*$ are asymptotically c-free with respect to $(tr_N,\varphi^{v_N})$ (normalized trace, vector state).
• The first-order asymptotic expansion for polynomial traces yields a cyclically c-free structure:

$$\mathbb{E}[tr_N(P(A_N,U_NB_NU_N^*))] = \psi_A*\psi_B(P) +\frac1N\,(\omega_A\circledast_{\psi_B}^{\phi_B}\omega_B)(P) +O(N^{-2})$$

This demonstrates convergence to conditional freeness with an explicit infinitesimal correction, encoding higher-order fluctuation effects in the spectra and eigenvectors (Cébron et al., 2022).

5. Abstract Correspondences and Cumulant Transformations

A deeper relation between c-freeness and infinitesimal freeness is available via the transformation $\Delta^*$ on families of cumulants. In the abstract vector-space setting (Fevrier et al., 2018):

• Given a vector space $V$ and a coproduct $\Delta:V\to V\otimes V$, for any family $\phi=(\phi_n)$ of multilinear functionals:

$$(\Delta^*\phi)_n(v_1,\ldots,v_n) = \sum_{m=1}^n \phi_{n+1}(v_1,\ldots,v_{m-1},\,\Delta(v_m),\,v_{m+1},\ldots,v_n)$$

• The central theorem:

$\Delta^*(K^{(c)}) = K'$

where $K^{(c)}$ are the c-free cumulants and $K'$ the infinitesimal cumulants. This machinery facilitates the translation between additive convolutions and free products in both frameworks.

6. Connections with Other Noncommutative Independences

Infinitesimal conditional freeness interpolates between various independence notions:

Framework	Characterization	Correspondence
Classical c-freeness	$(\psi,\varphi)$ only	Ordinary c-free cumulants, no infinitesimal corrections
Infinitesimal freeness	$(\varphi,\varphi')$	Special case: $\psi=\varphi$ in cyclic c-freeness
Cyclic-Boolean independence	$\psi=\delta$ ("augmentation" state)	Recovers cyclic-Boolean rules
Cyclic-monotone independence	$\psi=\delta$ (one side), $\psi=\varphi$	Moment-cumulant rules specialize accordingly

A plausible implication is that the unified cumulant approach enables combinatorial limit theorems for non-crossing partitions with cycles and facilitates the construction of random matrix models encoding ordered or indented independence (Cébron et al., 2022, Lenczewski, 27 Dec 2025).

7. Representative Examples and Applications

• Eigenvector statistics: The empirical spectral distribution of the sum $A_N+U_NB_NU_N^*$ converges to the c-free convolution in both the bulk (trace) law and the vector state law. Infinitesimal corrections manifest in first-order adjustments to the Cauchy transform of the limiting measure (Cébron et al., 2022).
• BBP transition: Outlier eigenvalues for rank-one deformations are explained by asymptotic conditional freeness, connecting to type B freeness and analytic subordination in free probability.
• Moment calculus: For polynomials in cyclically c-free variables, computation of the infinitesimal law via cyclic-conditional cumulants enables explicit formulas for low-degree moments. This suggests potential for extended combinatorial spectral analysis in random matrix theory (Cébron et al., 2022, Lenczewski, 27 Dec 2025).

In summary, infinitesimal conditional freeness introduces a robust and unifying mechanism for handling higher-order spectral fluctuations in noncommutative probability and random matrix ensembles, with combinatorial, analytic, and transformative correspondences to a spectrum of independence notions.