# Convergence Time to Equilibrium of the Metropolis dynamics for the GREM

## TL;DR

This paper analyzes how quickly the Metropolis dynamics for the GREM reach equilibrium by deriving bounds on the spectral gap using advanced mathematical tools, applicable to models with multiple hierarchical levels.

## Contribution

It provides new bounds on the convergence time of Metropolis dynamics for GREM with arbitrary hierarchy levels using Poincaré inequalities and convex analysis.

## Key findings

- Bounds on the inverse spectral gap are established.
- The convergence time depends on the hierarchical structure of the GREM.
- Methodology applies to general cases of the model.

## Abstract

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 Poincar\'e inequality and a path technique. We also apply convex analysis tools to give the bounds in the most general case of the model.