Logarithmic corrections at the critical point of the 4D Ising model
In plain words
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.
Precise statement
Nearest-neighbour Ising model on $Z^4$. Prove $\chi(\beta)\sim C(\beta_c-\beta)^{-1}\left|\log(\beta_c-\beta)\right|^{1/3}$ as $\beta\to \beta_c^{-}$ (the $n=1$ case of the exponent $(n+2)/(n+8)$), and the corresponding corrections for the correlation length and specific heat. Answer: a proof.
What would settle it
A proof of the $\mid\operatorname{log}\mid^{1/3}$ susceptibility correction for the nearest-neighbour 4D Ising model.
Status in the literature
Aizenman and Duminil-Copin (Annals of Mathematics, 2021, arXiv 1912.07973) proved Gaussian scaling limits with a logarithmic improvement of the tree bound; exact logarithmic exponents are proven for weakly coupled $4\mathrm{D}$ $\mid\phi\mid^{4}$ and weakly self-avoiding walk (Bauerschmidt, Brydges and Slade, 2014-2015).