Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

Abstract

Abstract. We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-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 the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

Local law and Tracy–Widom limit for sparse random matrices

“…Some of the negligible contributions can be identified by power counting, while others require further expansions using cumulant series and ideas inspired by the GOE computation in Remark 3.1 above. In Lemma 6.4, we will show that the remaining relevant terms stem from the term r = 3 and are, after further expansions, eventually identified to be s (4)…”

“…for arbitrary ℓ ′ ∈ N. By Corollary 6.2 and Remark 6.3, the two error terms E[Ω ℓ ′ (·)] in (6.65) and (6.66) are negligible for ℓ ′ ≥ 8D. With the extra factor N κ (4) t , we then write N κ…”

Bulk universality for generalized Wigner matrices with few moments

“…In particular, it suggests that m N (z) might converge to m sc at rate N −cε , instead of at the fastest possible rate N −1 , which was established by Götze-Tikhomirov [32] in the case when the fourth moments of h ij √ N are bounded. In view of the results of [8], we do not believe that (1.7) is optimal for small ε, when the h ij √ N only have (2 + ε) moments. Instead, we find it plausible that the error term N −cε in (1.7) should replaced by N −1/2−cε .…”

Empirical Spectral Distribution of a Matrix Under Perturbation

“…The following CLT extension lemma is borrowed from the paper of Shcherbina and Tirozzi [20]. We state here the version that can be found in the Appendix of [6].…”