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Sparse random matrices: spectral edge and statistics of rooted trees

Part of: Graph theory Limit theorems

Published online by Cambridge University Press:  01 July 2016

A. Khorunzhy
A. Khorunzhy*
Affiliation:
Institute for Low Temperature Physics, Ukraine
*
∗ Current address: Faculté de Mathematiques, Université Paris 7–Denis Diderot, 2 Place Jussieu, Paris 75251 Cedex 05, France. Email address: khorunjy@ilt.kharkov.ua
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Abstract

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices AN,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥AN,p∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥AN,p/√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of AN,p. They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

Keywords

random matricesspectral normuniversality conjectureuniversality conjecturerandom graphsErdos-Renyi partial sums

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 05C05: Trees
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 124 - 140
Copyright
Copyright © Applied Probability Trust 2001 

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