Abstract
An elliptic random matrix X is a square matrix whose -entry is a random variable independent of every other entry except possibly . Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with independent entries. When the entries of an elliptic random matrix have mean zero and unit variance, the empirical spectral distribution is known to converge to the uniform distribution on the interior of an ellipse determined by the covariance of the mirrored entries.
We consider elliptic random matrices whose entries fail to have two finite moments. Our main result shows that when the entries of an elliptic random matrix are in the domain of attraction of an α-stable random variable, for , the empirical spectral measure converges, in probability, to a deterministic limit. This generalizes a result of Bordenave, Caputo, and Chafaï for heavy-tailed matrices with independent and identically distributed entries. The key elements of the proof are (i) a general bound on the least singular value of elliptic random matrices under no moment assumptions; and (ii) the convergence, in an appropriate sense, of the matrices to a random operator on the Poisson Weighted Infinite Tree.
Funding Statement
A. Campbell has been supported in part by NSF grant ECCS-1610003. S. O’Rourke has been supported in part by NSF grants DMS-1810500 and ECCS-1610003.
Acknowledgments
The second author thanks Djalil Chafaï for answering his questions concerning [16, Appendix A]. We thank the anonymous referees for thorough readings and helpful comments.
Citation
Andrew Campbell. Sean O’Rourke. "Spectrum of heavy-tailed elliptic random matrices." Electron. J. Probab. 27 1 - 56, 2022. https://doi.org/10.1214/22-EJP849
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