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the model belongs to the DP universality class. One can observe that the curve
becomes broader in the vicinity of the critical point, in correspondence of the
divergence of critical fluctuations
χ ∼|p − p
c
|
−γ
,
γ
=0
.
54 [15]. By repeating
the same type of simulations for the kink dynamics (random initial condition),
we obtain slightly different curves, as shown in the right panel of Fig. 4. We
notice that all curves have roughly the same width. Indeed, the exponent
γ
for
systems in the PC universality class is believed to be exactly 0 [16], as given
by the scaling relation
γ
=
dν
⊥
−
2
β
[15]. Clearly, much more information can
be obtained from the knowledge of
P
(
c
), either by direct numerical simulations
or dynamical mean field trough finite scale analysis, as shown for instance in
Ref. [17].
4 Larger Neighborhoods
In order to study the effects of a larger neighborhood and different threshold
values
q
, let us start with the well known two-dimensional “Vote” model. It is
defined on a square lattice, with a Moore neighborhood composed by 9 neigh-
bors, synthetically denoted M in the following [8]. If a strict majority rule
q
=4
is applied (rule M56789, same convention as in Sec. 2) to a random initial con-
figuration, one observes the rapid quenching of small clusters of ones and zero,
similar to what happens with rule T23 in the one-dimensional case. A small noise
quickly leads the system to an absorbing state. On the other hand, a small frus-
tration
q
= 3 (rule M46789) for an initial density
c
0
=0
.
5 leads to a metastable
point formed by patches that evolve very slowly towards one of the two absorb-
ing states. However, this metastable state is given by the perfect balance among
the absorbing states. If one starts with a different initial “magnetization”, one
of the two absorbing phases quickly dominates, except for small imperfections
that disappear when a small noise is added.
In the general case, the mean-field equation is
2
r
+1
s
2
r
+1
c
=
c
s
(1
− c
)
2
r
+1
−s
p
s
,
(2)
s
=0
sketched in the left panel of Fig. 5.
We studied the asymptotic behavior of this map for different values of
r
and
q
. For a given
r
there is always a critical
q
c
value of
q
for which the active phase
disappears, with an approximate correspondence
q
c
4
/
5
r
. The active phase is
favored by the presence of frustrations and the absence of the external pressure,
i.e. for
J<
0 and
H
= 0. We performed extensive computations for the extreme
case
J
=
−∞
,
H
= 0 corresponding to a society of non-conformists without
television. As shown in the right panel of Fig. 5, by increasing the neighborhood
size
r
, one starts observing oscillations in the equilibrium process. This is evident
in the limit of infinite neighborhood: the parallel dynamics induced by elections
(in our model) makes individual tend to disalign from the marginal majority,
originating temporal oscillations that can have a chaotic character. Since the
