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We show that the notion of critical droplets is central to an understanding of the nature of ground states in the Edwards–Anderson–Ising model of a spin glass in arbitrary dimensions. Given a specific ground state, we suppose that the coupling value for a given edge is varied with all other couplings held fixed. Beyond some specific value of the coupling, a droplet will flip, leading to a new ground state; we refer to this as the critical droplet for that edge and ground state. We show that the distribution of sizes and energies over all edges for a specific ground state can be used to determine which of the leading scenarios for the spin glass phase is correct. In particular, the existence of low-energy interfaces between incongruent ground states, as predicted by replica symmetry breaking, is equivalent to the presence of critical droplets, whose boundaries comprise a positive fraction of edges in the infinite lattice. [ABSTRACT FROM AUTHOR]

We show that the notion of critical droplets is central to an understanding of the nature of ground states in the Edwards–Anderson–Ising model of a spin glass in arbitrary dimensions. Given a specific ground state, we suppose that the coupling value for a given edge is varied with all other couplings held fixed. Beyond some specific value of the coupling, a droplet will flip, leading to a new ground state; we refer to this as the critical droplet for that edge and ground state. We show that the distribution of sizes and energies over all edges for a specific ground state can be used to determine which of the leading scenarios for the spin glass phase is correct. In particular, the existence of low-energy interfaces between incongruent ground states, as predicted by replica symmetry breaking, is equivalent to the presence of critical droplets, whose boundaries comprise a positive fraction of edges in the infinite lattice.

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature β > 0 in a large finite box Λ β ⊂ Z 2 whose volume depends on β. Each pair of neighboring particles has a negative binding energy − U < 0 , while each particle has a positive activation energy Δ > 0. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime Δ ∈ (U , 2 U) and in the limit as β → ∞ , how and when the system nucleates, i.e., grows a supercritical droplet somewhere in Λ β. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyze how subcritical droplets form and dissolve on multiple space–time scales when the volume is moderately large , namely, | Λ β | = e Θ β with Δ < Θ < 2 Δ − U. In the third paper we consider the setting where the volume is very large , namely, | Λ β | = e Θ β with Δ < Θ < Γ − (2 Δ − U) , where Γ is the energy of the critical droplet in the local model, i.e., when Λ β has a fixed volume not depending on β and particles can be created and annihilated at the boundary, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative , i.e., particles move around and interact but are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach : the tube of typical trajectories leading to nucleation is described via a series of events, whose complements have negligible probability, on which the evolution of the gas can be captured by a coarse-grained Markov chain on a space of droplets, which we refer to as droplet dynamics. [ABSTRACT FROM AUTHOR]

This is the third in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involved three model-dependent quantities: the energy, the shape and the number of critical droplets. In the second paper we proved the first and the second hypothesis and identified the energy of critical droplets. In the third paper we prove the third hypothesis and identify the shape and the number of critical droplets, thereby completing our analysis. Both the second and the third paper deal with understanding the geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system, as explained in the first paper. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi-type particle systems. [ABSTRACT FROM AUTHOR]

We study the metastable behavior of the two-dimensional Ising model in the case of an alternate updating rule: parallel updating of spins on the even (odd) sublattice are permitted at even (odd) times. We show that although the dynamics is different from the Glauber serial case the typical exit path from the metastable phase remains the same. [ABSTRACT FROM AUTHOR]