# The semicircle law for matrices with ergodic entries

## TL;DR

This paper investigates how the spectral distribution of symmetric matrices with ergodic diagonal entries converges to the semicircle law under certain decay conditions of correlations, extending understanding of spectral behavior in correlated random matrices.

## Contribution

It demonstrates that the semicircle law applies to matrices with ergodic entries when correlations decay, and shows deviations when correlations do not decay, linking to previous exchangeable process results.

## Key findings

- Semicircle law holds when correlations decay to zero.
- Non-decaying correlations prevent convergence to the semicircle law.
- Results align with prior studies on exchangeable processes.

## Abstract

We study the empirical spectral distribution (ESD) of symmetric random matrices with ergodic entries on the diagonals. We observe that for entries with correlations that decay to 0, when the distance of the diagonal entries becomes large the limiting ESD is the well known semicircle law. If it does not decay to 0 (and have the same sign) the semicircle law cannot be the limit of the ESD. This is good agreement with results on exchangeable processes analysed in Friesen and L\"owe (2013a) and Hochst\"attler et al. (2016).