Harmonic-map metastability conjecture for O(N) models

Let $M$ be a compact Riemannian manifold, and define the energy functional on maps $f:M\to \mathbb{S}^{N-1}$ by

$$\mathcal{E}(f)=\int_M \|Df(x)\|^2\,\mathrm{d}x,$$

where the norm is taken with respect to the standard Riemannian metric on the sphere. Consider the $O(N)$ model on a graph $G$ discretizing $M$.

Harmonic-map metastability conjecture. The relaxation time of the corresponding Langevin dynamics grows exponentially fast with $\beta$ if and only if $\mathcal E$ has non-trivial, that is, non-constant, local minima.

This conjecture makes the proposed connection between metastability and the energy landscape of harmonic maps precise for the $O(N)$ model. The source leaves quantitative bounds in dimensions $d>1$ as an open problem.