Convergence Time to Equilibrium of the Metropolis Dynamics for the GREM

We study the convergence time to equilibrium of the Metropolis dynamics for the generalized random energy model with an arbitrary number of hierarchical levels, a finite and reversible continuous-time Markov process, in terms of the spectral gap of its transition probability matrix. This is done by deducing bounds to the inverse of the gap using a Poincaré inequality and a path technique. We also apply convex analysis tools to give the bounds in the most general case of the model.