Quenched Site Dilution Research Articles

Kauffman Cellular Automata on Quasicrystal Topology

In this paper we perform numerical simulations to study Kauffman cellular automata (KCA) on quasiperiod lattices. In particular, we investigate phase transition, magnetic entropy and propagation speed of the damage on these lattices. Both the critical threshold parameter $p_{c}$ and the critical exponents are estimated with good precision. In order to investigate the increase of statistical fluctuations and the onset of chaos in the critical region of the model, we have also defined a magnetic entropy to these systems. It is seen that the magnetic entropy behaves in a different way when one passes from the frozen regime ($p<p_{c}$) to the chaotic regime ($p>p_{c}$). For a further analysis, the robustness of the propagation of failures is checked by introducing a quenched site dilution probability $q$ on the lattices. It is seen that the damage spreading is quite sensitive when a small fraction of the lattice sites are disconnected. A finite-size scaling analysis is employed to estimate the critical exponents. From these numerical estimates, we claim that on both pure ($q=0$) and diluted ($q=0.05$) quasiperiodic lattices, the KCA model belongs to the same universality class than on square lattices. Furthermore, with the aim of comparing the dynamical behavior between periodic and quasiperiodic systems, the propagation speed of the damage is also calculated for the square lattice assuming the same conditions. It is found that on square lattices the propagation speed of the damage obeys a power law as $v\sim (p-p_{c})^{\alpha}$, whereas on quasiperiod lattices it follows a logarithmic law as $v \sim \ln(p-p_{c})^\alpha$.

Critical properties of the contact process with quenched dilution

We have studied the critical properties of the contact process on a square lattice with quenched site dilution by Monte Carlo simulations. This was achieved by generating in advance the percolating cluster, through the use of an appropriate epidemic model, and then by the simulation of the contact process on the top of the percolating cluster. The dynamic critical exponents were calculated by assuming an activated scaling relation and the static exponents by the usual power law behavior. Our results are in agreement with the prediction that the quenched diluted contact process belongs to the universality class of the random transverse-field Ising model. We have also analyzed the model and determined the phase diagram by the use of a mean-field theory that takes into account the correlation between neighboring sites.

Spatial evolutionary game with bond dilution

Abstract In this paper, we study numerically the prisoner’s dilemma game (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched and annealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at the critical occupation probability q ∗ , which is higher than the bond percolation transition point for a square lattice. In the critical region, the defined order parameter has a scaling form as P e ∼ ( q − q ∗ ) β for q q ∗ with the critical exponents β = 1.42 for PDG and β = 1.52 for SG, which differ from those with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperative behavior. We find that the cooperation is much enhanced in the range of small payoff parameters on a lattice with slightly annealed bond dilution.

Quenched-vacancy induced spin-glass order

The ferromagnetic phase of an Ising model in d=3 , with any amount of quenched antiferromagnetic bond randomness, is shown to undergo a transition to a spin-glass phase under sufficient quenched bond dilution. This result, demonstrated here with the numerically exact global renormalization-group solution of a d=3 hierarchical lattice, is expected to hold true generally, for the cubic lattice and for quenched site dilution. Conversely, in the ferromagnetic-spin-glass-antiferromagnetic phase diagram, the spin-glass phase expands under quenched dilution at the expense of the ferromagnetic and antiferromagnetic phases. In the ferromagnetic-spin-glass phase transition induced by quenched dilution, reentrance as a function of temperature is seen, as previously found in the ferromagnetic-spin-glass transition induced by increasing the antiferromagnetic bond concentration.

Dynamic hysteresis in ferroelectrics with quenched randomness

Dynamic hysteresis is simulated in two-dimensions for random ferroelectrics using a Ginzburg–Landau–Devonshire model with quenched site dilution. The frequency and dilution scaling of the hysteresis loop area and the coercive field are analyzed.

Critical behavior in the site-diluted three-dimensional three-state Potts model

We have studied numerically the effect of quenched site dilution on a first order phase transition in three dimensions. We have simulated the site diluted three states Potts model studying in detail the second order region of its phase diagram. We have found that the $\nu$ exponent is compatible with the one of the three dimensional diluted Ising model whereas the $\eta$ exponent is definitely different.

Fermionic structure of two-dimensional Ising model with quenched site dilution

We apply a new anticommuting path integral technique to clarify the fermionic structure of the 2D Ising model with quenched site dilution. In the N-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. An unusual feature is that the leading term of the interaction in the exact lattice theory is of order 2 N in the fermions, where N is the number of replicas. The continuum-limit approximation near T c for weak dilution produces, however, an effective four-fermion interaction. In particular, this implies the double-logarithmic singularity in the specific heat near T c for weak site dilution. The exact value of the initial slope is also obtained.

A dilute bootstrap percolation: Lattice model for unresponsiveness in T-cell immunology

The biased majority rule of cellular automata takes a spin up if and only if at least two of its four nearest neighbors on the square lattice are up. We generalize this type of bootstrap percolation by introducing quenched site dilution as well as a random birth and decay process. Our Monte Carlo simulations then give first-order transitions qualitatively similar to our results from meanfield reaction equations describing the induction of T-cell unresponsiveness in the immune system.

Finite-size-scaling analysis of subsystem data in the dilute Ising model.

Monte Carlo simulation results for the magnetization of subsystems of finite lattices are used to determine the critical temperature and a critical exponent of the simple-cubic Ising model with quenched site dilution, at a concentration of [ital p]=40%. Particular attention is paid to the effect of the finite size of the systems from which the subsystem results are obtained. This finiteness of the lattices involved is shown to be a source of large deviations of critical temperatures and exponents estimated from subsystem data from their values in the thermodynamic limit. By the use of different lattice sizes, the results [ital T][sub [ital c]](40%)=1.209[plus minus]0.002 and [nu](40%)=0.78[plus minus]0.01 could be extrapolated.

Transmission of order in a correlated spin glass.

We introduce a new family of models for spin-glass systems. This family has quenched site dilution with varying strengths of local correlations and modulated ordered phases in the pure limit. The correlated dilution mimics the experimental situation in Cu-Mn, Au-Fe, and Pt-Mn. Relatively weak correlations are conjectured to be sufficient to explain the observed interactions between regions of modulated spin order and coexisting smaller ferromagnetic regions in CuMn. The models are investigated by the transmission-of-order approach, which relates the zero-temperature line of the temperature-dilution phase diagram to a family of correlated percolation models. Exact results for the correlated percolation models on the Cayley tree and simulations on the square lattice are presented. We find novel phases in which percolation occurs on some but not all sublattices.

Free energy of weakly dilute Ising models

The effect of random quenched site dilution on the free energy of nearest-neighbor Ising models is considered. An exact relation between the distribution of local fields and the change in the free energy on dilution is derived. For frustrated systems at low temperatures, dilution leads to a lowering of the entropy, while for well-ordered systems dilution raises the internal energy.

Effect of quenched impurities on long-range order in systems with a frustrated ground state

Destruction of long-range order (LRO) in interacting systems with a frustrated ground state due to quenched dilution is discussed. In such systems there is an additional contribution to the disordering process when initially frustrated bonds become satisfied as dilution removes the competing bonds. It is therefore expected that at $T=0$, LRO vanishes at an impurity concentration ${q}_{L}$ which is lower than the corresponding percolation threshold impurity concentration ${q}_{c}$. To study this phenomenon the anisotropic Ising antiferromagnet on a triangular lattice with quenched site dilution is analyzed. An estimate for ${q}_{L}$ is obtained by means of series-expansion methods. This model may describe the magnetic phase of oxygen adsorbed on graphite which has been of considerable interest in recent years.

The magnetically dilute XY ferromagnet. III. Two-dimensional lattices

For pt.II see ibid., vol.9, p.2575 (1976). The two-dimensional spin-1/2 XY model is investigated for the two cases of quenched-site dilution and annealed-site dilution. On the square and triangular lattices, seven terms have been derived in the low-concentration expansion for the free energy and for the fluctuation in the perpendicular magnetization. Analysis of the fluctuation series provides evidence for a phase transition. High-temperature specific-heat curves are presented for possible comparison with experiment.