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

TL;DR

This paper proves that the spectral radius of a large random matrix with i.i.d. entries converges to the square root of its dimension, linking it to the circular law and using novel analytic techniques.

Contribution

It establishes the convergence of the spectral radius in probability and introduces a new proof approach via characteristic polynomial analysis and Gaussian analytic functions.

Findings

Spectral radius converges to the square root of the matrix dimension.

Reciprocal characteristic polynomial converges to a hyperbolic Gaussian analytic function.

Proof method is shorter and different from traditional spectral radius proofs.

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.