A class of Gibbs measures proportional to exp(−V/ε) whose potential satisfies a local Polyak-Łojasiewicz inequality, so its minima form a connected, compact, possibly non-contractible C² embedded submanifold. The first non-trivial eigenvalue of the induced Laplace-Beltrami operator on that manifold gives a temperature-independent lower bound on the measure's Poincaré constant, which implies overdamped Langevin dynamics converges to equilibrium at rate order 1/ε (up to logarithms) in the low-temperature regime, even for landscapes with non-isolated minima.
This paper studies Log-PL measures: Gibbs measures μ_ε ∝ exp(−V/ε) at temperature ε whose potential V satisfies a local Polyak-Łojasiewicz (PŁ) inequality. Under this condition the set of local minim…