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Can the two-scale sampling algorithm for local free energy minimizers be proven to achieve dimension-free convergence rates when the energy landscape satisfies a uniform Poincare inequality at the local level?

Related: Holley-Stroock perturbation lemma for log-Sobolev inequalities, Cheeger inequality for continuous Markov chains, Bovier-Eckhoff-Gayrard-Klein metastability theory

The two-scale approach for sampling local free energy minimizers constructs an algorithm that separates fast local fluctuations from slow transitions between metastable states. A core open question is whether such algorithms can achieve convergence rates that do not degrade exponentially or even polynomially with the dimension of the state space, provided the local energy wells each satisfy a Poincare or log-Sobolev inequality uniformly. Currently, convergence guarantees for metastable sampling methods either assume strong global conditions that fail in multimodal settings or produce dimension-dependent bounds that become vacuous in high-dimensional applications like molecular simulation.

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