Fluctuations of the magnetization in the Block Potts Model

TL;DR

This paper analyzes the fluctuations of magnetization in a block-structured mean-field Potts model, proving central limit theorems and deviation principles, and introduces a rotated magnetization to compare with related models.

Contribution

It establishes CLTs and deviation principles for block magnetization in a generalized high-temperature regime, including a novel rotated magnetization approach.

Findings

Magnetization concentrates around a uniform vector with Gaussian fluctuations.

Central limit theorems are proved for the block magnetization.

A moderate deviation principle describes fluctuations on lower scales.

Abstract

In this note we study the block spin mean-field Potts model, in which the spins are divided into $s$ blocks and can take $q\ge 2$ different values (colors). Each block is allowed to contain a different proportion of vertices and behaves itself like a mean-field Ising/Potts model which also interacts with other blocks according to different temperatures. Of particular interest is the behavior of the magnetization, which counts the number of colors appearing in the distinct blocks. We prove central limit theorems for the magnetization in the generalized high temperature regime and provide a moderate deviation principle for its fluctuations on lower scalings. More precisely, the magnetization concentrates around the uniform vector of all colors with an explicit, but singular, Gaussian distribution. In order to remove the singular component, we will also consider a rotated magnetization, which enables us to compare our results to various related models.