Metastability of Ising and Potts Models Without External Fields in Large Volumes at Low Temperatures

Kim, Seonwoo; Seo, Il Won

doi:10.1007/s00220-022-04465-z

Cited by 5 publications

(2 citation statements)

References 23 publications

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“…Therefore, one can apply this approach to study mixing of metastable systems such as particle systems [10,24,45], ferromagnetic systems [11,16,25,26,34], machine learning [1], and others. 1.5.…”

Section: Introductionmentioning

confidence: 99%

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

Lee

Seo

2022

J Stat Phys

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In this article, we study the mixing properties of metastable diffusion processes which possess a Gibbs invariant distribution. For systems with multiple stable equilibria, socalled metastable transitions between these equilibria are required for mixing since the unique invariant distribution is concentrated on these equilibria. Consequently, these systems exhibit slower mixing compared to those with a unique stable equilibrium, as analyzed in Barrera and Jara (Ann. Appl. Probab. 30:1164-1208, 2020). Our proof is based on the theory of metastability, which is a primary tool for studying systems with multiple stable equilibria. Within this framework, we compute the total variation distance between the distribution of the diffusion process and its invariant distribution for any time scale larger than ϵ −1 . Finally, we derive precise asymptotics for the mixing time.

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Section: Introductionmentioning

confidence: 99%

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

Lee

Seo

2022

J Stat Phys

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“…In the context of statistical physics, metastability corresponds to the so-called first-order phase transition, in that certain observables of a system exhibit discontinuity with respect to the intensive variables of the system. Typical examples are found in small random perturbations of dynamical systems [9-11, 38, 39, 43], interacting particle systems with sticky interactions [7,23,25,26,31,32], ferromagnetic spin systems in low temperatures [5,15,27,40,41], etc. From a dynamical point of view, metastability can be explained as follows.…”

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confidence: 99%

Correction: Second time scale of the metastability of reversible inclusion processes

Kim

2023

Probab. Theory Relat. Fields

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In this article, we study the hierarchical structure of metastability in the reversible inclusion process. We fully characterize the third time scale of metastability subject to any underlying geometry of the system and prove that this is the last time scale. We also demonstrate that there are no other meaningful time scales except the three identified ones. This work completes the verification of the conjecture made in [7] which was partially resolved on the first time scale in [7] and on the second time scale in [25]. Main tools are potential-theoretic approach and martingale approach to metastability; we thoroughly investigate the highly-complicated energy landscape of the system to construct suitable test objects to provide sharp asymptotics on capacities. Contents 53 Appendix B. Proof of results in Section 4.3 56 References 58

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Metastability for Kawasaki Dynamics on the Hexagonal Lattice

Baldassarri

Jacquier

2023

J Stat Phys

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In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice $${\mathbb {H}}^2$$ H 2 in the limit of vanishing temperature. Let $$\varLambda \subset {\mathbb {H}}^2$$ Λ ⊂ H 2 a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on $$\varLambda $$ Λ , but when they occupy neighboring sites they feel a binding energy $$-U<0$$ - U < 0 . Along each bond touching the boundary of $$\varLambda $$ Λ from the outside to the inside, particles are created with rate $$\rho =e^{-\varDelta \beta }$$ ρ = e - Δ β , while along each bond from the inside to the outside, particles are annihilated with rate 1, where $$\beta $$ β is the inverse temperature and $$\varDelta >0$$ Δ > 0 is an activity parameter. For the choice $$\varDelta \in {(U,\frac{3}{2}U)}$$ Δ ∈ ( U , 3 2 U ) we prove that the empty (resp. full) hexagon is the unique metastable (resp. stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice.

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Metastability of Ising and Potts Models Without External Fields in Large Volumes at Low Temperatures

Cited by 5 publications

References 23 publications

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

Correction: Second time scale of the metastability of reversible inclusion processes

Metastability for Kawasaki Dynamics on the Hexagonal Lattice

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