MATHPH In the literature: open

Non-Gaussian scaling limit of the critical 3D Ising model

In plain words

In four or more dimensions the critical Ising model looks like a free (Gaussian) field at large scales. In 3D it should not, and this difference is unproven.

Precise statement

Prove that every subsequential scaling limit of the critical spin field on $Z^{3}$ is non-Gaussian, for example that the rescaled fourth Ursell function does not vanish, and that $\eta > 0$ strictly (the infrared bound gives only $\eta \ge 0$). Answer: a proof.

What would settle it

A proof of a nonvanishing four-point Ursell function in the 3D scaling limit.

Status in the literature

Gaussianity is proven for $d \ge 5$ (Aizenman; Frohlich, 1982) and $d = 4$ (Aizenman and Duminil-Copin, 2021).

See also