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

TL;DR

This paper proves that for symmetric random matrices with zero mean, unit variance, and finite higher moments, the spectral radius concentrates near the square root of the matrix dimension, supporting a conjecture about the circular law.

Contribution

It establishes the conjecture for symmetric matrices with finite moments greater than two, using a novel truncation method for cycle weights.

Findings

Spectral radius is close to the square root of the dimension for the specified matrices.

The proof introduces a new truncation technique for cycle weights.

Supports the conjecture that no outliers exist in the circular law under these conditions.

Abstract

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 dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance, in other words that there are no outliers in the circular law. In this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.