Spreading of damage: An unexpected disagreement between the sequential and parallel updatings in Monte Carlo simulations

Spin-1 Ising model: exact damage-spreading relations and numerical simulations

The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.

Using exact relations in damage-spreading simulations: the Baxter line of the two-dimensional Ashkin-Teller model

The nearest-neighbor-interaction ferromagnetic Ashkin-Teller model is investigated on a square lattice through a powerful computational method for dealing with correlation functions in magnetic systems. This technique, which is based on damage-spreading numerical simulations, makes use of exact relations involving special kinds of damage and correlation functions, as well as the corresponding order parameters of the model. The computation of correlation functions, which represents usually a hard task in standard Monte Carlo simulations, due to large fluctuations, turns out to be much simpler within the present approach. We concentrate our analysis along the Baxter line, well known for its continuously varying critical exponents; seven different points along this line are investigated. The critical exponents associated with correlation functions along the Baxter line are successfully evaluated, by means of numerical methods, within damage-spreading simulations. The efficiency of this method is confirmed through precise estimates of the critical exponents associated with the order parameters (magnetization and polarization), as well as with their corresponding correlation functions, in spite of the small lattice sizes considered.

Damage-spreading simulations through exact relations for the two-dimensional Potts ferromagnet

A powerful computational method for dealing with correlation functions in magnetic systems, based on damage-spreading simulations, is reviewed and tested, by investigating the q-state Potts ferromagnet, on a square lattice, at criticality. Exact relations involving special kinds of damage and the spin-spin correlation function, as well as the magnetization, are used. The efficiency of the method arises with a significant reduction of the finite-size effects, with respect to conventional Monte Carlo simulations. Correlation functions, which represent usually a hard task within this latter procedure, appear to be much more easily estimated through the present damage-spreading simulations. The effectiveness of the technique is illustrated by an accurate estimate of the exponent eta, of the spin-spin correlation function, for q=2, 3, and 4, with rather small lattice sizes. In the cases q > or = 5, an analysis of the magnetization is consistent with the well-known first-order phase transition.

Damage spreading in two-dimensional trivalent cellular structures with competing Glauber and Kawasaki dynamics

The damage spreading of the Ising model on several two-dimensional trivalent structures, including soap froth, Voronoi, and hierarchical structures, are studied with competing Glauber and Kawasaki dynamics. The damage spreading transition temperature T(d) and the Curie temperature T(C) of these structures are compared. We find that T(d) of the hierarchical lattices decreases sharply as the probability of occurrence of Kawasaki dynamics increases, whereas for soap froth and Voronoi, T(d) for the Voronoi and soap froth remain nearly unchanged except when the dynamics is dominated by Kawasaki dynamics. T(d) and T(C) in our two-dimensional structures are nearly the same and they behave similarly as we change the relative probability of occurrence of the Glauber and Kawasaki dynamics. A heuristic argument is provided to explain the numerical results.

Damage spreading on two-dimensional trivalent structures with Glauber dynamics: hierarchical and random lattices

The damage spreading of the Ising model on various two-dimensional trivalent structures with Glauber dynamics is investigated. It is shown that topology plays an important role in determining the damage spreading transition temperatures of the trivalent structures. When damage is considered in terms of only the topological properties of the cellular patterns, the transition temperature above which damage is saturated is found to be determined by the cells with the highest edge number. When the area of cells is also taken into account in the computation of damage, the damage spreading transition temperatures are all lowered. These results are verified by simulation on a set of hierarchical lattices constructed by recursive application of the star-triangle transformation on the vertices of the hexagonal structure, as well as soap froth and randomized lattice structures using Voronoi construction.

Damage-spreading phase and damage-frozen phase in a solid-on-solid model

Based on the recently suggested scaling ansatz [Phys. Rev. E 62, 3376 (2000)] for damage spreading in the surface roughening phenomenon, the characteristics of the damage-spreading phase and damage-frozen phase in a two-dimensional solid-on-solid model that has a roughening transition at T=T(R) are studied. In the damage-spreading phase, which exists for T>T(R), the average vertical damage-spreading distance d(perpendicular)(d(//)=0,L,T) and the average lateral damage-spreading distance D(//)(L,T) are shown to satisfy d(perpendicular)(d(//)=0,L,T) approximately ln L and D(//)(L,T) approximately L, respectively. In the damage-frozen phase, which exists for T<T(R), it is shown that d( perpendicular)(d(//)=0,L-->infinity,T) approximately finite and D(//)(L-->infinity,T) approximately finite. From these results it is concluded that the damage-spreading phase describes the surface roughening phase well and the damage-frozen phase describe the smooth phase well.

Dynamical self-affinity of damage spreading in surface growth models

The dynamical anisotropic scaling properties of the surface growth models are restudied by use of the damage spreading concept. For that the vertical damage spreading distance d( perpendicular) of a damaged column as well as the lateral damage spreading distance d(||) is introduced. The scaling Ansatze for &dmacr;( perpendicular)(d(||),t), D(||) identical with<d(||)> and D( perpendicular) identical with<&dmacr;( perpendicular)> are suggested. The critical property of the probability distribution P(d(||),t) for the survived damages is also suggested. The suggested scaling relations are tested by simulating various growth models with substrate dimension d=1. From these results it can be concluded that the critical property or dynamical self-affinity of a surface growth model can also be determined by investigating the damage spreading.