Convergence Time to Equilibrium of the Metropolis Dynamics for the GREM

  • A. M. B. NascimentoEmail author
  • L. R. Fontes


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.


Spin glasses GREM Metropolis dynamics Convergence to equilibrium Spectral gap Poincaré inequality 

Mathematics Subject Classification

60K35 82B44 82C44 82D30 



This work is part of the Ph.D. thesis of the second author at IME-USP and was supported in part by CNPq 140762/2016-7. We warmfully thank Pierre Picco for suggesting this problem and for innumerable discussions concerning it in many occasions.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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