Cumulant Recursion Relations

• Cumulant recursion relations are formal equations that express higher-order cumulants in terms of lower-order ones or moments, enabling systematic decomposition of complex correlation structures.
• They leverage combinatorial methods such as partitions, trees, and Bell polynomials to translate observable moments into additive invariants useful in various analytical frameworks.
• Applications of these relations span random matrix theory, quantum field theory, and stochastic processes, providing the basis for renormalization, resummation, and analytic expansions.

A cumulant recursion relation is a formal, combinatorial, or algebraic equation expressing cumulants (or structurally analogous quantities) in terms of lower-order cumulants, moments, or other related functionals. Such recursive relations underpin the systematic decomposition of high-order correlations in probability theory, noncommutative probability, random matrix theory, algebraic combinatorics, and field theory. Their general structure is shaped by the underlying independence structure (classical, free, Boolean, monotone), the combinatorial lattice (partitions, noncrossing partitions), and the analytical or algebraic setting (Hopf algebras, pre-Lie algebras, forest formulas). Cumulant recursion is essential for the induction and computation of cumulants, the translation of observable quantities (moments) into additive invariants (cumulants), and the design of resummation, expansion, or renormalization schemes.

1. Cumulant Recursion in Probability and Noncommutative Settings

The classic context is the moment-cumulant relation, expressing the $n$-th order cumulant $K_n$ as a function of all moments $m_1,\ldots,m_n$ via the Möbius inversion on the partition lattice: $$K_n = \sum_{\pi\in \mathcal{P}_n} \mu(\pi,1_n) \prod_{B\in \pi} m_{|B|}$$ where $\mu(\cdot,\cdot)$ denotes the Möbius function on partitions. In noncommutative probability—the free, Boolean, monotone, and delta ($\Delta$) settings—the classical lattice is replaced by noncrossing or interval partitions, and the corresponding cumulative objects (free cumulants $R_n$, Boolean cumulants $B_n$, monotone cumulants $H_n$, Delta cumulants $C_n$) obey recursions indexed by these respective combinatorial structures.

For free cumulants, the recursion leverages noncrossing partitions: $$m_n = \sum_{\pi\in NC_n} \prod_{B\in \pi} R_{|B|} \quad \text{and} \quad R_n = \sum_{\pi\in NC_n} m_n \cdot \mu(\pi,1_n)$$ Free nested cumulant formulas generalize classical Brillinger's formula (law of total cumulance) by replacing set partitions with noncrossing partitions and constructing nested (noncommutative) products (Lehner, 2013).

Monotone and Boolean cumulants, as well as their interrelations, follow analogous but lattice-specific recursions and Möbius inversions (Arizmendi et al., 2014, Ebrahimi-Fard et al., 2017, Celestino et al., 2020).

2. Combinatorial and Algebraic Structures

Combinatorics governs cumulant recursion at every level. The primary objects are:

• Partitions/Noncrossing partitions: Set partitions in classical probability, noncrossing partitions in free and monotone theories, interval partitions for Boolean cumulants.
• Forests and trees: Tree expansions appear in forest formulas and are central for recursive expansions in pre-Lie algebraic frameworks (Celestino et al., 2022).
• Heaps, pyramids, Tutte polynomials: These enumerate specific partition contributions, determine polynomial weights, and are critical in expressing cumulant coefficients (Arizmendi et al., 2014).
• Bell polynomials: Bell polynomials encode the combinatorial essence of the recursion for both classical and generalized cumulants. In applications to martingales and Hawkes processes, cumulants of high-order increments are precisely governed by these objects (Fukasawa et al., 2020, Privault, 2020).

The recursive expansions for cumulants are often achieved via forest formulas: sums over labeled trees or forests, each weighted by symmetry factors (e.g., Connes–Moscovici or Murua coefficients), and with combinatorial rules for nestings, orderings, or colorings (Celestino et al., 2022, Burchardt, 2018).

3. Hopf Algebra and Magnus Expansion Formulations

The combinatorial Hopf algebraic framework treats cumulants as infinitesimal characters—linear functionals vanishing on the unit and concatenations, with moment-cumulant recursion encoded as logarithms (via shuffle exponentials) of the moment character: $$\phi = \exp^*(p) \quad (\text{monotone}),\quad \phi = \exp^{<}(K) \quad (\text{free}), \quad \phi = \epsilon + B > \phi \quad (\text{Boolean})$$ Half-shuffle exponentials and Magnus expansion provide the underlying algebraic machinery for relating monotone cumulants to free or Boolean cumulants: $$p = \Omega(K),\quad B = -W(-\Omega(K)),\quad \rho = \Omega'(\kappa)$$ where $\Omega$ and $W$ represent appropriate (pre-)Lie algebraic operators; such expansions organize all recursion relations for moments and cumulants (Ebrahimi-Fard et al., 2017, Celestino et al., 2020, Celestino et al., 2022).

4. Functional Recursion and Topological Recursion

In higher-order free probability, generating series for moments and cumulants satisfy exquisitely structured functional relations. For instance, second-order relations invoke “R-transform–like” equations connecting moment and cumulant generating series (Borot et al., 2021): $$M_2(1/x_1,1/x_2)/(x_1x_2) = \frac{dw_1}{dx_1}\frac{dw_2}{dx_2} \left[\frac{C_2(w_1,w_2)}{w_1w_2} + \frac{1}{(w_1-w_2)^2}\right] - \frac{1}{(x_1-x_2)^2}$$ with $w_i = M(1/x_i)/x_i$. Generalized to arbitrary order, these relations encode combinatorics of partitioned permutations, bicoloured trees, and convolution operators manifesting as graph sums (Borot et al., 2021, Hock, 2022, Hock, 2023).

Topological recursion further illuminates cumulant relations by connecting moments and cumulants via symplectic transformations (exchange of spectral curve coordinates $x$ and $y$), leading to universal operator–graph formulae. Laplace transforms of these relations convert differential recursion into exponential generating functions, sharply mirroring classical cumulant decomposition (Hock, 2022, Hock, 2023).

5. Recursive Expansions in Constructive Field Theory

Cumulant recursion appears in explicit expansions of the free energy and cumulants for random matrix models with polynomial interactions, especially via the Loop Vertex Representation (LVR). The expansion involves:

1. Organizing cumulant derivatives of the partition function into sums over labeled combinatorial maps (ribbon graphs with cilia) (Rivasseau, 2023).
2. Further reorganizing into tree (forest) expansions with analyticity in a cardioid domain for the coupling parameter and Borel–LeRoy summability at the origin, managed via remainder bounds with controlled factorial growth.
3. Employing forest formulas and holomorphic functional calculus, e.g., using the generating function of Fuss–Catalan numbers, $T_p(z)$, to linearize the interaction and write recursion relations for cumulants through matrix functions and resolvent integrals.

Explicitly, for matrix models with $$S(M,M^\dagger)=\operatorname{Tr}\big[M M^\dagger + \lambda (M M^\dagger)^p\big]$$,

$A(\lambda,X) = X T_p(-\lambda X^{p-1})$

and the cumulant expansion respects convergent structure, analytic domains, and summability via Borel transforms: $$F(\lambda) = \frac{1}{(p-1)\lambda} \int_0^\infty B(t) \left(\frac{t}{\lambda}\right)^{\frac{1}{p-1}-1} \exp\left[-\left(\frac{t}{\lambda}\right)^{1/(p-1)}\right]dt$$ with each expansion term recursively built from lower-order contributions and the combinatorial graph structure (Rivasseau, 2023).

6. Applications and Key Implications

Cumulant recursion relations have broad implications:

• Noncommutative probability: Characterization of independence, computation of distributional invariants, correspondence between moments and cumulants.
• Random matrix theory: Computation of higher-order fluctuations, explicit expansion of free energies and correlators, construction and analysis of stable models.
• Algebraic combinatorics: Enumeration via forest and tree expansions, computation of structure constants for Jack characters.
• Stochastic processes: Recursive computation of realized cumulants for martingales and Hawkes processes, with the aggregation property encoded by Bell polynomials (Fukasawa et al., 2020, Privault, 2020).
• Quantum field theory and numerical analysis: Renormalization via Zimmermann’s forest formula, connection to Runge–Kutta and Magnus methods.

These results underscore the universality of cumulant recursion for decomposing complex dependence structures and for inductive, combinatorial, and analytic methods of computation.

7. Representative Recursion Formulas and Tables

Cumulant Type	Partition Lattice	Recursion Formula / Expansion
Classical $K_n$	All partitions	$$K_n = \sum_{\pi} \mu(\pi,1_n) \prod_{B\in \pi} m_{|B|}$$
Free $R_n$	Noncrossing	$m_n = \sum_{\pi} \prod_{B\in \pi} R_{|B|}$
Boolean $B_n$	Interval	$m_n = \sum_{\pi} \prod_{B\in \pi} B_{|B|}$
Monotone $H_n$	NC, weighted trees	$B_n = \sum_{\pi} (-1)^{|\pi|-1}/T(\pi)! H_\pi$

For the forest/Magnus framework: $$p = \Omega(K), \quad B = -W(-\Omega(K)), \quad (\exp(a)|w_i) = \sum_T \frac{1}{T!} Q_T(w_i)$$ where $Q_T$ accounts for the combinatorial composition from each tree $T$.

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Cumulant recursion relations thus constitute a unifying principle and a computational backbone for advanced calculations in modern probability, combinatorics, field theory, and matrix models, systematically disentangling higher-order structures through combinatorial and algebraic recursion.