Modern aspects of random matrix theory : AMS Short Course, Random Matrices, January 6-7, 2013, San Diego, California

186 Pages·2014·2.02 MB·English

by  Van H. Vu

Volume 72
Modern Aspects of
Random Matrix Theory
AMS Short Course
Random Matrices
January 6–7, 2013
San Diego, California
Van H. Vu
Editor
AMS SHORT COURSE LECTURE NOTES
Introductory Survey Lectures
published as a subseries of
Proceedings of Symposia in Applied Mathematics

Volume 72
Modern Aspects of
Random Matrix Theory
AMS Short Course
Random Matrices
January 6–7, 2013
San Diego, California
Van H. Vu
Editor
EDITORIAL COMMITTEE
Mary Pugh (Chair) Daniel Rockmore
2010 Mathematics Subject Classification. Primary 11C20, 60B20, 15B52, 05D40, 60H25,
62-07.
Library of Congress Cataloging-in-Publication Data
AMSShortCourse,RandomMatrices(2013: SanDiego,Calif.)
Modern aspects of random matrix theory : AMS Short Course, Random Matrices, January
6–7,2013,SanDiego,California/VanH.Vu,editor.
pagescm. —(Proceedingsofsymposiainappliedmathematics;volume72)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-9471-2(alk.paper)
1.Randommatrices—Congresses. 2.Numbertheory—Congresses. I.Vu,Van,1970–editor
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Contents
Preface vii
Lecture notes on the circular law
Charles Bordenave and Djalil Chafa¨ı 1
Free probability and random matrices
Alice Guionnet 35
Random matrix theory, numerical computation and applications
Alan Edelman, Brian Sutton, and Yuyang Wang 53
Recent developments in non-asymptotic theory of random matrices
Mark Rudelson 83
Random matrices: The universality phenomenon for Wigner ensembles
Terence Tao and Van Vu 121
Index 173
v

Preface
The theory of random matrices is an amazingly rich topic in mathematics.
Beside being interesting in its own right, random matrices play a fundamental role
invariousareassuchasstatistics, mathematicalphysics, combinatorics, theoretical
computer science, number theory and numerical analysis, to mention a few. A
famousexampleistheworkofthephysicistEugeneWigner,whousedthespectrum
of random matrices to model energy levels of atoms, and consequently discovered
the fundamental semi-circle law which describes the limiting distribution of the
eigenvalues of a random hermitian matrix.
Specialrandommatricesmodelswheretheentriesareiidcomplexorrealgauss-
ian random variables (GUE, GOE or Wishart) have been studied in detail. How-
ever, much less was known about general models, as the above mentioned study
relies very heavily on properties of the gaussian distribution. In the last ten years
or so, we have witnessed considerable progresses on several fundamental problems
concerninggeneralmodels, suchastheCircularlawconjectureorUniversalitycon-
jectures. More importantly, these new results are proved using novel and robust
approaches which seem to be applicable to many other problems. Surprising con-
nectionstotheemergingfield offreeprobability have alsobeenmade andfortified.
Equally surprising is the discovery that many practical tricks for numerical prob-
lems (to make the computation of eigenvalues faster or more reliable, say) can also
be used as powerful theoretical tools to study spectral limits.
Another area where we see rapid progressions is the theory of computing and
applications (which includes numerical analysis, theoretical computer science, ma-
chine learning and data analysis). Here properties of random matrices have been
used for the purpose of designing and analyzing practical algorithms. As already
realized by von Neumann and Goldstine at the dawn of the computer era, bounds
on the condition number of large random matrices would play a central role in a
vast number of numerical problems. Their questions were posed 70 years ago, but
effective ways to estimate this number have only been found in recent years. As
a model for random noise/error, random matrices enter all problems concerning
large data, perhaps one of the most talked about subjects in applied science in
recent years. Today, random matrices are studied not only for their own mathe-
matical beauty, but also for a very real purpose of making digital images sharper
or computer networks more reliable. These new goals have motivated new lines of
research, such as non-asymptotic or large deviation theory for random matrices.
This volume contains surveys by leading researchers in the field, written in
introductory style to quickly provide a broad picture about this fascinating and
vii
viii PREFACE
rapidly developing topic. We aim totouchmost of the key points mentioned above
(andmany more)without puttingtoomuch technicalburden onthereaders. Most
ofthesurveysareaccessiblewithbasicknowledgeinprobabilityandlinearalgebra.
We also made an attempt to discuss a considerable amount of open problems.
Some of these are classical but many are new, motivated by current developments.
These problems may serve as a guideline for future research, especially for young
researchers who would like to study this wonderful subject.
Van H. Vu
New Haven, Fall 2013.
ProceedingsofSymposiainAppliedMathematics
Volume72,2014
http://dx.doi.org/10.1090/psapm/072/00617
Lecture notes on the circular law
Charles Bordenave and Djalil Chafa¨ı
Abstract. Thecircularlawtheoremstatesthattheempiricalspectraldistri-
butionofan×nrandommatrixwithi.i.d.entriesofvariance1/ntendstothe
uniform law on the unit disc of the complex plane as the dimension n tends
to infinity. This phenomenon is the non-Hermitian counterpart of the semi
circularlimitforWignerrandomHermitianmatrices,andthequartercircular
limit for Marchenko-Pastur random covariance matrices. In these expository
notes, we present a proof in a Gaussian case, due to Mehta and Silverstein,
basedonaformulabyGinibre,andaproofoftheuniversalcasebyrevisiting
the approach of Tao and Vu, based on the Hermitization of Girko, the loga-
rithmicpotential,andthecontrolofthesmallsingularvalues. Wealsodiscuss
somerelatedmodelsandopenproblems.
These notes constitute an abridged and updated version of the probability
survey[BC],preparedattheoccasionoftheAmericanMathematicalSocietyshort
courseonRandomMatrices,organizedbyVanH.Vuforthe2013AMS-MAAJoint
Mathematics Meeting held in January 9–13 in San Diego, CA, USA.
Section1introducesthenotionofeigenvaluesandsingularvaluesanddiscusses
their relationships. Section 2 states the circular law theorem. Section 3 is devoted
totheGaussianmodelknownastheComplexGinibreEnsemble, forwhichthelaw
ofthespectrumisknownandleadstothecircularlaw. Section4providestheproof
of the circular law theorem in the universal case, using the approach of Tao and
Vu based on the Hermitization of Girko and the logarithmic potential. Section 5
gathers finally some few comments on related problems and models.
All random variables are defined on a unique common probability space
(Ω,A,P). An element of Ω is denoted ω. We write a.s., a.a., and a.e. for al-
most surely, Lebesgue almost all, and Lebesgue almost everywhere respectively.
1. Two kinds of spectra
The eigenvalues of a matrix A ∈ M (C) are the roots in C of its character-
n
istic polynomial P (z) := det(A−zI). We label them λ (A),...,λ (A) so that
A 1 n
|λ (A)| ≥ ··· ≥ |λ (A)| with growing phases. The spectral radius is |λ (A)|. The
1 n 1
eigenvaluesform the algebraic spectrum of A. The singular values of Aare defined
2010 MathematicsSubjectClassification. Primary15B52(60B20;60F15).
Key words and phrases. Spectrum, singular values, eigenvalues, random matrices, random
graphs,circularlaw,ginibreensemble,nonHermitianmatrices,nonnormalmatrices.
(cid:3)c2014 Bordenave and Chafa¨ı
1