2012/09/11 by Male, Camille
#Combinatorics (math.CO) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.1209.2366
The model of heavy Wigner matrices generalizes the classical ensemble of Wigner matrices: the sub-diagonal entries are independent, identically distributed along to and out of the diagonal, and the moments its entries are of order 1/N, where N is the size of the matrices. Adjacency matrices of Erdös-Renyi sparse graphs and matrices with properly truncated heavy tailed entries are examples of heavy Wigner matrices. We consider a family XN of independent heavy Wigner matrices and a family YN of arbitrary random matrices, independent of XN, with a technical condition (e.g. the matrices of YN are deterministic and uniformly bounded in operator norm, or are deterministic diagonal). We characterize the possible limiting joint *-distributions of (XN,YN) in the sense of free probability. We find that they depend on more than the *-distribution of YN. We use the notion of distributions of traffics and their free product to quantify the information needed on YN and to infer the limiting distribution of (XN,YN). We give an explicit combinatorial formula for joint moments of heavy Wigner and independent random matrices. When the matrices of YN are diagonal, we give recursion formulas for these moments. We deduce a new characterization of the limiting eigenvalues distribution of a single heavy Wigner.