# A CLT for the total energy of the two-dimensional critical Ising model

## TL;DR

This paper establishes a central limit theorem for the total energy of the two-dimensional critical Ising model with specific boundary conditions, showing it converges to a Gaussian distribution under certain growth conditions for the lattice dimensions.

## Contribution

It proves a new CLT for the total energy of the 2D critical Ising model with mixed boundary conditions, extending understanding of its fluctuation behavior.

## Key findings

- Total energy fluctuations follow a Gaussian distribution asymptotically.
- The result holds for lattice dimensions satisfying a growth condition involving logarithms.
- Provides a quantitative description of energy distribution at criticality.

## Abstract

Consider the Ising model on $([1,2N]\times[1,2M])\cap\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\geq N/(\ln N)^{\alpha}$ for some $\alpha\in[0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\rightarrow\infty$.