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Metastability of the three-state Potts model with general interactions
Probability · Submitted 25 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Then, we identify all the critical configurations that are visited with high probability during the metastable transition.
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The abstract, sentence by sentence
We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants $J_{ij}>0$, where $i,j\in\{1,2,3\}$ 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 $\beta$ tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants $J_{ij}>0$, where $i,j\in\{1,2,3\}$ are the possible spin values (or colors). | method 0.52 | 0.45 | 0.02 | asserted 0.94 |
| s1 | The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. | background 0.48 | 0.37 | 0.04 | asserted 1.00 |
| s2 | The system evolves according to a Glauber-type spin-flipping dynamics. | method 0.98 | 0.17 | 0.02 | asserted 1.00 |
| s3 | 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. | contribution 0.42 | 0.49 | 0.08 | asserted 0.98 |
| s4 | 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 $\beta$ tends to infinity. | contribution 0.99 | 0.92 | 0.09 | asserted 0.92 |
| s5 | Then, we identify all the critical configurations that are visited with high probability during the metastable transition. | result 0.61 | 0.95 | 0.26 | asserted 1.00 |
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The paper
Original abstract
We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants $J_{ij}>0$, where $i,j\in\{1,2,3\}$ 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 $\beta$ tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition.
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How the literature uses it
3 arXiv papers cite this in 3 sentences
Papers citing it
Cites it as background 3 papers
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The pathwise approach was used to study the low-temperature behavior of finite-volume models with single-spin-flip Glauber dynamics, e.g. [ref] , with Kawasaki dynamics, e.g. [ref] , and with parallel dynamics, e.g. [ref] .
Ising model on clustered networks: A model for opinion dynamics · Introduction · 0.81
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Metastability was studied in [ref] .
Dynamical phase transition for the homogeneous multi-component Curie-Weiss-Potts model · Introduction · 0.92
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The pathwise approach has been widely adopted to the low-temperature behavior of finite-volume models with single-spin-flip Glauber dynamics, e.g. [ref] , with Kawasaki dynamics, e.g. [ref] , and with parallel dynamics, e.g. [ref] .
Critical configurations of the hard-core model on square grid graphs · Introduction · 0.81
What it cites · 13 sentences without a confident label
Compares with 2 papers
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Indeed, it is more likely to observe such a spin update in our setting than in the setting of [ref] ; see also Remark [ref].
Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics · Introduction · 0.97
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On the other hand, in [ref] , there is a unique metastable state and $q-1$ stable configurations.
Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics · Introduction · 0.84
Cites as support 1 paper
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This indicates that, with high probability, during a metastable transition, no other metastable configurations are visited, see Figure [ref]-(d) or [ref] .
Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics · Introduction · 0.96
Cites as background 4 papers
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In [ref] the authors investigate the tunneling behavior of the Potts model with zero external magnetic field.
Energy Landscape and Metastability of Stochastic Ising and Potts Models on Three-dimensional Lattices Without External Fields · Introduction · 0.97
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The $q$-Potts model with positive and negative external magnetic field has been studied in [ref] and [ref] , respectively.
Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics · Introduction · 0.91
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The $q$-Potts model with positive and negative external magnetic field has been studied in [ref] and [ref] , respectively.
Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics · Introduction · 0.91
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