# Local Semicircle Law for Curie-Weiss Type Ensembles

## TL;DR

This paper establishes local semicircle laws for a class of random matrices with exchangeable, slowly decaying correlations, called Curie-Weiss type ensembles, extending understanding of spectral properties in correlated settings.

## Contribution

It introduces and analyzes local semicircle laws for Curie-Weiss type ensembles with slow correlation decay, a novel class of correlated random matrices.

## Key findings

- Derived various forms of local semicircle laws.
- Showed decay rate of correlations as N^{-l/2}.
- Applicable to Curie-Weiss($eta$)-distributed entries with $eta \,\leq\, 1$.

## Abstract

We derive and compare various forms of local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay at a very slow rate. In fact, any $l$-point correlation will decay at a rate of $N^{-l/2}$. We call our ensembles \emph{of Curie-Weiss type}, and Curie-Weiss($\beta$)-distributed entries are admissible as long as $\beta\leq 1$.