MATHPH
In the literature: open
Existence of critical exponents for the 3D Ising model
In plain words
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.
Precise statement
Nearest-neighbour Ising model on $Z^3$ at $\beta_c$. Prove that $\langle s_0 s_x\rangle_{\beta_c}=\left|x\right|^{-(1+\eta)+o(1)}$ and that the correlation length $\xi(\beta)=(\beta_c-\beta)^{-\nu+o(1)}$, with numerical values eta approximately 0.0363 and nu approximately 0.6300. Answer: a proof of existence of the limits, with or without their values.
What would settle it
A proof that the exponents $\eta$ and $\nu$ exist as limits.
Status in the literature
Continuity of the phase transition in 3D was proven by Aizenman, Duminil-Copin and Sidoravicius (2015).