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… 

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References

SHOWING 1-10 OF 52 REFERENCES

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

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

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

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

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

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

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

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

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.

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

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
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