# Semicircle law for generalized Curie-Weiss matrix ensembles at subcritical temperature

## TL;DR

This paper extends the semicircle law to generalized Curie-Weiss matrix ensembles at subcritical temperatures by using an ensemble modification that accounts for stronger correlations, maintaining the spectral measure.

## Contribution

It introduces a method to handle stronger correlations at subcritical temperatures by removing average magnetization, extending the semicircle law proof.

## Key findings

- Semicircle law holds at subcritical temperatures with ensemble modification.
- The ensemble modification is of rank 1, preserving the spectral measure.
- The approach generalizes previous results to stronger correlation regimes.

## Abstract

Hochst\"attler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in the paper mentioned above. In order to do so one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1 thus not changing the limiting spectral measure.