The top eigenvalue of the random Toeplitz matrix and the sine kernel

TL;DR

This paper establishes a connection between the top eigenvalue of large random symmetric Toeplitz matrices and the sine kernel's operator norm, revealing asymptotic behavior as matrix size grows.

Contribution

It demonstrates that the scaled top eigenvalue converges to the square of the sine kernel's 2→4 operator norm, linking random matrix theory with integral kernel analysis.

Findings

Top eigenvalue scaled by √(2n log n) converges

Convergence to the square of the sine kernel's 2→4 norm

Provides asymptotic behavior of eigenvalues in Toeplitz matrices

Abstract

We show that the top eigenvalue of an $n\times n$ random symmetric Toeplitz matrix, scaled by $\sqrt{2n\log n}$, converges to the square of the $2\to4$ operator norm of the sine kernel.