# Mathematical Examples of Space-Time Phases

@article{Diakonova2011MathematicalEO, title={Mathematical Examples of Space-Time Phases}, author={Marina Diakonova and Robert S. MacKay}, journal={Int. J. Bifurc. Chaos}, year={2011}, volume={21}, pages={2297-2304} }

The space-time phases of a complex dynamic system are the probability distributions for state as a function of space and time which arise by evolving initial probability distributions from the distant past. Toom proved important results about space-time phases for a class of majority voter probabilistic cellular automata (PCA). Here, variants of the majority voter PCA are presented which are proved to exhibit a variety of types of space-time phase. These examples are expected to serve as useful…

## 14 Citations

### Management of complex dynamical systems

- Mathematics
- 2018

Complex dynamical systems are systems with many interdependent components which evolve in time. One might wish to control their trajectories, but a more practical alternative is to control just their…

### Convergence Time of Probabilistic Cellular Automata on the Torus

- Computer Science, Mathematics
- 2018

This work analyzes the correspondence for Percolation PCA, a class of probabilistic cellular automata which are closely related to oriented percolation.

### Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

- Mathematics
- 2011

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of Toom. They are defined as stochastic perturbations of cellular…

### Overview: PCA Models and Issues

- Computer Science
- 2018

This book is an attempt to present a wide panorama of the current status of PCA theory and applications, which cover important issues and applications in probability, statistical mechanics, computer science, natural sciences, and dynamical systems.

### Phase Diagrams of Majority Voter Probabilistic Cellular Automata

- Computer Science
- 2015

A detailed numerical analysis of the phase diagrams for some majority voter probabilistic cellular automata is presented and the results are connected with theory, enabling many of the observed features to be proved.

### Percolation Operators and Related Models

- Mathematics
- 2018

The theory and methods of the two basic stochastic models with absorbing states: percolation operators and contact processes are reviewed and the challenges associated with reconciling different mathematical descriptions of natural phenomena are demonstrated.

### Rotating States in Driven Clock- and XY-Models

- Physics
- 2011

We consider 3D active plane rotators, where the interaction between the spins is of XY-type and where each spin is driven to rotate. For the clock-model, when the spins take N≫1 possible values, we…

### Toward a Quantitative Formulation of Emergence

- Computer Science
- 2013

A quantitative approach to emergence is developed, based on the property that in complex multivariate systems one can achieve under certain conditions a reduced closed-form description in terms of a…

### Presidential Address : Complex Systems in Science and Society

- Business
- 2013

I had the pleasure to give the IMA’s Presidential Address at the Royal Society and five IMA branches around the country in 2012–13. This is a written version. The core message is that mathematics has…

### The Elusive Present: Hidden Past and Future Dependency and Why We Build Models

- PsychologyPhysical review. E
- 2016

How much information can be shared between the past and the future but not reflected in the present is investigated and the consequences, the most important of which is that when the present hides past-future correlation or dependency the authors must move beyond sequence-based statistics and build state-based models.

## References

SHOWING 1-10 OF 28 REFERENCES

### Statistical mechanics of probabilistic cellular automata

- Computer Science
- 1990

The behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, is investigated from a rigorous statistical mechanics point of view and uniqueness of the invariant state and exponential decay of correlations in a high-noise regime is demonstrated.

### Stochastic and spatial structures of dynamical systems

- Mathematics
- 1996

This collection of conference papers on dynamical systems is divided into three sections: the effect of noise on data generated by dynamical systems, and testing whether these systems adequately…

### Robustness of Markov processes on large networks

- Mathematics
- 2011

A metric on the space of probability measures on the state of a large network is introduced, with respect to which the stationary measure of a Markov process on the network is proved under suitable…

### Temporally periodic phases and kinetic roughening.

- Materials SciencePhysical review letters
- 1993

The analogy between temporally periodic phases of noisy extended driven systems and smooth interfaces in growth models is used to derive results for both problems, viz., stable, temporally…

### Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems

- Computer Science
- 2008

Clustering in Globally Coupled Maps.- Snychronisation in CML.- Dynamics of Frenkel-Kontorova Chains.- Attractors and Bifurcations in CML.- Dynamics of Genetic Regulation Networks.- Phase Transitions…

### Coupled map lattices with phase transition

- Mathematics
- 2000

Examples of coupled map lattices are presented for which we prove the existence of phase transitions, in the senses of a non-unique `natural' measure or Gibbs state. They are constructed from…

### Are attractors relevant to turbulence?

- PhysicsPhysical review letters
- 1988

The statistical hypothesis underlying the "strange attractor" explanation of fluid turbulence is suspect and the nature of the attractor is irrelevant to the observed behavior when such systems are of even moderate size.

### On Critical Values for Some Random Processes with Local Interaction in R2

- Mathematics
- 2002

We complete the proof of a necessary and sufficient condition for existence of non-trivial critical values for some classes of random processes with local interaction, where the space is a real…

### A non-linear eroder in presence of one-sided noise

- Mathematics
- 2006

We study a class of cellular automata, that is random oper- ators acting on normed measures on the space {0,...,m} ZZ d which can be presented as superpositions FrD, where D is a monotonic…

### Freezing in Ising ferromagnets.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

This work investigates the final state of zero-temperature Ising ferromagnets that are endowed with single-spin-flip Glauber dynamics, and finds that the ground state is generally not reached for zero initial magnetization.