Phase Transitions in Cellular Automata

2009 ◽  
pp. 6771-6782
Author(s):  
Nino Boccara
Keyword(s):  
Phase Transitions ◽  
Cellular Automata

scholarly journals Deterministic phase transitions and self-organization in logistic cellular automata

2019 ◽  
Vol 100 (4) ◽  
Author(s):  
M. Ibrahimi ◽  
O. Gulseren ◽  
S. Jahangirov
Keyword(s):  
Phase Transitions ◽  
Cellular Automata ◽  
Self Organization

scholarly journals Monotone Cellular Automata in a Random Environment

2015 ◽  
Vol 24 (4) ◽  
pp. 687-722 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
PAUL SMITH ◽  
ANDREW UZZELL
Keyword(s):  
Phase Transitions ◽  
Cellular Automata ◽  
Two Dimensions ◽  
Bootstrap Percolation ◽  
Unified Theory ◽  
Critical Probability ◽  
The Family ◽  
Update Rules ◽  
Critical Class

In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}$d with random initial configurations. Formally, we are given a set $\mathcal{U}$ = {X1,. . . , Xm} of finite subsets of $\mathbb{Z}$d \ {0}, and an initial set A0 ⊂ $\mathbb{Z}$d of ‘infected’ sites, which we take to be random according to the product measure with density p. At time t ∈ $\mathbb{N}$, the set of infected sites At is the union of At-1 and the set of all x ∈ $\mathbb{Z}$d such that x + X ∈ At-1 for some X ∈ $\mathcal{U}$. Our model may alternatively be thought of as bootstrap percolation on $\mathbb{Z}$d with arbitrary update rules, and for this reason we call it $\mathcal{U}$-bootstrap percolation.In two dimensions, we give a classification of $\mathcal{U}$-bootstrap percolation models into three classes – supercritical, critical and subcritical – and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on ($\mathbb{Z}$/n$\mathbb{Z}$)2 is (log n)−Θ(1) for all models in the critical class, and that it is n−Θ(1) for all models in the supercritical class.The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on $\mathbb{Z}$d.

Phase Transitions in 2-dimensional Stochastic Cellular Automata

1986 ◽  
pp. 103-111 ◽  
Author(s):  
Yukito Iba ◽  
Yasuhiro Akutsu ◽  
Kunihiko Kaneko
Keyword(s):  
Phase Transitions ◽  
Cellular Automata ◽  
Stochastic Cellular Automata

Topology-induced phase transitions in totalistic cellular automata

2013 ◽  
Vol 249 ◽  
pp. 16-24 ◽  
Author(s):  
J.M. Baetens ◽  
B. De Baets
Keyword(s):  
Phase Transitions ◽  
Cellular Automata
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