# Fluctuation results for general block spin Ising models

## TL;DR

This paper analyzes a block spin mean-field Ising model, establishing large deviation principles, central limit theorems, and convergence rates for block magnetizations under general conditions.

## Contribution

It provides the first comprehensive large deviation and CLT results for general block interaction matrices in the Ising model.

## Key findings

- Proved Large Deviation Principles for block magnetizations.
- Established Central Limit Theorems with convergence rates.
- Analyzed high temperature regime behavior.

## Abstract

We study a block spin mean-field Ising model, i.e. a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices, and where pair interactions are given according to their blocks. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein's method we establish a rate of convergence in the Central Limit Theorem for the block magnetization vector in the high temperature regime.