MATHPH
In the literature: open
Asymptotic-freedom growth of the O(N) correlation length
In plain words
Perturbation theory predicts exactly how fast the distance over which arrows stay aligned grows as the temperature goes to zero, an exponential in the inverse temperature. Proving this formula would confirm asymptotic freedom in a lattice model.
Precise statement
For the $O(N)$ model on $Z^2$ with $N \ge 3$, prove $\xi(\beta) = C_N \beta^{-1/(N-2)} \exp(2 \pi \beta/(N-2)) (1 + o(1))$ as $\beta \to \infty$, with $\xi = 1/m$ the exponential correlation length; the constant C_N was predicted by Hasenfratz, Maggiore and Niedermayer (1990) from the Bethe ansatz. Answer: a proof of the leading asymptotics, with or without the constant.
What would settle it
Matching upper and lower bounds on $\xi(\beta)$ with the two-loop exponential and power-law factors.