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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusEmpirical Spectral Distributions of Sparse Random Graphs

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Empirical Spectral Distributions of... [We study the spectrum of a random multigraph with a degree sequence Dn=(Di)i=1n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbf {D}}_n=(D_i)_{i=1}^n$$ \end{document} and average degree 1 ≪ ωn ≪ 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 ωn−1Dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\omega _n^{-1} {\mathbf {D}}_n $$ \end{document} converges weakly to a limit ν, under mild moment assumptions (e.g., Di∕ωn are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to ν⊠σSC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\nu \boxtimes \sigma _{{\text{SC}}}$$ \end{document}, 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusEmpirical Spectral Distributions of Sparse Random Graphs

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.

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References (21)

Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-60753-1
Pages
319 –345
DOI
10.1007/978-3-030-60754-8_15
Publisher site
See Chapter on Publisher Site

Abstract

[We study the spectrum of a random multigraph with a degree sequence Dn=(Di)i=1n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbf {D}}_n=(D_i)_{i=1}^n$$ \end{document} and average degree 1 ≪ ωn ≪ 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 ωn−1Dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\omega _n^{-1} {\mathbf {D}}_n $$ \end{document} converges weakly to a limit ν, under mild moment assumptions (e.g., Di∕ωn are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to ν⊠σSC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\nu \boxtimes \sigma _{{\text{SC}}}$$ \end{document}, 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.]

Published: Nov 4, 2020

Keywords: Random matrices; Empirical spectral distribution; Random graphs; 05C80; 60B20

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