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Empirical Spectral Distributions of Sparse Random Graphs

Part of the Progress in Probability book series (PRPR,volume 77)

Abstract

We study the spectrum of a random multigraph with a degree sequence \({\mathbf {D}}_n=(D_i)_{i=1}^n\) and average degree 1 ≪ ω n ≪ n, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of \(\omega _n^{-1} {\mathbf {D}}_n \) converges weakly to a limit ν, under mild moment assumptions (e.g., D iω n are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to \(\nu \boxtimes \sigma _{{\text{SC}}}\), the free multiplicative convolution of ν with the semicircle law. Relating this limit with a variant of the Marchenko–Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous Erdős-Rényi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.

Keywords

  • Random matrices
  • Empirical spectral distribution
  • Random graphs

MSC Subject Class

  • 05C80
  • 60B20

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Acknowledgements

The authors wish to thank Nick Cook, Alice Guionnet, Allan Sly and Ofer Zeitouni for many helpful discussions. A.D. was supported in part by NSF grant DMS-1613091. E.L. was supported in part by NSF grants DMS-1513403 and DMS-1812095.

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Correspondence to Eyal Lubetzky .

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Dembo, A., Lubetzky, E., Zhang, Y. (2021). Empirical Spectral Distributions of Sparse Random Graphs. In: Vares, M.E., Fernández, R., Fontes, L.R., Newman, C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius. Progress in Probability, vol 77. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60754-8_15

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