Critical spectral-gap scaling at the spinodal thresholds

Determine the precise spectral gap of the local Davies generator for the mean-field interchange model with local dimension d≥3 at the two spinodal inverse temperatures β_min(d) and β=d, as well as in temperature windows that shrink toward either threshold with the system size n, where the free-energy Hessian is degenerate and nontrivial critical scaling is expected.

Background

The paper establishes polynomial spectral-gap bounds outside the metastable window β_min(d)<β<d and exponential gap decay inside that window, but its estimates explicitly exclude the boundary points β=β_min(d) and β=d. At these spinodal thresholds, the free-energy Hessian becomes degenerate, so the nondegenerate-well conductance and canonical-path arguments used away from the boundaries do not determine the critical finite-size behavior.

The unresolved problem is to identify the exact asymptotic scaling of the Davies generator's spectral gap at both thresholds and in temperature regimes approaching them as n grows. Such results would complete the dynamical phase diagram and characterize the crossover between polynomial mixing and metastable exponential slowdown.