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

@article{Bordenave2021ConvergenceOT, title={Convergence of the spectral radius of a random matrix through its characteristic polynomial}, author={Charles Bordenave and Djalil CHAFA{\"I} and David Garc'ia-Zelada}, journal={Probability Theory and Related Fields}, year={2021}, volume={182}, pages={1163-1181} }

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…

## 8 Citations

### Sparse matrices: convergence of the characteristic polynomial seen from infinity

- Mathematics
- 2021

We prove that the reverse characteristic polynomial det(In − zAn) of a random n×nmatrixAn with iidBernoulli(d/n) entries converges in distribution towards the random infinite product ∞ ∏ `=1 (1− z)`…

### Reliability Analysis of Sports Training Evaluation Index Based on Random Matrix

- EducationMathematical Problems in Engineering
- 2022

This paper presents an in-depth study and analysis of the reliability of indicators for the evaluation of sports training using the algorithm of random matrices. Random matrix theory is used to…

### Directional extremal statistics for Ginibre eigenvalues

- MathematicsJournal of Mathematical Physics
- 2022

We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel…

### The characteristic polynomial of sums of random permutations and regular digraphs

- Mathematics
- 2022

. Let A n be the sum of d permutations matrices of size n × n , each drawn uniformly at random and indepen-dently. We prove that det(I n − zA n / √ d ) converges when n → ∞ towards a random analytic…

### Density of Small Singular Values of the Shifted Real Ginibre Ensemble

- MathematicsAnnales Henri Poincaré
- 2022

We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away…

### On the condition number of the shifted real Ginibre ensemble

- MathematicsSIAM Journal on Matrix Analysis and Applications
- 2022

An accurate lower tail estimate is derived on the lowest singular value σ1(X − z) of a real Gaussian (Ginibre) random matrixX shifted by a complex parameter z and an improved upper bound on the eigenvalue condition numbers for real Ginibre matrices is obtained.

### Aspects of Coulomb gases

- Physics
- 2020

Coulomb gases are special probability distributions, related to potential theory, that appear at many places in pure and applied mathematics and physics. In these short expository notes, we focus on…

### Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

- Mathematics, Computer Science
- 2020

The eigenvalues of P(Y,A) are investigated outside the spectrum of $P(c,a)$ where $c$ is a circular system which is free from $a$ and a sufficient condition is provided to guarantee that these eigen values coincide asymptotically with those of £P(0,A).

## References

SHOWING 1-10 OF 52 REFERENCES

### On the spectral radius of a random matrix: An upper bound without fourth moment

- MathematicsThe Annals of Probability
- 2018

Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high…

### Spectral radius of random matrices with independent entries

- MathematicsProbability and Mathematical Physics
- 2019

We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the…

### A limit theorem at the edge of a non-Hermitian random matrix ensemble

- Mathematics
- 2003

The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the…

### On the Characteristic Polynomial¶ of a Random Unitary Matrix

- Mathematics
- 2001

Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at…

### RANDOM MATRICES: THE CIRCULAR LAW

- Mathematics
- 2007

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of . Define the…

### Circular law for random matrices with unconditional log-concave distribution

- Mathematics
- 2015

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…

### Gaussian fluctuations for non-Hermitian random matrix ensembles

- Mathematics
- 2006

Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry…

### Circular law for random matrices with exchangeable entries

- MathematicsRandom Struct. Algorithms
- 2016

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the…

### Universality and the circular law for sparse random matrices.

- Computer Science
- 2012

This paper proves a universality result for sparse random n by n matrices where each entry is nonzero with probability $1/n^{1-\alpha}$ where $0<\alpha\le1$ is any constant.

### Around the circular law

- Mathematics
- 2012

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the…