Zero-temperature Glauber Dynamics Research Articles

From classical to quantum stochastic processes.

In this paper, we construct quantum analogs starting from classical stochastic processes, by replacing random "which path" decisions with superpositions of all paths. This procedure typically leads to nonunitary quantum evolution, where coherences are continuously generated and destroyed. In spite of their transient nature, these coherences can change the scaling behavior of classical observables. Using the zero temperature Glauber dynamics in a linear Ising spin chain, we find quantum analogs with different domain growth exponents. In some cases, this exponent is even smaller than for the original classical process, which means that coherence can play an important role to speed up the relaxation process.

Nonequilibrium dynamics in Ising-like models with biased initial condition.

We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number z in the Ising model. For the generalized voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of z are also obtained. A related study is the behavior of the exit probability E(x_{0}), defined as the probability that a configuration ends up with all spins up starting with x_{0} fraction of up spins. An interesting result is E(x_{0})=x_{0} in the mean field approximation when z=2, which is consistent with the conserved magnetization in the system. For larger values of z,E(x_{0}) shows the usual finite size dependent nonlinear behavior both in the mean field model and in the Ising model with nearest neighbor interaction on different two dimensional lattices. For such a behavior, a data collapse of E(x_{0}) is obtained using y=(x_{0}-x_{c})/x_{c}L^{1/ν} as the scaling variable and f(y)=1+tanh(λy)/2 appears as the scaling function. The universality of the exponent and the scaling factor is investigated.

Zero-temperature Glauber dynamics on the 3-regular tree and the median process

In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d.~initial spins $\sigma_x(0)$ from $\{-1,+1\}$ with $\mathbb{P}_p(\sigma_x(0) = +1)=p$, and they update their spins at the arrival times of i.i.d. Poisson processes to agree with a majority of their neighbors. We study this process on the 3-regular tree $\mathbb{T}_3$, where it is known that the critical threshold $p_c$, below which $\mathbb{P}_p$-a.s. all spins fixate to $-1$, is strictly less than $1/2$. Defining $\theta(p)$ to be the $\mathbb{P}_p$-probability that a vertex fixates to $+1$, we show that $\theta$ is a continuous function on $[0,1]$, so that, in particular, $\theta(p_c)=0$. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures $\mathbb{P}_p$. Along the way, we study the time-infinity agreement clusters of the median process, show that they are a.s. finite, and deduce that all continuous spins flip finitely often. In the second half of the paper, we show a correlation decay statement for the discrete spins under $\mathbb{P}_p$ for a.e. value of $p$. The proof relies on finiteness of a vertex's "trace" in the median process to derive a stability of discrete spins under finite resampling. Last, we use our methods to answer a question of C. Howard (2001) on the emergence of spin chains in $\mathbb{T}_3$ in finite time.

Hysteresis in the zero-temperature random-field Ising model on directed random graphs

We use zero-temperature Glauber dynamics to study hysteresis in the random-field Ising model on directed random graphs. The critical behavior of the model depends on the connectivity $z$ of the graph rather differently from that on undirected graphs. Directed graphs and zero-temperature dynamics are relevant to a wide class of social phenomena including opinion dynamics. We discuss the efficacy of increasing external influence in inducing a first-order phase transition in opinion dynamics. The numerical results are supported by an analytic solution of the model.

Coarsening Model on $${\mathbb{Z}^{d}}$$ with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP

We study the coarsening model (zero-temperature Ising Glauber dynamics) on $\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a Markov process on the state space $\{-1,+1\}^{\mathbb{Z}^d}$ of "spin configurations" in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$ with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial state of this Markov chain is distributed according to a product measure with probability $p$ for a spin to be $+1$. In this paper, we show that for any given $p>0$, there exist $q$ close enough to 1 such that a.s. every spin has a limit of $+1$. This is of particular interest for small values of $p$, for which it is known that if $q=1/2$, a.s. all spins have a limit of $-1$. For dimension $d=2$, we also obtain near-exponential convergence rates for $q$ sufficiently large, and for general $d$, we obtain stretched exponential rates independent of $d$. Two important ingredients in our proofs are refinements of block arguments of Fontes-Schonmann-Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.

Fate of Ising ferromagnets and antiferromagnets by zero-temperature Glauber dynamics on the two-dimensional Archimedean and 2-uniform lattices

The fate of the Ising ferromagnet and antiferromagnet by the zero-temperature Glauber dynamics from random initial spin configuration is investigated in two-dimensional Archimedean and 2-uniform lattices. Blinker states are found in addition to the ground state and metastable state. We show that an even-coordinated lattice can arrive at a blinker state or a metastable state without a striped structure, in contrast to common expectations. The universal relationship between the critical percolation and the probability of a striped final state is confirmed for six lattices. Results about the fate of the antiferromagnetic Ising model show that the geometric frustration suppresses ordering more and promotes blinker state.

Fragmentation transition in a coevolving network with link-state dynamics.

We study a network model that couples the dynamics of link states with the evolution of the network topology. The state of each link, either A or B, is updated according to the majority rule or zero-temperature Glauber dynamics, in which links adopt the state of the majority of their neighboring links in the network. Additionally, a link that is in a local minority is rewired to a randomly chosen node. While large systems evolving under the majority rule alone always fall into disordered topological traps composed by frustrated links, any amount of rewiring is able to drive the network to complete order, by relinking frustrated links and so releasing the system from traps. However, depending on the relative rate of the majority rule and the rewiring processes, the system evolves towards different ordered absorbing configurations: either a one-component network with all links in the same state or a network fragmented in two components with opposite states. For low rewiring rates and finite-size networks there is a domain of bistability between fragmented and nonfragmented final states. Finite-size scaling indicates that fragmentation is the only possible scenario for large systems and any nonzero rate of rewiring.

The expanded triangular Kitaev–Heisenberg model in the full parameter space

Abstract The classical Kitaev–Heisenberg model on the triangular lattice is investigated by simulation in its full parameter space together with the next-nearest neighboring Heisenberg interaction or the single-ion anisotropy. The variation of the system is demonstrated directly by the joint density of states (DOS) depending on energy and magnetization obtained from Wang–Landau algorithm. The Metropolis Monte Carlo simulation and the zero-temperature Glauber dynamics are performed to show the internal energy, the correlation functions and spin configurations at zero temperature. It is revealed that two types of DOS (U and inverse U) divide the whole parameter range into two main parts with antiferromagnetic and ferromagnetic features respectively. In the parameter range of U type DOS, the mixed frustration from the triangular geometry and the Kitaev interaction produces rich phases, which are influenced in different ways by the next-nearest neighboring Heisenberg interaction and the single-ion anisotropy.

Particle-current fluctuations in a variant of the asymmetric Glauber model

We study the total particle-current fluctuations in a one-dimensional stochastic system of classical particles consisting of branching and death processes which is a variant of asymmetric zero-temperature Glauber dynamics. The full spectrum of a modified Hamiltonian, whose minimum eigenvalue generates the large deviation function for the total particle-current fluctuations through a Legendre-Fenchel transformation, is obtained analytically. Three examples are presented and numerically exact results are compared to our analytical calculations.

Phase transition in a two-dimensional Ising ferromagnet based on the generalized zero-temperature Glauber dynamics

At zero temperature, based on the Ising model, the phase transition in a two-dimensional square lattice is studied using the generalized zero-temperature Glauber dynamics. Using Monte Carlo (MC) renormalization group methods, the static critical exponents and the dynamic exponent are studied; the type of phase transition is found to be of the first order.

Fixation for coarsening dynamics in 2D slabs

We study zero-temperature Ising Glauber Dynamics, on $2D$ slabs of thickness $k \geq 2$. In this model, $\pm 1$-valued spins at integer sites update according to majority vote dynamics with two opinions. We show that all spins reaches a final state (that is, the system fixates) for $k=2$ under free boundary conditions and for $k=2$ or $3$ under periodic boundary conditions. For thicker slabs there are sites that fixate and sites that do not.

Phase diagram for a zero-temperature Glauber dynamics under partially synchronous updates

We consider generalized zero-temperature Glauber dynamics under a partially synchronous updating mode for a one-dimensional system. Using Monte Carlo simulations, we calculate the phase diagram and show that the system exhibits phase transition between the ferromagnetic and active antiferromagnetic phases. Moreover, we provide analytical calculations that allow us to understand the origin of the phase transition and confirm simulation results obtained earlier for synchronous updates.

Absorbing states of zero-temperature Glauber dynamics in random networks

We study zero-temperature Glauber dynamics for Ising-like spin variable models in quenched random networks with random zero-magnetization initial conditions. In particular, we focus on the absorbing states of finite systems. While it has quite often been observed that Glauber dynamics lets the system be stuck into an absorbing state distinct from its ground state in the thermodynamic limit, very little is known about the likelihood of each absorbing state. In order to explore the variety of absorbing states, we investigate the probability distribution profile of the active link density after saturation as the system size N and (k) vary. As a result, we find that the distribution of absorbing states can be split into two self-averaging peaks whose positions are determined by (k), one slightly above the ground state and the other farther away. Moreover, we suggest that the latter peak accounts for a nonvanishing portion of samples when N goes to infinity while (k) stays fixed. Finally, we discuss the possible implications of our results on opinion dynamics models.

Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$

We study zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\) , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define \({p_c(\mathbb{Z}^d)}\) to be the infimum over p such that the system fixates at ‘ + ’ with probability 1. It is a folklore conjecture that \({p_c(\mathbb{Z}^d) = 1/2}\) for every \({2 \le d \in \mathbb{N}}\) . We prove that \({p_c(\mathbb{Z}^d) \to 1/2}\) as d → ∞.

Zero-temperature Glauber dynamics on small-world networks

The zero-temperature Glauber dynamics of the ferromagnetic Ising model on small-world networks, rewired from a two-dimensional square lattice, has been studied by numerical simulations. For increasing disorder in finite networks, the nonequilibrium dynamics becomes faster so that the ground state is found more likely. For any finite value of the rewiring probability p, the likelihood of reaching the ground state goes to zero in the thermodynamic limit, similarly to random networks. The spin correlation ξ(r) is found to decrease with distance as ξ(r) ∼ exp(−r/λ), λ being a correlation length scaling with p as λ ∼ p−0.73. These results are compared with those obtained earlier for addition-type small-world networks.

Percolation framework in Ising-spin relaxation

We introduce a framework based on the percolation idea to investigate the relaxation under zero-temperature Glauber and outflow dynamics on L x L square and triangular lattices. This helps us to understand the appearance of a double time regime in the survival probability. We show that the first, short-time, regime corresponds to relaxation through droplets and the second, long-time, regime corresponds to relaxation through stripes. For both dynamics the probability that the system becomes ordered through droplets (which indicates fast relaxation) is about 2/3 .

PERSISTENCE IN THE ZERO-TEMPERATURE DYNAMICS OF THE Q-STATES POTTS MODEL ON UNDIRECTED-DIRECTED BARABÁSI–ALBERT NETWORKS AND ERDÖS–RÉNYI RANDOM GRAPHS

The zero-temperature Glauber dynamics is used to investigate the persistence probability P(t) in the Potts model with Q = 3, 4, 5, 7, 9, 12, 24, 64, 128, 256, 512, 1024, 4096, 16 384, …, 230 states on directed and undirected Barabási–Albert networks and Erdös–Rényi (ER) random graphs. In this model, it is found that P(t) decays exponentially to zero in short times for directed and undirected ER random graphs. For directed and undirected BA networks, in contrast it decays exponentially to a constant value for long times, i.e., P(∞) is different from zero for all Q values (here studied) from Q = 3, 4, 5, …, 230; this shows "blocking" for all these Q values. Except that for Q = 230 in the undirected case P(t) tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.

A class of stochastic games with infinitely many interacting agents related to Glauber dynamics on random graphs

We introduce and study a class of infinite-horizon non-zero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove ‘fixation’, i.e. players will adopt a constant strategy after a finite time. The resulting dynamics is related to zero-temperature Glauber dynamics on random graphs of possibly infinite volume.

Comparison of voter and Glauber ordering dynamics on networks

We study numerically the ordering process of two very simple dynamical models for a two-state variable on several topologies with increasing levels of heterogeneity in the degree distribution. We find that the zero-temperature Glauber dynamics for the Ising model may get trapped in sets of partially ordered metastable states even for finite system size, and this becomes more probable as the size increases. Voter dynamics instead always converges to full order on finite networks, even if this does not occur via coherent growth of domains. The time needed for order to be reached diverges with the system size. In both cases the ordering process is rather insensitive to the variation of the degree distribution from sharply peaked to scale free.

Voter dynamics on an Ising ladder: coarsening and persistence

Coarsening and persistence of Ising spins on a ladder is examined under voter dynamics. The density of domain walls decreases algebraically with time as $t^-{1/2}$ for sequential as well as parallel dynamics. The persistence probability decreases as $t^{-\theta_{s}}$ under sequential dynamics, and as $t^{-\theta_{p}}$ under parallel dynamics where $\theta_{p} = 2 \theta_{s} \approx .88$. Numerical values of the exponents are explained. The results are compared with the voter model on one and two dimensional lattices, as well as Ising model on a ladder under zero-temperature Glauber dynamics.