Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions

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 $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $\Lambda$. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\beta>0$ is the inverse temperature and $\Delta>0$ is an activity parameter. Thus, the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place with particular attention to the critical configurations that asymptotically have to be crossed with probability 1. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates and their boundaries for the nucleation in the strongly anisotropic case. We observe very different behaviors for this case with respect to the isotropic ($U_1=U_2$) and weakly anisotropic ($U_1<2U_2$) ones. Moreover, we derive mixing time, spectral gap and sharp estimates for the asymptotic transition time for the strongly anisotropic case.Comment: 51 pages and 14 figures. In this paper we consider, with the same authors, the same model as our previous work arXiv:2108.02017, but in a complementary range of parameter values. Thus sections 1, 2, 3 and 4.1 in the present paper are similar to sections 1, 2, 3.1 and 4.1 in the previous paper arXiv:2108.0201

Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field

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 asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.Comment: 44 pages, 15 figures. All comments are welcome

Metastability for the Ising model on the hexagonal lattice

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 and study the relaxation 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

Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with Statistical Mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In this paper, we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non--Metropolis systems such as Probabilistic Cellular Automata

Crossover times in bipartite networks with activity constraints and time-varying switching rates

In this paper we study the performance of a bipartite network in which customers arrive at the nodes of the network, but not all nodes are able to serve their customers at all times. Each node can be either active or inactive, and two nodes connected by a bond cannot be active simultaneously. This situation arises in wireless random-access networks where, due to destructive interference, stations that are close to each other cannot use the same frequency band. We consider a model where the network is bipartite, the active nodes switch themselves off at rate 1, and the inactive nodes switch themselves on at a rate that depends on time and on which half of the bipartite network they are in. An inactive node cannot become active when one of the nodes it is connected to by a bond is active. The switching protocol allows the nodes to share activity among each other. In the limit as the activation rate becomes large, we compute the crossover time between the two states where one half of the network is active and the other half is inactive. This allows us to assess the overall activity of the network depending on the switching protocol. Our results make use of the metastability analysis for hard-core interacting particle models on finite bipartite graphs derived in an earlier paper. They are valid for a large class of bipartite networks, subject to certain assumptions. Proofs rely on a comparison with switching protocols that are not time-varying, through coupling techniques.Comment: 32 pages, 2 figur

Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions

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 with periodic boundary conditions. Let $\beta>0$ be the inverse temperature and let $\Lambda\subset\Lambda^\beta\subset\mathbb{Z}^2$ be two boxes. We consider the asymptotic regime corresponding to the limit as $\beta\rightarrow\infty$ for finite volume $\Lambda$ and $\lim_{\beta\rightarrow\infty}\frac{1}{\beta}\log|\Lambda^\beta|=\infty$. We study the simplified model, in which particles perform independent random walks on $\Lambda^\beta\setminus\Lambda$ and inside $\Lambda$ particles perform simple exclusion, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $\Lambda^\beta$. The initial configuration is chosen such that $\Lambda$ is empty and $\rho|\Lambda^\beta|$ particles are distributed randomly over $\Lambda^\beta\setminus\Lambda$. Our results will use a deep analysis of a local model, i.e., particles perform Kawasaki dynamics inside $\Lambda$ and along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\Delta>0$ is an activity parameter. Thus, in the local model the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We take $\Delta\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place in the local model with particular attention to the critical configurations that asymptotically have to be crossed with probability 1.Comment: 74 pages and 26 figure

Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics

In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $\Lambda$. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\beta$ is the inverse temperature and $\Delta>0$ is an activity parameter. Thus, the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta\in{(U_1,U_1+U_2)}$ and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature $\beta$. We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \emph{critical droplets}, as well as some of their properties. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the nucleation pattern.Comment: 73 pages, 22 figure