# Metastability for the Ising model on the hexagonal lattice

@article{Apollonio2021MetastabilityFT, title={Metastability for the Ising model on the hexagonal lattice}, author={Valentina Apollonio and Vanessa Jacquier and Francesca R. Nardi and Alessio Troiani}, journal={Electronic Journal of Probability}, year={2021} }

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations…

## 7 Citations

### Metastability for Kawasaki dynamics on the hexagonal lattice

- Mathematics
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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 H 2 in the limit of vanishing temperature. Let Λ ⊂ H 2 a ﬁnite set…

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- Mathematics
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We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants J ij > 0, where i, j ∈ { 1 , 2 , 3 } are the possible spin values (or colors). The…

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

- MathematicsCommunications in Mathematical Physics
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In this article, we investigate the energy landscape and metastable behavior of Ising and Potts models on two-dimensional square or hexagonal lattices in the low-temperature regime, especially in the…

### Critical Droplets and Sharp Asymptotics for Kawasaki Dynamics with Strongly Anisotropic Interactions

- PhysicsJournal of Statistical Physics
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<jats:p>In this paper we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature. Let…

### Shaken Dynamics: An Easy Way to Parallel Markov Chain Monte Carlo

- Materials ScienceJournal of Statistical Physics
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We define a class of Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbor interaction described by a Hamiltonian function H(σ)\documentclass[12pt]{minimal}…

### Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics

- Mathematics
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We consider the ferromagnetic q -state Potts model on a ﬁnite grid with non-zero external ﬁeld and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the…

### Critical Configurations and Tube of Typical Trajectories for the Potts and Ising Models with Zero External Field

- MathematicsJournal of Statistical Physics
- 2021

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature…

## References

SHOWING 1-10 OF 64 REFERENCES

### Polyiamonds and Polyhexes with Minimum Site-Perimeter and Achievement Games

- MathematicsElectron. J. Comb.
- 2010

The site-perimeter of an animal is the number of empty cells connected to the animal by an edge, and the minimum site- perimeter with a given cell size is found for animals on the triangular and hexagonal grid.

### Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

- Mathematics
- 2020

We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an…

### Shaken dynamics for 2d ising models.

- Mathematics, Physics
- 2019

We define a Markovian parallel dynamics for a class of nearest neighbor spin systems. In the dynamics, beside the two usual parameters $J$, the strength of the interaction, and $\lambda$, the…

### On the metastability in three modifications of the Ising model

- Mathematics
- 2019

We consider three extensions of the standard 2D Ising model with Glauber dynamics on a finite torus at low temperature. The first model (see Chapter 2) is an anisotropic version, where the…

### Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

- MathematicsJournal of Statistical Physics
- 2019

On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter…

### On the Essential Features of Metastability: Tunnelling Time and Critical Configurations

- Mathematics
- 2004

AbstractWe consider Metropolis Markov chains with finite state space and transition probabilities of the form
$$P(\eta ,\eta ')=q(\eta ,\eta ')e^{- \beta [H(\eta ') - H(\eta)]_+}$$
for given energy…

### Critical droplets and metastability for a Glauber dynamics at very low temperatures

- Physics
- 1991

We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two dimensional torus under a positive magnetic field in the…

### Metropolis Dynamics Relaxation via Nucleation and Growth

- Physics
- 1997

Abstract:We consider the Ising model with Metropolis dynamics on under a small positive external field h. We show that the relaxation time, i.e., the time it takes for the system to reach the…

### Behavior of droplets for a class of Glauber dynamics at very low temperature

- Physics
- 1992

SummaryWe consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy…