THE SPECTRAL GAP OF THE REM UNDER METROPOLIS DYNAMICS

We derive upper and lower bounds for the spectral gap of the random en ergy model under Metropolis dynamics which are sharp in exponential or der. They are based on the Variational characterization of the gap. Fo r the lower bound, a Poincare inequality derived by Diaconis and Stroo ck is used. The scaled asymptotic expression is a linear function of t he temperature. The corresponding function for a global version of the dynamics exhibits phase transition instead. We also study the depende nce of lower order terms on the volume. In the global dynamics, we obs erve a phase transition. For the local dynamics, the expressions we ha ve, which are possibly not sharp, do not change their order of depende nce on the volume as the temperature changes.