Information Technology Reference
In-Depth Information
one inverse and one equal bit in these positions. As only these two bits play the
crucial role of determining the pattern, they shall be called determinant bits. In
summary we have
occupied vertices
S
1
···
b
k
···
b
l
···
···
b
k
···
b
l
···
potential hubs
S
2
.
(2)
···
b
k
···
b
l
···
stable holes
S
3
···
b
k
···
b
l
···
These very few principles allow to explain all observations made in the sim-
ulations. We can construct a perfect 2-cluster pattern, a configuration in which
all nodes of group
S
1
are occupied and the others remain empty. It is perfect
in the sense that there are no defects but also no hubs. Such a configuration is
2
2
×
2
-fold degenerated where the first factor represents the choice of the two
determinant bits, and the second factor gives the number of possible positions
of these bits in the bitstring of length
d
.
We further can compute the number of occupied neighbors
n
(
∂v
)ofavertex
v
of any group. Since all nodes of
S
1
are occupied in the perfect pattern,
n
(
∂v
)
is given by the number of links between
v
and the other elements of
S
1
. A link
between two nodes exists if their bitstrings are complementary except for up to
two mismatches. If
v
S
1
, it has two bits in common with all other vertices in
S
1
,namely
b
k
and
b
l
. Thus, all remaining bits must be exactly complementary.
There is only
one
vertex
w
∈
∈
S
1
,w
=
v
, which obeys the constraints. If
v
∈
S
2
or
v
S
3
, there is one pre-determined mismatch or none, respectively. The
remaining mismatches can be distributed among the
d
∈
−
2 non-determinant bits.
Thus
d
1
−
2
n
(
∂v
)=
∀
v
∈
S
2
and
n
(
∂v
)=11for
d
=12
,
(3)
j
j
=0
d −
2
j
2
n
(
∂v
)=
∀
v
∈
S
3
and
n
(
∂v
)=56for
d
=12
,
(4)
j
=0
which is in good agreement with the simulations, cf. Table 1.
This regularity encouraged us to the following concept. Considering the two
determinant bits as coordinates of a two-dimensional space, they will define the
corners of a two-dimensional hypercube, which is called a pattern module.
The corner with coordinates (
b
k
,
b
l
) represents an occupied vertex, the op-
posite corner (
b
k
,
b
l
) is a stable hole, and the neighboring corners of (
b
k
,
b
l
)
are potential hubs. The module is the building block for the entire regular con-
figuration which can be understood as consisting of 2
d−
2
congruently occupied
'parallel worlds'. Any choice of the two determining bits is of course possible,
all corresponding patterns are equivalent, the 2-cluster pattern is 2
2
×
2
-fold
degenerated. The individual history (the realization of the random influx) selects
the determining bits. Thus the degeneracy is lifted, a symmetry breaking has
occurred.
