Can Extra Updates Delay Mixing?

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.     

Metastable behavior of low-temperature glauber dynamics with stirring

We consider the metastable behavior of a superposition of a ferromagnetic spin system with a Glauber dynamics and stirring dynamics. Starting from configuration –1, minus spins at all lattice sites in a fixed volume under periodic boundary conditions, the process stays close to this configuration for an unpredictable time until the formation of a droplet, of spins +1, with a certain critical size and decays to configuration +1 in a relatively short time. We observe that the size of the droplet depends on the rate of exclusion.     

Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree

Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.     

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

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 limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins +1 in a sea of spins −1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down (−1) the pattern of evolution leading to the more stable configuration with all spins up (+1) approaches, as the temperature vanishes, a metastable behavior: the system stays close to −1 for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to +1 in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability. Partially supported by CNPq. Part of this work was done while RHS was visiting Rome, supported by an agreement between CNPq and CNR     

Stochastic model for the dynamics of a spin-oscillator coupled system

We construct a stochastic model for the dynamics of a one-dimensional system consisting of bilinearly coupled harmonic oscillators and spins. The spin dynamics is defined as a Glauber model where the spins are effectively coupled through their interaction with the oscillators. To maintain internal thermal equilibrium in the composite system, which does not exhibit Onsager symmetry, we introduce a phenomenological retarded friction in the oscillator equation of motion and relate it to the spin correlation function through a fluctuation-dissipation theorem. The oscillator susceptibility is derived and the behavior of its poles as functions of wavevector and temperature is studied. The results are compared to those obtained by other authors who have studied similar systems, using irreversible thermodynamics. In contrast to ours, these treatments do not give an explicit result for the wavevector dependence of the poles.     

Degenerate ferrimagnetic ground state of frustrated kagome lattice

The frustrated Ising model on kagome lattice with nearest-neighboring antiferromagnetic interaction is investigated by using Monte Carlo simulation of the Wang-Landau algorithm and Glauber dynamics. The geometrical frustration leads to a particularly high degeneracy of ground states in this system. A small magnetic field applied can lift the degeneracy partially, and produce the magnetization plateau of 1/3 saturate value (Ms), which is analogous to the magnetic behavior in triangular antiferromagnetic system. However, different from the long-range ferrimagnetic state responsible for 1/3 Ms plateau in triangular lattice, the ferrimagnetic ground state corresponding to 1/3 Ms plateau in kagome lattice is short-ranged and still highly degenerate. Furthermore, the spin configuration of these degenerate ferrimagnetic ground states show an inherent characteristic that the spins along the magnetic field must be aligned on the closed loops, which can be well understood in terms of geometrical frustration.     

A Note on the Metastability of the Ising Model: The Alternate Updating Case

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.     

Injected Power Fluctuations in 1D Dissipative Systems

Using fermionic techniques, we compute exactly the large deviation function (ldf) of the time-integrated injected power in several one-dimensional dissipative systems of classical spins. The dynamics are T=0 Glauber dynamics supplemented by an injection mechanism, which is taken as a poissonian flipping of one particular spin. We discuss the physical content of the results, specifically the influence of the rate of the Poisson process on the properties of the ldf.     

Damage spreading on the 3–12 lattice with competing Glauber and Kawasaki dynamics

The damage spreading of the Ising model on the 3–12 lattice with competing Glauber and Kawasaki dynamics is studied. The difference between the two kinds of nearest-neighboring spin interactions (interaction between two 12-gons, or interaction between a 12-gon and a triangle) are considered in the Hamiltonian. It is shown that the ratio of the interaction strengthF between the two kinds of interactions plays an important role in determining the critical temperature T_d of phase transition from frozen to chaotic. Two methods are used to introduce the bond dilution on the Ising model on the 3–12 lattice: regular and random. The maximum of the average damage spreading 〈D〉_max can approach values lower than 0.5 in both cases and the reason can be attributed to the ’survivors’ among the spins. We have also, for the first time, presented the phase diagram of the mixed G-K dynamics in the 3–12 lattice which shows what happens when going from pure Glauber to pure Kawasaki     

Spectrum of relaxation times for ising spin clusters in random fields

Summary Exact results on the single-spin-flip Glauber dynamics of six-coupled random field Ising spins with the coordination number of four are presented. Two distributions of random fields (RF), binary (BD) and Gaussian (GD) ones, are investigated. The effects of the static magnetic field are discussed. In the zero-magnetic-field case, the number of diverging relaxation times is equal to the number of energy minima minus one. This rule breaks in the presence of a magnetic field. The longest relaxation times in the absence of the field verify the Arrhenius law with the energy barrier determined by the energy needed to invert the ground-state spin configuration. At low temperature, according to the Arrhenius law, the spectrum of relaxation times shows a two-peaked distribution on a logarithmic scale. In the GD case of RF, the energy barrier distribution is continuous, while it is quasi-discrete in the BD case.     

Reduced Markov chains at stochastic spin models

From Glauber's stochastic spin model in discrete time, reduced Markov-chain models are constructed. The transition matrices of the reduced models utilize equilibrium correlation functions of the full N-spin system; however, the reduced models involve the time-dependent behavior of only a cluster of spins. The reduced models have as an invariant vector the exact marginal equilibrium probability for the spins in the cluster. In that sense, the reduced models have the same equilibrium as the N-spin Glauber model, but will, in general, display a different time dependence. One of the reduced models is solved exactly here for a one-dimensional lattice, a square lattice, and a simple-cubic lattice.     

Vitrification in a 2D Ising model with mobile bonds

A bond-disordered two-dimensional Ising model is used to simulate Kauzmann's mechanism of vitrification in liquids, by a Glauber Monte Carlo simulation. The rearrangement of configurations is achieved by allowing impurity bonds to hop to nearest neighbors at the same rate as the spins flip. For slow cooling, the theoretical minimum energy configuration is approached, characterized by an amorphous distribution of locally optimally arranged impurity bonds. Rapid cooling to low temperatures regularly finds bond configurations of higher energy, which are both a priori rare and severely restrictive to spin movement, providing a simple realization of kinetic vitrification. A supercooled liquid regime is also found, and characterized by a change in sign of the field derivative of the spin-glass susceptibility at a finite temperature. Received 3 August 2000 and Received in final form 9 March 2001     

Stark resonances in disordered systems

We study further the metastable behavior of Metropolis dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model, with positive and small external field, in the limit as the temperature vanishes (see [NS]). We focus on the typical features of the escape (nucleation) from the (metastable) configuration with all spins –1, to the (stable) configuration with all spins +1. Using the reversibility of the process as the main tool, we prove (for the discrete time version of the model) that the first step of a typical escaping path is the time reverse of a typical time evolution of a shrinking subcritical rectangular droplet, which is one slice smaller than a critical droplet. This subcritical droplet then evolves in a time of order 1 to a critical droplet, which finally grows with features described in [NS].Work partially supported by the Brazilian CNPq and by the American NSF, under grant DMS91-00725     

Magnetization dynamics in isolated Ising chains

The Glauber dynamics of an Ising chain or ring is shown to be determined by two characteristic times: τ₁ for relaxation of the average magnetization per spin and τ₂ for dynamical spontaneous symmetry breaking. An analytical solution for magnetization dynamics in a finite chain with fixed spins at both ends is found by the method of images. This solution is then used to calculate the spin-spin correlation functions for rings and chains. At low temperatures, since τ₁ ? τ₂, there must exist a range of times when the chain is in one of two ordered states.     

Metastability for the Ising Model on the Hypercube

We consider Glauber dynamics for the low-temperature, ferromagnetic Ising Model on the n-dimensional hypercube. We derive precise asymptotic results for the crossover time (the time it takes for the dynamics to go from the configuration with a “\(-1\)” at every vertex, to the configuration with a “\(+1\)” at each vertex) in the limit as the inverse temperature \(\beta \rightarrow \infty \).     

Dynamics of the finite SK spin-glass

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system sizeN. For the largest of these barriers we find a behavior proportional toN with =1/3.     

Strong decay to equilibrium in one-dimensional random spin systems

We consider a spin system on a lattice with finite-range, possibly unbounded random interactions. We show that for such systems the Glauber dynamics cannot decay to equilibrium exponentially fast inL ₂ even at high temperatures. Additionally, for one-dimensional systems with unbounded random couplings we prove that with probability one the corresponding Glauber dynamics has a fast (subexponential) decay to equilibrium in the uniform norm, provided that the distribution of random couplings satisfies some exponential bound.     

Large Deviations for the Macroscopic Motion of an Interface

We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale. Furthermore, using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time, we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough.     

One-dimensional Kinetic Ising Model with Nonuniform Coupling Constants

A nonuniform extension of the Glauber model on a one-dimensional lattice with boundaries is investigated. The static behavior of the system is investigated. It is shown that there are cases where the system exhibits a static phase transition, which is a change of behavior of the static profile of the expectation values of the spins near end points.     

Erratum: “Magnetization dynamics in isolated Ising chains” [<Emphasis Type="Italic">JETP</Emphasis> 110, 360 (2010)]

The Glauber dynamics of an Ising chain or ring is shown to be determined by two characteristic times: τ₁ for relaxation of the average magnetization per spin and τ₂ for dynamical spontaneous symmetry breaking. An analytical solution for magnetization dynamics in a finite chain with fixed spins at both ends is found by the method of images. This solution is then used to calculate the spin-spin correlation functions for rings and chains. At low temperatures, since τ₁ ≫ τ₂, there must exist a range of times when the chain is in one of two ordered states.