Access the full text.
Sign up today, get an introductory month for just $19.
W. Hachem, A. Hardy, J. Najim (2014)
Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edgesarXiv: Probability
(2017)
Spectrum of random graphs. In Advanced topics in random matrices, volume 53 of Panor. Synthèses, pages 91–150
C. Bordenave, M. Lelarge (2007)
Resolvent of large random graphsRandom Structures & Algorithms, 37
Simon Coste, J. Salez (2018)
Emergence of extended states at zero in the spectrum of sparse random graphsThe Annals of Probability
J. Salez (2016)
Spectral atoms of unimodular random treesJournal of the European Mathematical Society
(1981)
Constrained switchings in graphs. In Combinatorial mathematics, VIII (Geelong, 1980), volume 884 of Lecture Notes in Math., pages 314–336
E-mail address: amir@math.stanford.edu Eyal Lubetzky Courant Institute
R. Taylor (1981)
Contrained switchings in graphs
L. Tran, V. Vu, Ke Wang (2010)
Sparse random graphs: Eigenvalues and eigenvectorsRandom Structures & Algorithms, 42
N. Rao, R. Speicher (2007)
Multiplication of free random variables and the S-transform: the case of vanishing meanArXiv, abs/0706.0323
R. Arratia, T. Liggett (2005)
How likely is an i.i.d. degree sequence to be graphicalAnnals of Applied Probability, 15
J. Silverstein, S. Choi (1995)
Analysis of the limiting spectral distribution of large dimensional random matricesJournal of Multivariate Analysis, 54
C. Tracy, H. Widom (1992)
Introduction to Random Matrices
(1987)
Multiplication of certain noncommuting random variables
J. Silverstein, Z. Bai (1995)
On the empirical distribution of eigenvalues of a class of large dimensional random matricesJournal of Multivariate Analysis, 54
Ioana Dumitriu, Soumik Pal (2009)
Sparse regular random graphs: Spectral density and eigenvectorsAnnals of Probability, 40
T. Tao (2012)
Topics in Random Matrix Theory
(1960)
Graphs with prescribed degrees of vertices
E. OctavioArizmendi, V. Pérez-Abreu (2009)
The S-transform of symmetric probability measures with unbounded supports, 137
V. Marčenko, L. Pastur (1967)
DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICESMathematics of The Ussr-sbornik, 1
Catherine Greenhill, Matteo Sfragara (2017)
The switch Markov chain for sampling irregular graphs and digraphsTheor. Comput. Sci., 719
[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
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get an introductory month for just $19.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.