The limiting distributions of large heavy Wigner and arbitrary random matrices

Abstract : 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 X_N of independent heavy Wigner matrices and a family Y_N of arbitrary random matrices, independent of X_N, with a technical condition (e.g. the matrices of Y_N are deterministic and uniformly bounded in operator norm, or are deterministic diagonal). We characterize the possible limiting joint *-distributions of (X_N,Y_N) in the sense of free probability. We find that they depend on more than the *-distribution of Y_N. We use the notion of distributions of traffics and their free product to quantify the information needed on Y_N and to infer the limiting distribution of (X_N,Y_N). We give an explicit combinatorial formula for joint moments of heavy Wigner and independent random matrices. When the matrices of Y_N are diagonal, we give recursion formulas for these moments. We deduce a new characterization of the limiting eigenvalues distribution of a single heavy Wigner.
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Contributor : Camille Male <>
Submitted on : Wednesday, September 19, 2012 - 3:18:37 PM
Last modification on : Saturday, May 25, 2019 - 1:32:41 AM

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  • HAL Id : hal-00733792, version 1
  • ARXIV : 1209.2366


Camille Male. The limiting distributions of large heavy Wigner and arbitrary random matrices. Journal of Functional Analysis, Elsevier, 2016, 271 (1), pp.46. ⟨hal-00733792⟩



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