Circular law for random matrices with unconditional log-concave distribution

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

[1]  V. Milman,et al.  Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space , 1989 .

[2]  Rafal Latala.  On Weak Tail Domination of Random Vectors , 2007 .

[3]  H. Nguyen.  Random doubly stochastic matrices: The circular law , 2012, 1205.0843.

[4]  Fedor Nazarov,et al.  On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis , 2003 .

[5]  A. Zee,et al.  Non-gaussian non-hermitian random matrix theory: Phase transition and addition formalism , 1997 .

[6]  F. Barthe,et al.  Invariances in variance estimates , 2011, 1106.5985.

[7]  M. Talagrand.  An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities , 1988 .

[8]  C. Bordenave,et al.  Spectrum of Markov Generators on Sparse Random Graphs , 2012, 1202.0644.

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

[10]  J. Lindenstrauss,et al.  Geometric Aspects of Functional Analysis , 1987 .

[11]  R. Adamczak.  On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries , 2011 .

[12]  D. Hensley.  Slicing convex bodies—bounds for slice area in terms of the body’s covariance , 1980 .

[13]  R. Lata,et al.  SOME ESTIMATES OF NORMS OF RANDOM MATRICES , 2004 .

[14]  Terence Tao,et al.  Random matrices: Universality of ESDs and the circular law , 2008, 0807.4898.

[15]  Charles Bordenave,et al.  Circular law theorem for random Markov matrices , 2008, 0808.1502.

[16]  Djalil Chafaï,et al.  Interactions between compressed sensing, random matrices, and high dimensional geometry , 2012 .

[17]  Tiefeng Jiang,et al.  Circular law and arc law for truncation of random unitary matrix , 2012 .

[18]  Van Vu,et al.  Circular law for random discrete matrices of given row sum , 2012, 1203.5941.

[19]  Alexander E. Litvak,et al.  Condition number of a square matrix with i.i.d. columns drawn from a convex body , 2012 .

[20]  C. Bordenave,et al.  Spectrum of Non-Hermitian Heavy Tailed Random Matrices , 2010, 1006.1713.

[21]  Ofer Zeitouni,et al.  The single ring theorem , 2009, 0909.2214.

[22]  B. Klartag,et al.  A Berry-Esseen type inequality for convex bodies with an unconditional basis , 2007, 0705.0832.

[23]  C. Bordenave,et al.  Around the circular law , 2011, 1109.3343.

[24]  Miklós Simonovits,et al.  Isoperimetric problems for convex bodies and a localization lemma , 1995, Discret. Comput. Geom..

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

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

[27]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[28]  R. Latala.  Some estimates of norms of random matrices , 2005 .

[29]  S. Bobkov,et al.  On Concentration of Empirical Measures and Convergence to the Semi-circle Law , 2010 .

[30]  Madan Lal Mehta,et al.  Random Matrices and the Statistical Theory of Energy Levels , 2014 .

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

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

[33]  Large deviations from the circular law , 1998 .

[34]  R. Adamczak.  SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES , 2013 .

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

[36]  Tiefeng Jiang,et al.  Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles , 2009 .

[37]  T. Tao,et al.  From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices , 2008, 0810.2994.

[38]  C. Bordenave,et al.  The circular law , 2012 .

[39]  C. Borell.  Convex measures on locally convex spaces , 1974 .

[40]  T. Tao.  Outliers in the spectrum of iid matrices with bounded rank perturbations , 2010 .

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

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