From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices

The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.

[1]  Kenneth B. Huber.  Department of Mathematics , 1894 .

[2]  J. Lindeberg.  Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung , 1922 .

[3]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[4]  J. Littlewood,et al.  On the Number of Real Roots of a Random Algebraic Equation , 1938 .

[5]  P. Erdös.  On a lemma of Littlewood and Offord , 1945 .

[6]  J. Neumann,et al.  Numerical inverting of matrices of high order , 1947 .

[7]  P. Moran,et al.  Mathematics of Statistics , 1948, Nature.

[8]  J. Neumann,et al.  Numerical inverting of matrices of high order. II , 1951 .

[9]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[10]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[11]  András Sárközy,et al.  Über ein Problem von Erdös und Moser , 1965 .

[12]  D. Kleitman.  On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors , 1970 .

[13]  L. Arnold,et al.  On Wigner's semicircle law for the eigenvalues of random matrices , 1971 .

[14]  L. Pastur.  On the spectrum of random matrices , 1972 .

[15]  V. Uppuluri,et al.  Asymptotic distribution of eigenvalues of random matrices , 1972 .

[16]  G. Halász.  Estimates for the concentration function of combinatorial number theory and probability , 1977 .

[17]  Richard P. Stanley,et al.  Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property , 1980, SIAM J. Algebraic Discret. Methods.

[18]  Gene H. Golub,et al.  Matrix computations , 1983 .

[19]  Jeffrey C. Lagarias,et al.  On the Tightest Packing of Sums of Vectors , 1983, Eur. J. Comb..

[20]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[21]  Zoltán Füredi,et al.  Solution of the Littlewood-Offord problem in high dimensions , 1988 .

[22]  E. Szemerédi,et al.  On the probability that a random ±1-matrix is singular , 1995 .

[23]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[24]  M. Talagrand.  A new look at independence , 1996 .

[25]  A. Edelman.  The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law , 1997 .

[26]  V. L. GIRKO.  Strong Circular Law , 1997 .

[27]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[28]  V. Girko.  A Refinement of the Central Limit Theorem for Random Determinants , 1998 .

[29]  Shang-Hua Teng,et al.  Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time , 2001, STOC '01.

[30]  D. Spielman,et al.  Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time , 2004 .

[31]  The Strong Circular Law. Twenty years later. Part II , 2004 .

[32]  T. Tao,et al.  Inverse Littlewood-Offord theorems and the condition number of random discrete matrices , 2005, math/0511215.

[33]  Daniel A. Spielman.  The Smoothed Analysis of Algorithms , 2005, FCT.

[34]  S. Chatterjee.  A simple invariance theorem , 2005, math/0508213.

[35]  T. Tao,et al.  On the singularity probability of random Bernoulli matrices , 2005, math/0501313.

[36]  M. Rudelson.  Invertibility of random matrices: norm of the inverse , 2005, math/0507024.

[37]  Terence Tao,et al.  On random ±1 matrices: Singularity and determinant , 2006, Random Struct. Algorithms.

[38]  M. Rudelson,et al.  Smallest singular value of random matrices and geometry of random polytopes , 2005 .

[39]  Kevin P. Costello,et al.  Random symmetric matrices are almost surely nonsingular , 2005, math/0505156.

[40]  D. Spielman,et al.  Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices , 2003, SIAM Journal on Matrix Analysis and Applications.

[41]  Additive Combinatorics: Contents , 2006 .

[42]  P. Deift.  Universality for mathematical and physical systems , 2006, math-ph/0603038.

[43]  T. Tao,et al.  On random ±1 matrices: Singularity and determinant , 2006 .

[44]  T. Tao,et al.  RANDOM MATRICES: THE CIRCULAR LAW , 2007, 0708.2895.

[45]  M. Rudelson,et al.  The Littlewood-Offord problem and invertibility of random matrices , 2007, math/0703503.

[46]  G. Freiman.  Foundations of a Structural Theory of Set Addition , 2007 .

[47]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[48]  Terence Tao,et al.  The condition number of a randomly perturbed matrix , 2007, STOC '07.

[49]  Golub Gene H. Et.Al.  Matrix Computations, 3rd Edition , 2007 .

[50]  F. Gotze,et al.  On the Circular Law , 2007, math/0702386.

[51]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices , 2007 .

[52]  M. Rudelson,et al.  The least singular value of a random square matrix is O(n−1/2) , 2008, 0805.3407.

[53]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[54]  Van H. Vu,et al.  Concentration of Random Determinants and Permanent Estimators , 2009, SIAM J. Discret. Math..

[55]  Alexander Tikhomirov,et al.  The circular law for random matrices , 2007, 0709.3995.

[56]  Wang Zhou,et al.  Circular law, extreme singular values and potential theory , 2010, J. Multivar. Anal..

[57]  ACTA ARITHMETICA , 2022 .

[58]  T. Tao,et al.  Random Matrices: a General Approach for the Least Singular Value Problem , 2022 .