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ε
cr
ν
β
γ
Glauber CA asynchronous
0
.
7197
1
.
012
0
.
126
1
.
771
Glauber CA synchronous
0
.
6580
0
.
93
0
.
116
1
.
627
(14)
Toom CA asynchronous
0
.
8658
1
.
05
0
.
131
1
.
837
Toom CA synchronous
0
.
8224
0
.
87
0
.
109
1
.
522
Let us recall that the exponents found for synchronized coupled map lattices are
ν
CML
=0
.
887,
β
CML
=0
.
111 and
γ
CML
=1
.
55 [14].
4.5Data Collapse
Indirect support ofthe validity ofestimates can be provided by observing the
collapse ofthe magnetization and susceptibility data. The ollowing universal
functions
ˆ
|m|
and
χ
are expected to emerge:
ˆ
|m|
((
ε − ε
cr
)
L
1
/ν
)=
L
β/ν
|m|
L
(
ε
)
(15)
χ
((
ε − ε
cr
)
L
1
/ν
)=
L
−γ/ν
χ
L
(
ε
)
(16)
It turns out that our sets ofdata lead to the acceptable collapse in all systems,
though when the Glauber CA with synchronous updating is considered then
some discrepancy is visible, see Fig. 4.
5Discussion
Ising-like phase transitions studied by probabilistic cellular automata are well
described by scaling and finite-size scaling laws valid at equilibrium. Deriving
accurate numerical estimates ofcritical exponents is a di
I
cult task [5]. We be-
lieve that the methodology applied here is reliable for three main reasons: (1)
the density ofdata collected in simulations allowed us to use linear approxima-
tions ofhigh accuracy (the Pearson coe
I
cient
r
2
is always greater than 0
.
90 );
(2) since only lattices ofsizes greater than 20 were taken into account, it was
allowed to neglect corrections to dominant scalings; (3) except estimates for
γ
all other quantities were derived from two or more functions.
Our simulations have justified the validity ofhyperscaling relation which for
d
-dimensional system takes form:
2
β
+
γ
=
νd
and which is known to hold at equilibrium in case offluctuation-dominated
transition, see, e.g., [22]. Moreover, the ratios
β/ν
and
γ/ν
follow the Ising
behavior. However, the mode ofupdating divides systems into two classes:
ν
=1
and
ν ≈
0
.
90. The weak universality denotes the independence of
β/ν
and
γ/
ν
on microscopic details [23]. Therefore, we can say that the synchronously
updated CA belong to the weak universality class ofthe Ising model. Furthemore,
the same probabilistic cellular automata but updated randomly possess critical
properties that are identical with the Ising thermodynamic model.
