Fluctuations for block spin Ising models

TL;DR

This paper studies the magnetization fluctuations in the block spin Ising model at high temperatures, proving CLTs, and identifying a critical line where these theorems no longer hold, with alternative non-standard CLTs applicable.

Contribution

It establishes the first CLTs for the block spin Ising model's magnetization and characterizes the critical temperature line where classical CLTs fail.

Findings

CLTs proven for high temperature magnetization fluctuations

Breakdown of CLTs at a critical temperature line

Non-standard CLTs valid at the critical line

Abstract

We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet, Rigollet and Srivastava. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we show the validity of a non-standard Central Limit Theorems for the magnetization.