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Register a new userClusters and Recurrence in the Two-Dimensional Zero-Temperature Stochastic Ising Model
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We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or -1 to each site in ${bf Z}^2$, is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate one, polls its 4 neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times $t< infty$, but the cluster of a fixed site diverges (in diameter) as $t to infty$; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.
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We investigate the nonequilibrium dynamics following a quench to zero temperature of the non-conserved Ising model with power-law decaying long-range interactions $propto 1/r^{d+sigma}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $alpha$, the persistence exponent $theta$, and the fractal dimension $d_f$. It is found that the growth exponent $alphaapprox 3/4$ is independent of $sigma$ and different from $alpha=1/2$ as expected for nearest-neighbor models. In the large $sigma$ regime of the tunable interactions only the fractal dimension $d_f$ of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponent $theta$ this is a direct consequence of the different growth exponents $alpha$ as can be understood from the relation $d-d_f=theta/alpha$; they just differ by the ratio of the growth exponents $approx 3/2$. This relation has been proposed for annihilation processes and later numerically tested for the $d=2$ nearest-neighbor Ising model. We confirm this relation for all $sigma$ studied, reinforcing its general validity.
We reconsider the criticality of the Ising model on two-dimensional dynamical triangulations based on the N-by-N hermitian two-matrix model with the introduction of a loop-counting parameter and linear terms in the potential. We show that in the large-N limit even though the Ising model is classical, the critical temperature can reach absolute zero by tuning the loop-counting parameter, and the corresponding continuum theory turns out to be the quantised theory of neither pure gravity nor gravity coupled to conformal matter with central charge being 1/2.
We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes.
The dynamic evolution at zero temperature of a uniform Ising ferromagnet on a square lattice is followed by Monte Carlo computer simulations. The system always eventually reaches a final, absorbing state, which sometimes coincides with a ground state (all spins parallel), and sometimes does not (parallel stripes of spins up and down). We initiate here the numerical study of ``Chaotic Time Dependence (CTD) by seeing how much information about the final state is predictable from the randomly generated quenched initial state. CTD was originally proposed to explain how nonequilibrium spin glasses could manifest equilibrium pure state structure, but in simpler systems such as homogeneous ferromagnets it is closely related to long-term predictability and our results suggest that CTD might indeed occur in the infinite volume limit.
We study the zero-temperature criticality of the Ising model on two-dimensional dynamical triangulations to contemplate its physics. As it turns out, an inhomogeneous nature of the system yields an interesting phase diagram and the physics at the zero temperature is quite sensitive about how we cool down the system. We show the existence of a continuous parameter that characterizes the way we approach the zero-temperature critical point and it may enter in a critical exponent.
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