On the anti-commutator of two free random variables

Abstract

Let pκnpaqqně1 denote the sequence of free cumulants of a random variable a in a non-commutative probability space pA, ϕq. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants pκnpab `baqqně1 in terms of pκnpaqqně1 and pκnpbqqně1, 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 Y2n of noncrossing partitions of the form σ " tB1, B3, . . . , B2n´1, E1, . . . , Eru, with r ě 0, such that i P Bi for i " 1, 3, . . . , 2n ´1 and |Ej| even for j ď r. Therefore, by studying the sets Y2n we obtain new results regarding the distribution of ab `ba. For instance, the size |Y2n| 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.

Boolean cumulants and subordination in free probability

“…Recently it was noticed that despite their simplicity Boolean cumulants are useful for noncommutative probability in general [18] and free probability in particular [5,14,23,37]. Boolean cumulants were used for the first combinatorial solution to the problem of the free anticommutator in [14] (an analytic solution was found earlier by Vasilchuk [41]; a solution in terms of free cumulants was presented recently in [32]), and for the identification of the coefficients of power series expansion of subordination functions [23] (implicitly also in [48]).…”