Convergence of the spectral radius of a random matrix through its characteristic polynomial
arXiv:2012.05602 · doi:10.1007/s00440-021-01079-9
Abstract
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.
arXiv commentMinor revision
References in corpus (11)
- Around the circular law
- Universality and the circular law for sparse random matrices
- Edge Universality for non-Hermitian Random Matrices
- Gaussian fluctuations for non-Hermitian random matrix ensembles
- Spectrum of non-Hermitian heavy tailed random matrices
- Circular law for random matrices with exchangeable entries
- Spectral radius of random matrices with independent entries
- Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs
- Random doubly stochastic matrices: The circular law
- On the spectral radius of a random matrix: an upper bound without fourth moment
- Circular law for random matrices with unconditional log-concave distribution