Open Access
August 2020 Local law and Tracy–Widom limit for sparse stochastic block models
Jong Yun Hwang, Ji Oon Lee, Wooseok Yang
Bernoulli 26(3): 2400-2435 (August 2020). DOI: 10.3150/20-BEJ1201


We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy–Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.


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Jong Yun Hwang. Ji Oon Lee. Wooseok Yang. "Local law and Tracy–Widom limit for sparse stochastic block models." Bernoulli 26 (3) 2400 - 2435, August 2020.


Received: 1 September 2019; Revised: 1 January 2020; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193965
MathSciNet: MR4091114
Digital Object Identifier: 10.3150/20-BEJ1201

Keywords: Local law , random matrices , Stochastic block model , Tracy–Widom distribution

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 3 • August 2020
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