Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix

TL;DR

This paper proves that the eigenvalue distribution of large random symmetric Toeplitz matrices is absolutely continuous with a bounded density, using spectral averaging techniques, partially answering a longstanding question in random matrix theory.

Contribution

It establishes the absolute continuity of the eigenvalue distribution for symmetric Toeplitz matrices, a significant step forward in understanding their spectral properties.

Findings

Eigenvalue distribution is absolutely continuous with density bounded by 8.

Uses spectral averaging technique from random Schrödinger operators.

Partially answers a question posed by Bryc, Dembo, and Jiang (2006).

Abstract

We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schr\"{o}dinger operators. The similar question for Hankel matrices remains open.