For the critical 2D Ising model on a 2N by 2M rectangle with periodic-free boundaries, the total energy, after centering by (4/π)N ln N and scaling by sqrt(4MN ln N), converges to a Gaussian with variance 8/π as N and M grow together.
Billingsley (1995)
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A CLT for the total energy of the two-dimensional critical Ising model
For the critical 2D Ising model on a 2N by 2M rectangle with periodic-free boundaries, the total energy, after centering by (4/π)N ln N and scaling by sqrt(4MN ln N), converges to a Gaussian with variance 8/π as N and M grow together.