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Probability Theory and Related Fields

, Volume 171, Issue 1–2, pp 543–616 | Cite as

Local law and Tracy–Widom limit for sparse random matrices

  • Ji Oon Lee
  • Kevin Schnelli
Article

Abstract

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(Np). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than \(N^{-2/3}\) with a deterministic shift of order \((Np)^{-1}\).

Keywords

Local law Sparse random matrices Erdős–Rényi graph 

Mathematics Subject Classification

60B20 62H10 

Notes

Acknowledgements

We thank László Erdős for useful comments and suggestions. Ji Oon Lee is grateful to the department of mathematics, University of Michigan, Ann Arbor, for their kind hospitality during the academic year 2014–2015.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.KAISTDaejeonSouth Korea
  2. 2.KTH Royal Institute of TechnologyStockholmSweden
  3. 3.IST AustriaKlosterneuburgAustria

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