Extreme eigenvalues of sparse, heavy tailed random matrices

2015/06/19 by Antonio Auffinger, Auffinger, Antonio, Si Tang +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1506.06175

openalex publication_date 2015/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the statistics of the largest eigenvalues of p × p sample covariance matrices Σ_p,n = M_p,nM_p,n^* when the entries of the p × n matrix M_p,n are sparse and have a distribution with tail t^-α, α>0. On average the number of nonzero entries of M_p,n is of order n^μ+1, 0 ≤ μ≤ 1. We prove that in the large n limit, the largest eigenvalues are Poissonian if α<2(1+μ^-1) and converge to a constant in the case α>2(1+μ^-1). We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.

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