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 and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

We consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$X={-1,+1}Λ where $$\varLambda \subset {\mathbb {Z}}^2$$Λ⊂Z2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$${-1̲,c̲o,c̲e} to the stable state $$\underline{+1}$$+1̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$c̲o,c̲e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

Metastability is a ubiquitous phenomenon in nature, which interests several fields of natural sciences. Since metastability is a genuine non-equilibrium phenomenon, its description in the framework of thermodynamics and statistical mechanics has progressed slowly for a long time. Since the publication of the first seminal paper in which the metastable behavior of the mean field Curie–Weiss model was approached by means of stochastic techniques, this topic has been largely studied by the scientific community. Several papers and books have been published in which many different spin models were studied and different approaches were developed. In this review, we focus on the comparison between the metastable behavior of synchronous and asynchronous dynamics, namely, stochastic processes in discrete time in which, at each time, either all the spins or one single spin is updated. In particular, we discuss how two different stochastic implementations of the very same Hamiltonian give rise to different metastable behaviors.

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.

In the present manuscript, we address and solve for the first time a nonlocal discrete isoperimetric problem. We consider indeed a generalization of the classical perimeter, what we call a nonlocal bi-axial discrete perimeter , where not only the external boundary of a polyomino \mathcal{P} contributes to the perimeter, but all internal and external components of \mathcal{P} . Furthermore, we find and characterize its minimizers in the class of polyominoes with fixed area n . Moreover, we explain how the solution of the nonlocal discrete isoperimetric problem is related to the rigorous study of the metastable behavior of a long-range bi-axial Ising model .