Asynchronous Cellular Automata and Brownian Motion

International audience This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.

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Condensation of SIP Particles and Sticky Brownian Motion

Journal of Statistical Physics ◽

10.1007/s10955-021-02775-5 ◽

2021 ◽

Vol 183 (3) ◽

Author(s):

Mario Ayala ◽

Gioia Carinci ◽

Frank Redig

Keyword(s):

Brownian Motion ◽

Interacting Particle Systems ◽

Particle Systems ◽

Self Duality ◽

Inclusion Process ◽

Interacting Particle ◽

New Information ◽

Homogeneous Product ◽

Coarsening Dynamics ◽

The Difference

AbstractWe study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time t. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.

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Pseudorandom Number Generators Based on Asynchronous Cellular Automata and Cellular Automata With Inhomogeneous Cells

Advances in Systems Analysis, Software Engineering, and High Performance Computing - Formation Methods, Models, and Hardware Implementation of Pseudorandom Number Generators ◽

10.4018/978-1-5225-2773-2.ch006 ◽

2017 ◽

pp. 127-138

Keyword(s):

Cellular Automata ◽

Cellular Automaton ◽

Local State ◽

Active Cell ◽

State Function ◽

Local Function ◽

Time Step ◽

Random Number Generators ◽

Asynchronous Cellular Automata ◽

A Cell

The sixth chapter deals with the construction of pseudo-random number generators based on a combination of two cellular automata, which were considered in the previous chapters. The generator is constructed based on two cellular automata. The first cellular automaton controls the location of the active cell on the second cellular automaton, which realizes the local state function for each cell. The active cell on the second cellular automaton is the main cell and from its output bits of the bit sequence are formed at the output of the generator. As the first cellular automaton, an asynchronous cellular automaton is used in this chapter, and a synchronous cellular automaton is used as the second cellular automaton. In this case, the active cell of the second cellular automaton realizes another local function at each time step and is inhomogeneous. The algorithm for the work of a cell of a combined cellular automaton for implementing a generator and its hardware implementation are presented.

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EXTENDED APPLICATION OF CONSTRUCTIVE CRITERIA FOR ERGODICITY OF INTERACTING PARTICLE SYSTEMS

International Journal of Modern Physics C ◽

10.1142/s0129183196000028 ◽

1996 ◽

Vol 07 (01) ◽

pp. 1-18 ◽

Cited By ~ 2

Author(s):

HANS DE JONG ◽

CHRISTIAN MAES

Keyword(s):

Cellular Automata ◽

Discrete Time ◽

Continuous Time ◽

Interacting Particle Systems ◽

Particle Systems ◽

Probabilistic Cellular Automata ◽

Interacting Particle ◽

Spin Flip ◽

Extended Application ◽

Computational Aspects

We discuss computational aspects of verifying constructive criteria for ergodicity of interacting particle systems. Both discrete time (probabilistic cellular automata) and continuous time spin flip dynamics are considered. We also investigate how the criteria have to be adapted if stirring is added to the dynamics.

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