We derive upper and lower bounds for the spectral gap of the random energy model under Metropolis dynamics which are sharp in exponential order. They are based on the variational characterization of the gap. For the lower bound, a Poincaré inequality derived by Diaconis and Stroock is used. The scaled asymptotic expression is a linear function of the temperature. The corresponding function for a global version of the dynamics exhibits phase transition instead.
We also study the dependence of lower order terms on the volume. In the global dynamics, we observe a phase transition. For the local dynamics, the expressions we have, which are possibly not sharp, do not change their order of dependence on the volume as the temperature changes.
"The spectral gap of the REM under Metropolis dynamics." Ann. Appl. Probab. 8 (3) 917 - 943, August 1998. https://doi.org/10.1214/aoap/1028903457