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with
1. In this way the configurations
x
=
−
1
and
x
=
1
are no longer
absorbing. This brings the model back into the class of equilibrium models for
which there is no phase transition in one dimension but metastable states can
nevertheless persist for long times. The width of the hysteresis cycle, shown in
the right panel of Fig. 2, depends on the value of
and the relaxation time
T
.
We study the asymptotic density as
p
1
and
p
2
move on a line with slope
1 inside the dashed region of Fig. 1. For
p
1
close to zero, the model has only
one stable state, close to the state
c
=0.As
p
1
increases adiabatically, the new
asymptotic density will still assume this value even when the state
c
= 1 become
stable. Eventually the first state become unstable, and the asymptotic density
jumps to the stable fixed point close to the state
c
= 1. Going backwards on
the same line, the asymptotic density will be close to one until that fixed point
disappears and it will jump back to a small value close to zero.
Although not very interesting from the point of view of opinion formation
models, the problem of universality classes in the presence of absorbing states
have attracted a lot of attention by the theoretical physics community in recent
years [11,12]. For completeness we report here the main results [6].
It is possible to show that on the symmetry line one can reformulate the
problem in terms of the
dynamics of kinks
between patches of empty and oc-
cupied sites. Since the kinks are created and annihilated in pairs, the dynamics
conserves the initial number of kinks modulo two. In this way we can present an
exact mapping between a model with symmetric absorbing states and one with
parity conservation.
Outside the symmetry line the system belongs to the directed percolation
universality class [13]. We performed simulations starting either from one and
two kinks. In both cases
p
t
=0
.
460(2), but the exponents were found to be
different. Due to the conservation of the number of kinks modulo two, starting
from a single site one cannot observe the relaxation to the absorbing state, and
thus
δ
= 0. In this case
η
=0
.
292(5),
z
=1
.
153(5). On the other hand, starting
with two neighboring kinks, we find
η
=0
.
00(2),
δ
=0
.
285(5), and
z
=1
.
18(2).
These results are consistent with the parity conservation universality class [3,4].
Let us now turn to the sensitivity of the model to a variation in the initial
configuration, i.e. to the study of
damage spreading
or, equivalently, to the lo-
cation of the chaotic phase. Given two replicas
x
and
y
, we define the difference
w
as
w
=
x
⊕
y
, where the symbol
⊕
denotes the sum modulus two.
The damage
h
is defined as the fraction of sites in which
w
= 1, i.e. as the
Hamming distance between the configurations
x
and
y
. We study the case of
maximal correlations by using just one random number per site, corresponding
to the smallest possible chaotic region [14].
In Fig. 3 the region in which the damage spreads is shown near the lower-
right corner (chaotic domain). Outside this region small spots appear near the
phase boundaries, due to the divergence of the relaxation time (second-order
transitions) or because a small difference in the initial configuration can bring
the system to a different absorbing state (first-order transition). The chaotic
