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Lévy-Khintchine random matrices and the Poisson weighted infinite skeleton tree


Author: Paul Jung
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 15B52, 60B20, 60G51
DOI: https://doi.org/10.1090/tran/6977
Published electronically: July 7, 2017
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Abstract: We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdős-Rényi random graphs with $ p_n\sim \frac 1 n$. Our $ n\times n$ random matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on $ n$, and we require row sums to converge in distribution. It is then well-known that the limit distribution must be infinitely divisible.

We show that a limiting empirical spectral distribution (LSD) exists and, via local weak convergence of associated graphs, that the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges called ``cords to infinity''. One example covered by the results are matrices with i.i.d. entries having infinite second moments but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is $ \mathbb{N}$ rooted at $ 1$, so the LSD is the semicircle law.


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Additional Information

Paul Jung
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea
Email: paul.jung@gmail.com

DOI: https://doi.org/10.1090/tran/6977
Keywords: Empirical spectral distribution, Wigner matrices, L\'evy matrices, heavy-tailed random matrices, sparse random matrices, Erd\H{o}s-R\'enyi graph, local weak convergence.
Received by editor(s): August 4, 2014
Received by editor(s) in revised form: March 12, 2015, February 13, 2016, and April 18, 2016
Published electronically: July 7, 2017
Additional Notes: The author’s research was partially supported by NSA grant H98230-14-1-0144 while at the University of Alabama Birmingham.
Article copyright: © Copyright 2017 American Mathematical Society