4,369 research outputs found
Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata
Cellular automata are widely used to model natural or artificial systems.
Classically they are run with perfect synchrony, i.e., the local rule is
applied to each cell at each time step. A possible modification of the updating
scheme consists in applying the rule with a fixed probability, called the
synchrony rate. For some particular rules, varying the synchrony rate
continuously produces a qualitative change in the behaviour of the cellular
automaton. We investigate the nature of this change of behaviour using
Monte-Carlo simulations. We show that this phenomenon is a second-order phase
transition, which we characterise more specifically as belonging to the
directed percolation or to the parity conservation universality classes studied
in statistical physics
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
Sum of exit times in series of metastable states in probabilistic cellular automata
Reversible Probabilistic Cellular Automata are a special class
of automata whose stationary behavior is described by Gibbs--like
measures. For those models the dynamics can be trapped for a very
long time in states which are very different from the ones typical
of stationarity.
This phenomenon can be recasted in the framework of metastability
theory which is typical of Statistical Mechanics.
In this paper we consider a model presenting two not degenerate in
energy
metastable states which form a series, in the sense that,
when the dynamics is started at one of them, before reaching
stationarity, the system must necessarily visit the second one.
We discuss a rule for combining the exit times
from each of the metastable states
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