Convergence of the spectral radius of a random matrix through its characteristic polynomial

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.

[1]  Giorgio Cipolloni,et al.  Edge universality for non-Hermitian random matrices , 2021, Probability theory and related fields.

[2]  N. Simm,et al.  Secular coefficients and the holomorphic multiplicative chaos , 2020, The Annals of Probability.

[3]  Djalil CHAFAÏ,et al.  Circular law for random matrices with unconditional log-concave distribution , 2013, 1303.5838.

[4]  Charles Bordenave,et al.  Outlier Eigenvalues for Deformed I.I.D. Random Matrices , 2014, 1403.6001.

[5]  Neil O'Connell,et al.  On the Characteristic Polynomial¶ of a Random Unitary Matrix , 2001 .

[6]  O. Zeitouni,et al.  Outliers of random perturbations of Toeplitz matrices with finite symbols , 2019, Probability Theory and Related Fields.

[7]  J. W. Silverstein,et al.  Gaussian fluctuations for non-Hermitian random matrix ensembles , 2005, math/0502400.

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

[9]  Nicholas A. Cook,et al.  Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs , 2016 .

[10]  L. Erdős,et al.  Spectral radius of random matrices with independent entries , 2019, Probability and Mathematical Physics.

[11]  M. Rudelson,et al.  The circular law for sparse non-Hermitian matrices , 2017, The Annals of Probability.

[12]  S. Janson,et al.  The asymptotic distributions of generalized U-statistics with applications to random graphs , 1991 .

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

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

[15]  S. Péché,et al.  A Note on the Second Order Universality at the Edge of Coulomb Gases on the Plane , 2013, Journal of Statistical Physics.

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

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

[18]  Yuval Peres,et al.  Zeros of Gaussian Analytic Functions and Determinantal Point Processes , 2009, University Lecture Series.

[19]  Philip Matchett Wood.  Universality and the circular law for sparse random matrices. , 2010, 1010.1726.

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

[21]  Raphael Butez,et al.  Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background , 2018, The Annals of Applied Probability.

[22]  From the first rigorous proof of the Circular Law in 1984 to the Circular Law for block random matrices under the generalized Lindeberg condition , 2018, Random Operators and Stochastic Equations.

[23]  On the spectral radius of a random matrix: An upper bound without fourth moment , 2016, The Annals of Probability.

[24]  M. Rudelson,et al.  The sparse circular law under minimal assumptions , 2018, Geometric and Functional Analysis.

[25]  C. Bordenave,et al.  Detection Thresholds in Very Sparse Matrix Completion , 2020, Foundations of Computational Mathematics.

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

[27]  Z. D. Bai,et al.  Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang , 1986 .

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

[29]  J. W. Silverstein.  THE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES , 1994 .

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

[31]  Djalil Chafaï,et al.  Circular law for random matrices with exchangeable entries , 2014, Random Struct. Algorithms.

[32]  B. Rider.  A limit theorem at the edge of a non-Hermitian random matrix ensemble , 2003 .

[33]  Eric Kostlan,et al.  On the spectra of Gaussian matrices , 1992 .

[34]  S. Geman.  THE SPECTRAL RADIUS OF LARGE RANDOM MATRICES , 1986 .

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