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

References in corpus (3)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Functional limit theorems for random regular graphs

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• 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