Central limit theorems for linear statistics of heavy tailed random matrices

Abstract

We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $\alpha$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

49 pages. In this fifth version, we have corrected a mistake in the fixed point equations for the limit covariance. To appear in Comm. Math. Phys

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• Gaussian fluctuations for linear spectral statistics of large random covariance matrices
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• Localization transition in random Levy matrices : multifractality of eigenvectors in the localized phase and at criticality
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Global eigenvalue fluctuations of random biregular bipartite graphs
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with heavy-tailed interaction