2019/12/20 by Arizmendi, Octavio, Mingo, James A.
#46L54 (15B52 46L53) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.1912.09873
We introduce R-diagonal and even operators of second order. We give a formula for the second order free cumulants of the square x2 of a second order even element in terms of the second order free cumulants of x. Similar formulas are proved for the second order free cumulants of aa^*, when a is a second order R-diagonal operator. We also show that if r is second order R-diagonal and b is second order free from r, then rb is also second order R-diagonal. We present a large number of examples, in particular the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.