scholarly journals Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

2013 ◽  
Vol 45 (04) ◽  
pp. 960-980 ◽  
Author(s):  
Ana Bušić ◽  
Jean Mairesse ◽  
Irène Marcovici
Keyword(s):  
Markov Chain ◽  
Cellular Automaton ◽  
Invariant Measures ◽  
Sampling Procedure ◽  
Monotonicity Property ◽  
Local Rule ◽  
Probabilistic Cellular Automata ◽  
Perfect Sampling ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

scholarly journals Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

2013 ◽  
Vol 45 (4) ◽  
pp. 960-980 ◽  
Author(s):  
Ana Bušić ◽  
Jean Mairesse ◽  
Irène Marcovici
Keyword(s):  
Markov Chain ◽  
Cellular Automaton ◽  
Invariant Measures ◽  
Sampling Procedure ◽  
Monotonicity Property ◽  
Local Rule ◽  
Probabilistic Cellular Automata ◽  
Perfect Sampling ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

CRITICAL BEHAVIOR OF A PROBABILISTIC LOCAL AND NONLOCAL SITE-EXCHANGE CELLULAR AUTOMATON

1994 ◽  
Vol 05 (03) ◽  
pp. 537-545 ◽  
Author(s):  
N. BOCCARA ◽  
J. NASSER ◽  
M. ROGER
Keyword(s):  
Cellular Automaton ◽  
Critical Behavior ◽  
Exchange Process ◽  
Mean Field ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton ◽  
Site Exchange ◽  
Two Parameters ◽  
Automata Network ◽  
Dimensional Range

We study the critical behavior of a probabilistic automata network whose local rule consists of two subrules. The first one, applied synchronously, is a probabilistic one-dimensional range-one cellular automaton rule. The second, applied sequentially, exchanges the values of a pair of sites. According to whether the two sites are first-neighbors or not, the exchange is said to be local or nonlocal. The evolution of the system depends upon two parameters, the probability p characterizing the probabilistic cellular automaton, and the degree of mixing m resulting from the exchange process. Depending upon the values of these parameters, the system exhibits a bifurcation similar to a second order phase transition characterized by a nonnegative order parameter, whose role is played by the stationary density of occupied sites. When m is very large, the correlations created by the application of the probabilistic cellular automaton rule are destroyed, and, as expected, the behavior of the system is then correctly predicted by a mean-field-type approximation. According to whether the exchange of the site values is local or nonlocal, the critical behavior is qualitatively different as m varies.

A Probabilistic Cellular Automaton for Evolution

1995 ◽  
Vol 5 (9) ◽  
pp. 1129-1134 ◽  
Author(s):  
Nikolaus Rajewsky ◽  
Michael Schreckenberg
Keyword(s):  
Cellular Automaton ◽  
Probabilistic Cellular Automaton

scholarly journals Markov chain approximation of one-dimensional sticky diffusions

2021 ◽  
Vol 53 (2) ◽  
pp. 335-369
Author(s):  
Christian Meier ◽  
Lingfei Li ◽  
Gongqiu Zhang
Keyword(s):  
Markov Chain ◽  
Approximate Solutions ◽  
Price Model ◽  
One Dimensional ◽  
Markov Chain Approximation ◽  
Alternative Approaches ◽  
Second Order Convergence ◽  
Chain Approximation ◽  
Matrix Exponentials ◽  
Interior Points

AbstractWe develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.

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