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In-Depth Information
5.1
Consistency of Distributions
In order to justify the approximation of formula (16), we have to check whether
it is consistent. There are two conditions which have to be satisfied:
(18)
;
i
+1
;j
Furthermore
has to be the same if computed by different marginalization.
There are four ways to calculate
, given by the four neighborhoods in
which it takes part:
i
1
;j
0
i
1;j
i;j
1
0
i;j
1
We can show that if these conditions are fulfilled in the initial distributions,
then our approximation preserves the consistency. The proof of consistency is
much harder for approximations using higher order distributions.
The voter model with five neighbors is currently very popular. Instead of a
point bifurcation a “phase separation” in the space generated by
is
predicted [14]. In this paper we have shown that the “bifurcation” predicted for
1-D automata is an artifact of the approximation. We conjecture that the same
result will be true for the 2-D voter model.
and
6
Conclusion and Outlook
We have reported about our first attempts to make a stochastic analysis of cel-
lular automata. We have shown for the 1-D voter model the observation of a bi-
furcation is an artifact of the mean-field approximation. The exact Markov pro-
cess analysis shows a completely different behavior. The stochastic automaton
behaves smooth concerning changes of
. Our approximation using tri-variate
distributions gives good results for
,but it also shows a wrong behavior
in the limit for
. This indicates the problem of our
approach. We are sure that a higher order approximation is better than an ap-
proximation of lower order, but we do not know how good the approximation
and
