Rigorous renormalization group and Ising criticality

5 problems

• Rotational and conformal invariance of the critical 3D Ising scaling limit

  Open

  At its critical point the 3D Ising model is believed to look the same under rotations and angle-preserving maps (conformal symmetry), which physicists use to compute exponents. In 2D this was proven; in 3D even rotation invariance is unproven.

• Existence of critical exponents for the 3D Ising model

  Open

  Prove that quantities such as the decay of spin correlations at the critical temperature follow exact power laws in 3D. Their numerical values are known very precisely, but even their existence is unproven.

• Logarithmic corrections at the critical point of the 4D Ising model

  Open

  In four dimensions the critical Ising model is almost free, and its quantities deviate from simple power laws only by slowly varying logarithms with predicted powers. These powers are proven for related weakly interacting models but not for the Ising model itself.

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

  Open

  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.

• Endpoint-distance exponent of the 3D self-avoiding walk

  Open

  A random path that never visits the same point twice models a polymer chain, and in 3D its size should grow as the number of steps to the power 0.588. Even a power-law improvement over straight-line growth is unproven.