The low-temperature ratio of the log-Sobolev constant to temperature converges to the Polyak-Lojasiewicz constant of the potential.
Metastability in reversible diffusion processes
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The ballistic limit of the log-Sobolev constant equals the Polyak-{\L}ojasiewicz constant
The low-temperature ratio of the log-Sobolev constant to temperature converges to the Polyak-Lojasiewicz constant of the potential.