On the anti-commutator of two free random variables
Daniel Perales
TL;DR
This work develops a graph- and cactus-based combinatorial framework to compute the free cumulants of the anti-commutator ab+ba for freely independent variables a and b. The central result expresses κ_n(ab+ba) as a sum over partitions indexed by X_{2n}, with a complementary description via Kreweras, and a parallel cactus-graph formulation that introduces oriented cacti with a 2^f_C weighting. The approach extends naturally to quadratic forms in k free variables, yielding k-colored oriented cacti formulas, and provides concrete applications to the case of two free Poisson(1) variables and to even elements, including connections to Catalan numbers and to known results for symmetric distributions. Altogether, the framework unifies combinatorial, graph-theoretic, and analytic aspects of cumulants for quadratic forms in free probability. The results yield both exact cumulant expressions and generating-function recursions that illuminate the distributional behavior of ab+ba and its moments.
Abstract
Let $(κ_n(a))_{n\geq 1}$ denote the sequence of free cumulants of a random variable $a$ in a non-commutative probability space $(\mathcal{A},\varphi)$. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants $(κ_n(ab+ba))_{n\geq 1}$ in terms of $(κ_n(a))_{n\geq 1}$ and $(κ_n(b))_{n\geq 1}$, where $a$ and $b$ are freely independent. Our formula expresses the $n$-th free cumulant of $ab+ba$ as a sum indexed by partitions in the set $\mathcal{Y}_{2n}$ of non-crossing partitions of the form \[ σ=\{B_1,B_3,\dots, B_{2n-1},E_1,\dots,E_r\}, \quad \text{with }r\geq 0, \] such that $i\in B_{i}$ for $i=1,3,\dots,2n-1$ and $|E_j|$ even for $j\leq r$. Therefore, by studying the sets $\mathcal{Y}_{2n}$ we obtain new results regarding the distribution of $ab+ba$. For instance, the size $|\mathcal{Y}_{2n}|$ is closely related to the case when $a,b$ are free Poisson random variables of parameter 1. Our formula can also be expressed in terms of cacti graphs. This graph theoretic approach suggests a natural generalization that allows us to study quadratic forms in $k$ free random variables.