IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v129y2019i11p4556-4575.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918306938
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2018.12.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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).
    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).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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).
    2. Martschink, Bastian, 2014. "Bounds on convergence for the empirical vector of the Curie–Weiss–Potts model with a non-zero external field vector," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 118-126.
    3. Cerioli, Andrea, 2002. "Tests of homogeneity for spatial populations," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 123-130, June.
    4. 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.
    5. Landim, C., 2015. "A topology for limits of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 1058-1088.
    6. Peruggi, Fulvio, 1987. "First-order transitions in percolation models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 141(1), pages 140-150.
    7. Andrea Cerioli, 2002. "Testing Mutual Independence Between Two Discrete-Valued Spatial Processes: A Correction to Pearson Chi-Squared," Biometrics, The International Biometric Society, vol. 58(4), pages 888-897, December.
    8. Bianchi, Alessandra & Gaudillière, Alexandre, 2016. "Metastable states, quasi-stationary distributions and soft measures," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1622-1680.
    9. Bakchich, A. & Benyoussef, A. & Touzani, M., 1993. "Antiferromagnetic Potts model in a magnetic field: a finite size scaling study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 192(3), pages 516-524.
    10. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4556-4575. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.