scholarly journals Asymptotic distribution of entry times in a cellular automaton with annihilating particles

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Petr Kůrka ◽  
Enrico Formenti ◽  
Alberto Dennunzio
Keyword(s):  
Cellular Automaton ◽  
Asymptotic Distribution ◽  
Bernoulli Measure ◽  
International Audience

International audience This work considers a cellular automaton (CA) with two particles: a stationary particle $1$ and left-going one $\overline{1}$. When a $\overline{1}$ encounters a $1$, both particles annihilate. We derive asymptotic distribution of appearence of particles at a given site when the CA is initialized with the Bernoulli measure with the probabilities of both particles equal to $1/2$.

scholarly journals The number of Euler tours of a random $d$-in/$d$-out graph

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Páidí Creed ◽  
Mary Cryan
Keyword(s):  
Asymptotic Distribution ◽  
Directed Graph ◽  
Polynomial Time ◽  
Simple Approach ◽  
Uniform Sampling ◽  
Concentration Result ◽  
International Audience ◽  
De Bruijn ◽  
Euler Tours

International audience In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.

scholarly journals Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Henryk Fukś ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Initial Density ◽  
Response Curves ◽  
Asynchronous Cellular Automata ◽  
Final Density ◽  
Short Cylinder ◽  
Bernoulli Measure ◽  
Single Transition ◽  
International Audience ◽  
Transition Rules

International audience We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.

scholarly journals Larger than Life: Digital Creatures in a Family of Two-Dimensional Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Kellie M. Evans
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Nearest Neighbor ◽  
Two Dimensional ◽  
Game Of Life ◽  
Large Neighborhood ◽  
International Audience ◽  
Parameter Values

International audience We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway's celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life's famous gliders and spaceships. We show that, as is becoming well known for nearest neighbor cellular automaton rules, these ``digital creatures'' are ubiquitous for certain parameter values.

scholarly journals Representing Reversible Cellular Automata with Reversible Block Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Jérôme Durand-Lose
Keyword(s):  
System Theory ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Linear Time ◽  
Fixed Size ◽  
Mathematical System ◽  
Reversible Cellular Automaton ◽  
International Audience ◽  
Infinite Lattices ◽  
Reversible Cellular Automata

International audience Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus <i>(Physica D 45)</i> improved by Kari in 1996 <i>(Mathematical System Theory 29)</i>.

scholarly journals Measure rigidity for algebraic bipermutative cellular automata

2007 ◽  
Vol 27 (6) ◽  
pp. 1965-1990 ◽  
Author(s):  
MATHIEU SABLIK
Keyword(s):  
Cellular Automata ◽  
Probability Measure ◽  
Cellular Automaton ◽  
Compact Group ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Invariant Subgroup ◽  
Shift Map ◽  
Bernoulli Measure ◽  
Measure Rigidity

AbstractLet $({\mathcal {A}^{\mathbb {Z}}} ,F)$ be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the $ {\mathbb {N}} \times {\mathbb {Z}} $-action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group $ {\mathcal {A}^{\mathbb {Z}}} $. This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on $ {\mathcal {A}^{\mathbb {N}}} $.

scholarly journals A weakly universal cellular automaton in the hyperbolic $3D$ space with three states

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Maurice Margenstern
Keyword(s):  
Cellular Automaton ◽  
3D Space ◽  
International Audience

International audience In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic $3D$-space. Here, we get such a cellular automaton with three states only.

scholarly journals Periodic Patterns in Orbits of Certain Linear Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
André Barbé ◽  
Fritz Haeseler
Keyword(s):  
Cellular Automata ◽  
Finite Field ◽  
Cellular Automaton ◽  
Periodic Patterns ◽  
Periodic Behavior ◽  
Universal Behavior ◽  
International Audience

International audience We discuss certain linear cellular automata whose cells take values in a finite field. We investigate the periodic behavior of the verticals of an orbit of the cellular automaton and establish that there exists, depending on the characteristic of the field, a universal behavior for the evolution of periodic verticals.

scholarly journals Conservation Laws and Invariant Measures in Surjective Cellular Automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Jarkko Kari ◽  
Siamak Taati
Keyword(s):  
Cellular Automata ◽  
Conservation Laws ◽  
Arbitrary Dimension ◽  
Close Link ◽  
One Dimensional ◽  
Full Support ◽  
Markov Measure ◽  
Bernoulli Measure ◽  
International Audience ◽  
Markov Measures

International audience We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

scholarly journals On the Exit Time of a Random Walk with Positive Drift

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Wojciech Szpankowski
Keyword(s):  
Random Walk ◽  
Asymptotic Distribution ◽  
Infinite Number ◽  
Exit Time ◽  
Saddle Points ◽  
Tauberian Theorems ◽  
Precise Asymptotics ◽  
First Exit Time ◽  
Mellin Transforms ◽  
International Audience

International audience We study a random walk with positive drift in the first quadrant of the plane. For a given connected region $\mathcal{C}$ of the first quadrant, we analyze the number of paths contained in $\mathcal{C}$ and the first exit time from $\mathcal{C}$. In our case, region $\mathcal{C}$ is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree is the parsing tree of the Tunstall or Khodak variable-to-fixed code, then the exit time of the underlying random walk corresponds to the phrase length not exceeding a given length. We derive precise asymptotics of the number of paths and the asymptotic distribution of the exit time. Even for such a simple walk, the analysis turns out to be quite sophisticated and it involves Mellin transforms, Tauberian theorems, and infinite number of saddle points.

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
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