scholarly journals The spectral gap and the dynamical critical exponent of an exact solvable probabilistic cellular automaton

2015 ◽  
Vol 438 ◽  
pp. 56-65
Author(s):  
M.J. Lazo ◽  
A.A. Ferreira ◽  
F.C. Alcaraz
Keyword(s):  
Cellular Automaton ◽  
Critical Exponent ◽  
Spectral Gap ◽  
Probabilistic Cellular Automaton

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

Mapping the rule table of a 2-D probabilistic cellular automaton to the chemical physics of etching and deposition

1999 ◽  
Vol 290 (1-2) ◽  
pp. 216-229 ◽  
Author(s):  
James A. Gurney ◽  
Edward A. Rietman ◽  
Matthew A. Marcus ◽  
Mark P. Andrews
Keyword(s):  
Cellular Automaton ◽  
Chemical Physics ◽  
Probabilistic Cellular Automaton

On Representation of Integers from Thin Subgroups of SL(2, ℤ) with Parabolics

2018 ◽  
Vol 2020 (18) ◽  
pp. 5611-5629 ◽  
Author(s):  
Xin Zhang
Keyword(s):  
Critical Exponent ◽  
Lattice Point ◽  
Fuchsian Group ◽  
Spectral Gap ◽  
Circle Method ◽  
Inner Product ◽  
Point Counting ◽  
Finitely Generated ◽  
Exceptional Set ◽  
Representation Of Integers

Abstract Let $\Lambda <SL(2,\mathbb{Z})$ be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and $\mathbf{v},\mathbf{w}$ be two primitive vectors in $\mathbb{Z}^2\!-\!\mathbf{0}$. We consider the set $\mathcal{S}\!=\!\{\left \langle \mathbf{v}\gamma ,\mathbf{w}\right \rangle _{\mathbb{R}^2}\!:\!\gamma\! \in\! \Lambda \}$, where $\left \langle \cdot ,\cdot \right \rangle _{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy–Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich, and Sarnak, together with Gamburd’s 5/6 spectral gap, we show that if $\Lambda $ has parabolic elements, and the critical exponent $\delta $ of $\Lambda $ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain–Kontorovich, which proves a density-one statement for the case when $\Lambda $ is free, finitely generated, has no parabolics, and has critical exponent $\delta>0.999950$.

Short Time Dynamics of an Irreversible Probabilistic Cellular Automaton

1998 ◽  
Vol 12 (21) ◽  
pp. 873-879 ◽  
Author(s):  
T. Tomé ◽  
J. R. Drugowich de Fel Icio
Keyword(s):  
Ising Model ◽  
Cellular Automaton ◽  
Early Time ◽  
Universality Class ◽  
Kinetic Ising Model ◽  
Time Dynamics ◽  
Probabilistic Cellular Automaton ◽  
Time Regime ◽  
The One ◽  
Short Time

We study the short-time dynamics of a three-state probabilistic cellular automaton. This automaton, termed TD model, possess "up-down" symmetry similar to Ising models, and displays continuous kinetic phase transitions belonging to the Ising model universality class. We perform Monte Carlo simulations on the early time regime of the two-dimensional TD model at criticality and obtain the dynamic exponent θ associated to this regime, and the exponents β/ν and z. Our results indicate that, although the model do not possess microscopic reversibility, it presents short-time universality which is consistent with the one of the kinetic Ising model.

scholarly journals New insights into traffic dynamics: a weighted probabilistic cellular automaton model

2008 ◽  
Vol 17 (7) ◽  
pp. 2366-2372 ◽  
Author(s):  
Li Xing-Li ◽  
Kuang Hua ◽  
Song Tao ◽  
Dai Shi-Qiang ◽  
Li Zhi-Peng
Keyword(s):  
Cellular Automaton ◽  
Cellular Automaton Model ◽  
Traffic Dynamics ◽  
Probabilistic Cellular Automaton
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