Open Access
2023 Metastability of the three-state Potts model with general interactions
Gianmarco Bet, Anna Gallo, Seonwoo Kim
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Electron. J. Probab. 28: 1-37 (2023). DOI: 10.1214/23-EJP1003


We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants Jij>0, where i,j{1,2,3} are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature β tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition. Our main tool is the so-called pathwise approach to metastability, which requires a detailed analysis of the energy landscape.


The authors would like to express their deepest gratitude to Prof. Francesca Romana Nardi, who encouraged the collaboration that led to this manuscript, but sadly passed away shortly after. This work is dedicated to her memory


SK was supported by NRF-2019-Fostering Core Leaders of the Future Basic Science Program/Global Ph.D. Fellowship Program and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2017R1A5A1015626, 2022R1F1A106366811 and 2022R1A5A6000840). SK would like to thank the University of Florence for the warm hospitality during his two-week stay to finalize this work.


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Gianmarco Bet. Anna Gallo. Seonwoo Kim. "Metastability of the three-state Potts model with general interactions." Electron. J. Probab. 28 1 - 37, 2023.


Received: 25 August 2022; Accepted: 3 August 2023; Published: 2023
First available in Project Euclid: 4 October 2023

MathSciNet: MR4649980
Digital Object Identifier: 10.1214/23-EJP1003

Primary: 60J28
Secondary: 60K35 , 82C22

Keywords: energy landscape , general interactions , Glauber dynamics , metastability , Potts model

Vol.28 • 2023
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