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Tunneling behavior of Ising and Potts models in the low-temperature regime

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  • Nardi, Francesca R.
  • Zocca, Alessandro

Abstract

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume and assume that its stochastic evolution is described by a Glauber-type dynamics parametrized by the inverse temperature β. Our analysis concerns the low-temperature regime β→∞, in which this multi-spin system has q stable equilibria. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the q-state Potts model, characterizing the asymptotic behavior of the first hitting times between stable equilibria as β→∞ in probability, in expectation, and in distribution and obtaining tight bounds on the mixing time as side-result.

Suggested Citation

  • Nardi, Francesca R. & Zocca, Alessandro, 2019. "Tunneling behavior of Ising and Potts models in the low-temperature regime," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4556-4575.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4556-4575
    DOI: 10.1016/j.spa.2018.12.001
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    References listed on IDEAS

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    1. Wang, Kongming, 1994. "Solutions of the variational problem in the Curie--Weiss--Potts model," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 245-252, April.
    2. Ellis, Richard S. & Wang, Kongming, 1992. "Limit theorems for maximum likelihood estimators in the Curie-Weiss-Potts model," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 251-288, March.
    3. Peruggi, Fulvio & di Liberto, Francesco & Monroy, Gabriella, 1987. "Phase diagrams of the q-state potts model on Bethe lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 141(1), pages 151-186.
    4. Beltrán, J. & Landim, C., 2011. "Metastability of reversible finite state Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1633-1677, August.
    5. Ellis, Richard S. & Wang, Kongming, 1990. "Limit theorems for the empirical vector of the Curie-Weiss-Potts model," Stochastic Processes and their Applications, Elsevier, vol. 35(1), pages 59-79, June.
    6. Gandolfo, Daniel & Ruiz, Jean & Wouts, Marc, 2010. "Limit theorems and coexistence probabilities for the Curie-Weiss Potts model with an external field," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 84-104, January.
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    Cited by:

    1. Baldassarri, Simone & Gallo, Anna & Jacquier, Vanessa & Zocca, Alessandro, 2023. "Ising model on clustered networks: A model for opinion dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 623(C).
    2. Landim, C., 2023. "Metastability from the large deviations point of view: A Γ-expansion of the level two large deviations rate functional of non-reversible finite-state Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 275-315.
    3. Bet, Gianmarco & Gallo, Anna & Nardi, F.R., 2024. "Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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