Information Technology Reference
In-Depth Information
Ta b l e 2 . Tr i-Variate approximation, initial distribution
,
=(0 ; 0 ; 1 ; 0 ; 1)
0 0
0.000
0.000
1.000
p ( x
=1) p ( x
2
=1) p ( x
5
0
1
1.000
1.000
1.000
0
2
1.000
1.000
0.000
0
3
0.000
0.000
1.000
0.01 12
0.082
0.082
0.910
0.01 13
0.885
0.885
0.879
0.01 14
0.837
0.837
0.141
0.01 15
0.203
0.203
0.790
0.01 50
0.501
0.501
0.501
Next we have a look at the expected passage times. Here we have the result
that nearby
the passage time from configuration
to
. This is still a
large number, if one takes into account that the number of configurations is only
is about
. It decreases to
nearby
.
Wesummarize the results: Fo r
is no bifurcation
point, as the mean-field approximations indicates. The point plays a unique role
in so far that at this point the uni-variate marginal frequencies are not changed.
Fo r
the point
we have a unique stationary distribution because of
theorem 1. But the stationary distribution is in many cases very uninteresting.
5
Approximations of the Probability Distribution of 2-D CA
For two dimensions the approximation problem gets much harder. We restrict
ourselves to the von Neumann neighborhood and bivariate approximations. Be-
cause of the symmetry assumed the voter model is now defined by three pa-
rameters, usually called
. Since the transition depends on the von
Neumann neighborhood, the distribution
,
and
can be
expressed by means of a distribution of the six surrounding variables. For no-
tational convenience, let
+1);x
i +1 ;j
be a possible value of the variable
and
a possible value of
, respectively. We start our analysis as before with
(12)
(13)
Search WWH ::




Custom Search