Dynamic exponents of a probabilistic three-state cellular automaton

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Physica A 350 (2005) 263–276

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Dynamic exponents of a probabilisticthree-state cellular automaton

Roberto da Silvaa,�, Nelson Alves Jr.b

aDepartamento de Informatica Teorica, Instituto de Informatica,Universidade Federal do Rio Grande do Sul.

Av. Bento Gonc-alves, 9500, CEP 91501-970 Porto Alegre RS, BrazilbDepartamento de Fısica e Matematica, Faculdade de Filosofia, Ciencias e Letras,

Universidade de Sao Paulo, Av. Bandeirantes, 3900-CEP 014040-901 Ribeirao Preto SP, Brazil

Received 9 September 2004; received in revised form 29 October 2004

Available online 24 December 2004

Abstract

In this work, a three-state cellular automaton proposed to describe part of a biological

immune system is revisited. We obtain the dynamic critical exponent z of the model by means

of a recent technique that mixes different initial conditions. Moreover, by using two distinct

approaches, we have also calculated the global persistence exponent yg; related to theprobability that the order parameter of the model does not change its sign up to time t

[PðtÞ / t�yg ].

r 2004 Elsevier B.V. All rights reserved.

PACS: 05.10.�a; 64.60.Ht; 02.50.�r

1. Introduction

Cellular automata are statistical–mechanical models which present a complexbehavior, despite their relatively simple dynamic rules. In general, these models aredefined in d-dimensional lattices with linear dimension L; in which each site i

ð1pipLdÞ is occupied by a variable si: The dynamic rules of a typical cellular

see front matter r 2004 Elsevier B.V. All rights reserved.

.physa.2004.11.052

nding author. Tel.: +5551 3316 7772; fax: +5551 3316 7308.

dresses: [email protected] (R. da Silva), [email protected] (N. Alves Jr.).

www.elsevier.com/locate/physa

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R. da Silva, N. Alves Jr. / Physica A 350 (2005) 263–276264

automaton have two basic characteristics: Firstly, the update of a variable si

depends only on its neighborhood (short-range interactions). Secondly, given aconfiguration of the system at instant t; at instant t þ 1 all variables si ð1pipLdÞ areupdated at once.Brass et al. [1,2] devised a simple cellular automaton model of T-helper (TH) cell

interactions to mimic the immune system of mice exposed to parasitic infections. T-helper cells that have not yet been presented with antigen are known as TH0 cells. THcells interact with fragments of antigen presented to them by specialized antigenpresenting cells. Once activated by the antigen, T-helper cells produce a range ofsoluble factors (cytokines), which play a critical role on the type of immune responseagainst the pathogen.The target of the model of Brass et al. [1] was to explain the polarization of the T-

helper cells, i.e., the splitting of T-helper cells not yet exposed to the antigen (TH0)between two kinds of mature cells: TH1 and TH2 T-helper cells. These two kinds ofTH cells are distinguished by the cytokines (soluble factors) they produce, as well asby the effects that such cocktail of soluble factors causes on the rest of the immunesystem. The immune response will depend on which kind of T-helper cell willpredominate on high-level infection. For example, in the case of infection withleishmania, the individual may be either protected (TH1 response) or susceptible(predominance of TH2 cells) [1]. On the other hand, when mice are infected with thenematode parasite Trichuris muris they are protected if TH2 cells predominate orare susceptible otherwise (TH1 response) [1]. Thus, understanding the dynamicsof the polarization of T-helper cells is a very important task in the study of theimmune system.The model proposed in Ref. [1] is defined in a cubic (d ¼ 3) lattice in which each

site is occupied by one of three different TH cell types: TH0, TH1 and TH2 cells. T-helper cells that have not yet been presented to the antigen are denoted by TH0. Inthe model, two distinct routes (equivalent to two populations of antigen presentingcells) govern the maturing of a TH0 cell: Antigen presented by the first (second) routeelicit a TH1 (TH2) response. In order to take into account the competition betweenmature TH cells, the induction TH0! TH1 (TH2) is forbidden when theneighborhood of the TH0 cell has an overall majority of TH2 (TH1) cells. At last,a mature cell dies (it is substituted by a TH0 cell) if such a cell is not re-stimulated bythe appropriate antigen within the time interval NT : The model can display aspontaneous symmetry breaking as one varies the antigen density or the cutoff NT ;in agreement with experimental results [3].A probabilistic version of the model was proposed by Tome and Drugowich de

Felıcio [4] by allowing the death of cells TH1 and TH2 at each time step with aprobability r: Thus, in this modified version of the model, the lifetime NT wassubstituted by a mean lifetime related to the probability r: Also, the development ofTH0 cells into either TH1 or TH2 cells occurs with a probability that depends on theneighborhood of the TH0 cell and on a parameter p; related to the antigen density.Such probabilistic version presents the intrinsic spontaneous symmetry breakingfound in the Brass et al. automaton, besides being amenable to an analyticalapproach [4].

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R. da Silva, N. Alves Jr. / Physica A 350 (2005) 263–276 265

The critical behavior of the probabilistic model proposed in Ref. [4] was studied ina subsequent work by Ortega et al. [5]. By using a finite-size scaling analysis, fromMonte-Carlo simulations of square lattices, Ortega et al. determined the ratios ofexponents b=n and g=n; as well as the critical point of the model. Their results suggestthat the probabilistic automaton, although not satisfying detailed balance, belongsto the same universality class of the two-dimensional kinetic Ising model [5], thus,supporting the up–down conjecture introduced by Grinstein et al. [6]. In a subsequentwork, Tome and Drugowich [7] obtained the dynamic critical exponent z of theprobabilistic automaton in two dimensions, by performing short-time Monte-Carlosimulations and studying the collapse of the fourth-order Binder’s cumulant fordifferent lattice sizes.In the present work we have revisited the probabilistic automaton of Ref. [4]. We

have re-obtained the dynamic critical exponent z of the automaton by using a recenttechnique that mixes different initial conditions [8]. Such a technique is based uponthe short-time critical dynamics introduced by Janssen et al. [9], who showed thateven far from equilibrium the short-time relaxation of the order parameter follows auniversal scale form

MðtÞ ¼ m0ty ; (1)

where MðtÞ is the order parameter at instant t (measured in Monte Carlo steps perSpin—MCS), m0 ¼ Mð0Þ is a small initial ðt ¼ 0Þ value of the order parameter, and yis the dynamic critical exponent, related to the increasing of the order parameterafter the quenching of the system. Eq. (1) demands working with sharply preparedinitial states with a precise value of m0: After obtaining the critical exponent y for anumber of m0 values, the final value for y is obtained from the limit m0 ! 0:Starting from an ordered state ðm0 ¼ 1Þ; the order parameter MðtÞ decays in time,

at the critical temperature, according to the power law [10]

hMðtÞim0¼1 t�b=nz ; (2)

where hð� � �Þi is the average of the quantity ð� � �Þ over different samples with initialorder parameter value m0 ¼ Mð0Þ; b and n are the usual static critical exponents,related to the order parameter and to the correlation length, respectively, and z is thedynamic critical exponent, defined as t xz; where t and x are time and spatialcorrelation lengths, respectively.Starting from a disordered configuration with m0 ¼ 0; the second moment of the

order parameter increases after the power law

hM2ðtÞim0¼0 to ; (3)

where the exponent o is given by

o ¼ d �2bn

� �1

z; (4)

and d is the dimension of the system.

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By combining Eqs. (2) and (3), da Silva et al. [8] obtained the ratio

F2 ¼hMðtÞ2im0¼0

hMðtÞi2m0¼1

td=z ; (5)

which corresponds to a function with mixed initial conditions. At this point, it isimportant to stress here that, from Eq. (5) above, two independent runs arenecessary in order to calculate the ratio F2: In one of them m0 ¼ 0 (numerator),while in the other one m0 ¼ 1 (denominator). The ratio F2 has proven to beuseful in determining the exponent z; according to recent studies of the 2DIsing model [8], q ¼ 3 and q ¼ 4 states Potts models [8], Ising model withnearest- and next-nearest-neighbor interactions [11], Baxter-Wu model [12], at thetricritical point of the 2D Blume-Capel model [13], at the Lifshitz point of the3D ANNNI model [14] and in other studies concerning models with one absorbentstate [15].In addition, in this work we have also calculated the global persistence exponent

yg [16], related to the probability that the order parameter of the model, MðtÞ; doesnot change its sign up to time t; after a quench of the system to the criticaltemperature.The layout of this paper is as follows. In Section 2 we explain the model. In

Section 3 we define the order parameter MðtÞ and we describe the methodology usedin order to obtain the dynamic critical exponents yg and z: In Section 4 the resultsobtained for the exponents z and yg are shown. Finally, in Section 5 we present themain concluding remarks.

2. The model

In this work we have studied a two-dimensional probabilistic cellular automatonin which dynamics is governed by local stochastic rules. At each site i of the squarelattice we have attached a variable si assuming the value 0;þ1 or �1; depending onwhether the site is occupied by a TH0, a TH1 or a TH2 cell, respectively. ConsideringN the total number of sites in the lattice, we have defined the set s ¼ ðs1; s2; . . . ;sN Þ

to represent the microscopic state of the system.The probability of state s at time t; PtðsÞ; evolves in time according to the

equation

Ptþ1ðs0Þ ¼Xs

W ðs0jsÞPtðsÞ ; (6)

where the transition probability W ðs0jsÞ from state s to state s0 obeys theconditionX

s0W ðs0jsÞ ¼ 1 : (7)

On the other hand, for a system that evolves at discrete time steps, in which all thesites are updated at once, as is the case for the cellular automaton in this work, the

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transition probability W ðs0jsÞ is written in the form

W ðs0jsÞ ¼YNi¼1

oiðs0ijsÞ ; (8)

where oiðs0ijsÞ is the conditional probability that site i be in the state s0i at time t þ 1;given that the state of the system is s at instant t: Such conditional probabilitysatisfies the conditionX

s0i

oiðs0ijsÞ ¼ 1 ; (9)

which implies immediately that Eq. (7) is fulfilled. The cellular automatoninvestigated in this work belongs to the class of totalistic cellular automata [17].Thus, the transition probability oiðs0ijsÞ depends on si and on the sum x ¼

Ptsiþt;

where the sum runs over the neighborhood of site i: More specifically, we areconsidering a particular kind of totalistic automaton for which oiðs0ijsÞ depends onlyon the sign of the sum x: By defining

si ¼ signðxÞ ¼

1 if x40 ;

0 if x ¼ 0 ;

�1 if xo0 ;

8><>: (10)

we may use the notation oiðs0ijsi; siÞ in order to explicit the transition probabilitydependence on si and si:The transition probabilities (dynamical rules) are given by

oiðþ1jsi; siÞ ¼ pdðsi; 0Þfdðsi;þ1Þ þ 12dðsi; 0Þg þ ð1� rÞdðsi;þ1Þ ; (11)

oið�1jsi; siÞ ¼ pdðsi; 0Þfdðsi;�1Þ þ 12dðsi; 0Þg þ ð1� rÞdðsi;�1Þ ; (12)

oið0jsi; siÞ ¼ ð1� pÞdðsi; 0Þ þ rfdðsi;þ1Þ þ dðsi;�1Þg ; (13)

where, as discussed in Section 1, r is the death probability of TH1 and TH2 cells, andp is a parameter related to the antigen density [4].The dynamical rules (11)–(13 ) above have ‘‘up–down’’, i.e.,

oiðs0ijsi; siÞ ¼ oið�s0ij � si;�siÞ : (14)

Therefore, following Grinstein et al. [6], we expect that the probabilistic cellularautomaton investigated in this work be at the same universality class of kineticIsing models.

3. The order parameter and the dynamic exponent yg

3.1. Order parameter

In the short-time Monte Carlo simulations performed in this work the orderparameter MðtÞ; as well as its higher moments MðkÞðtÞ; were obtained from averages

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over a certain number of samples, Ns: By defining the kth momentum of the orderparameter in sample number j at instant t as

Mkj ðtÞ ¼

1

L2k

XL2i¼1

siðtÞ

!k

; (15)

the order parameter is written in the form

MðkÞðtÞ ¼ Mkj ðtÞ

D E¼

1

NsL2k

XNs

j¼1

XL2i¼1

sijðtÞ

!k

; (16)

where sij denotes the variable at site i of sample number j: As defined in Eqs. (15)and (16) above, for k ¼ 1; the order parameter is exactly the mean magnetization,M ð1ÞðtÞ ¼ MðtÞ

� �:

Eq. (16) defines the order parameter and its higher moments for a set of Ns samples.Thus, if Nb sets of samples are considered at instant t; there are Nb measurements ofthe magnetization (and its higher moments), M

ðkÞl ðtÞ; where 1plpNb: By considering

such sets of samples, the final value of the magnetization and its higher moments areobtained from the average over the Nb sets of samples, i.e.,

MðkÞðtÞ ¼ ð1=NbÞXNb

l¼1

MðkÞl ðtÞ ; (17)

whereMðkÞl ðtÞ is given by Eq. (16) and the corresponding standard deviation (�) is given

by

� MðkÞl ðtÞ

h i¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNbðNb � 1Þ

2q XNb

l¼1

MðkÞl ðtÞ � M ðkÞðtÞ

� �2 !1=2: (18)

3.2. Dynamic critical exponent yg

In this section we define the dynamic critical exponent yg and we describe twomethods used in this work to obtain estimates for yg:In the first method, we have performed short-time Monte Carlo simulations in

order to calculate the global persistence probability PðtÞ; i.e., the probability that themagnetization does not change its sign up to time t: On the other hand, theprobability PðtÞ is numerically equal to the complement of the accumulateddistribution pðtÞ; according to which the magnetization changes its sign for the firsttime exactly at instant t; i.e.,

PðtÞ ¼ 1�Xt

t0¼1

pðt0Þ ¼ 1�Xt

t0¼1

nðt0Þ

Ns

; (19)

where nðt0Þ is the number of samples for which the magnetization changes its sign forthe first time at instant t0 and Ns is the total number of samples. The exponent yg

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may be obtained directly from the power-law scale relation [16]

PðtÞ t�yg ; (20)

from which we obtain ln PðtÞ ¼ c � yg ln t; where c is constant and each run requiresa sharply prepared initial state, with a precise small value of m0; as discussed inEq. (1).In a second way to obtain the exponent yg we have used the fact that the initial

magnetization dependence of PðtÞ can be cast in the following finite-size scalingrelation [16],

PðtÞ ¼ t�yg f ðt=LzÞ ¼ L�ygz ~f ðt=LzÞ ; (21)

which renders a different method to obtain the exponent yg from lattice sizes L1 andL2 [16]. For this end we defineW ðt;LÞ ¼ LygzPðtÞ; which turns out to be a function oft=Lz: Therefore, if we fix the dynamic exponent z; the exponent yg can be obtained bycollapsing the time series W ðt2;L2Þ ¼ f ðt2=Lz

2Þ onto W ðt1;L1Þ ¼ f ðt1=Lz1Þ as follows.

Under re-scaling, with b ¼ L2=L1; ðL24L1Þ; we obtain

W ðt2;L2Þ ¼ eW ðbzt1; bL1Þ ; (22)

and the best estimate for yg corresponds to the minimization of

w2ðygÞ ¼X

t

W ðt;LÞ � eW ðbzt; bLÞ

W ðt;LÞ�� þ eW ðbzt; bLÞ

�� 0B@

1CA2

(23)

by interpolating eW to the time values bzt: In order to obtain the exponent yg

using the collapse method described above, it is not necessary to fix a precise valueof the initial magnetization m0 in the short-time simulations, once the scalingrelation in Eq. (21) does not take into account the initial conditions of the system.So, we have used initial states in which m0h i 0: On the other hand, thecollapse method demands the dynamic exponent z to be known beforehand.Therefore, in this work we have used the scaling relation of Eq. (5) in order toobtain estimates for the exponent z: Although both methods described by Eqs. (20)and (21) were proposed in order to calculate the exponent yg of the two-dimensionalIsing model [16], such methods were used recently for estimates of yg

along the critical line and at the tricritical point of the 2D Blume-Capelmodel [18].

4. Results from short-time Monte Carlo simulations

In this section we present details about the short-time Monte Carlo simulationsperformed for the cellular automaton considered in this work, as well as the resultsobtained for both dynamic critical exponents z and yg:

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4.1. Critical parameters

Initially, we refine the critical parameter r ¼ 0:190 obtained in Ref. [4] for p ¼ 0:3:From Eq. (2) we expected a straight line for the log–log plot of MðtÞ against t at thecritical point ðp ¼ 0:3; r ¼ 0:190Þ: However, from log–log plots of Eq. (2) obtainedfor different values of r; we observed that more accurate straight lines are obtainedfor ra0:190; as depicted in Fig. 1 for r ¼ 0:190; 0:192; 0:194; 0:196 and 0:198: In theshort-time simulations performed in order to obtain the curves shown in Fig. 1 wehave used square lattices ðd ¼ 2Þ with linear dimensions L ¼ 160; Ns ¼ 10 000samples, Nb ¼ 5 sets of samples and 1000 MCS.For each value of r used in Fig. 1 we calculated the goodness of fit Q for the same

time interval ½t1; t2� and obtained the values shown in Table 1, from which we obtainthe critical value r ¼ 0:194: In order to confirm the critical value obtained in Table 1,we have also calculated the critical value of r for p ¼ 0:3 by using the effectiveexponent, which is given by [19]

lðtÞ ¼1

Dtlog

MðtÞ

Mðt=DtÞ

� �: (24)

Such exponent takes into account finite-time corrections (finite-time scaling) and, inthe limit t ! 1; Eq. (24) behaves as [19]

lðtÞ ¼ c1 þ c2=t ; (25)

where c1 and c2 are numerical constants and Dt is a fixed time step.Log–log plots of Eq. (24) are shown in Fig. 2, for Dt ¼ 10; p ¼ 0:3 and r ¼

0:190; 0:192; 0:194; 0:196 and 0:198: From Fig. 2 it is clear that the asymptotic

3 4 5 6 7

-0.5

-0.6

-0.7

-0.8

-0.9

p = 0.3

lnt

r = 0.198

r = 0.196

r = 0.194

r = 0.192

r = 0.190

1n M

(t)

Fig. 1. Log–log plots of the order parameter (magnetization) MðtÞ versus t; constructed for p ¼ 0:3 anddifferent values of r: The best linear fitting yields the best estimate for the critical value of the parameter r:From the results summarized in Table 1 (see text), we have located the critical point ðp ¼ 0:3; r ¼ 0:194Þ:

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Table 1

Values of Q for different values of r obtained from linear fitting of log–log plots of the magnetizationMðtÞ

versus t

Time interval r Q

½50; 300� 0:190 2:31� 10�7

½50; 300� 0:192 0:166½50; 600� 0:194 0:468½50; 300� 0:196 1:04� 10�28

½50; 300� 0:198 1:09� 10�76

0.05 0.10 0.15 0.20 0.25 0.30 0.35-0.07

-0.06

-0.05

-0.04p = 0.3

1/t

r = 0.198

r = 0.196

r = 0.194

r = 0.192

r = 0.190

λ(t)

Fig. 2. Effective exponent lðtÞ obtained for p ¼ 0:3 and different values of r: From this graph, we have

obtained the critical point ðp ¼ 0:3; r ¼ 0:194Þ (see text for details).


behavior of lðtÞ given by Eq. (25) is verified for r ¼ 0:194; thus, confirming the resultpreviously obtained from log–log plots of Eq. (2).Before presenting the estimates obtained for the critical exponents z and yg in the

next subsections, it is important to emphasize here that the Monte Carlo simulationswere performed only at the critical point ðp ¼ 0:3; r ¼ 0:194Þ:

4.2. Critical dynamic exponent z

In order to obtain the dynamic critical exponent z we have perfomed short-timeMonte Carlo simulations for square lattices ðd ¼ 2Þ with linear dimensions L ¼ 160;Ns ¼ 20 000 samples, Nb ¼ 5 sets of samples and 200 MCS. From the definition ofthe ratio F2 given by Eq. (5), we have constructed the log–log plot of F2ðtÞ versus t

shown in Fig. 3, obtained from Monte Carlo simulations performed for one set of

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2 3 4 5 6

-8.5

-8.0

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

L = 160

r = 0.194p = 0.3

lnt

1nF 2

(t)

Fig. 3. Typical log–log plot of F2ðtÞ versus t (straight line), obtained from short-time Monte Carlo

simulations at the critical point ðp ¼ 0:3; r ¼ 0:194Þ:

Table 2

Dynamic critical exponent z and the corresponding value of Q obtained for different time intervals

Time interval z Q

½10; 120� 2:0853ð7Þ 0:2½10; 160� 2:0856ð5Þ 0:49½10; 200� 2:0869ð4Þ 0:54½30; 200� 2:084ð4Þ 0:98½100; 200� 2:097ð8Þ 0:99


samples. The straight line depicted in Fig. 3 is in accordance with the linear behaviorpredicted in the scaling relation of Eq. (5).In Table 2 we summarize the estimates for z obtained from different time intervals

½t1; t2�; together with the corresponding values of Q:From Table 2, we have obtained the best estimate for the dynamic critical

exponent z ¼ 2:097ð8Þ; once the goodness of fit Q ¼ 0:99 is maximum at thecorresponding time interval. However, such value of z is somewhat smaller thanz ¼ 2:17ð2Þ; which was obtained from the fourth-order Binder’s cumulant [7]. On theother hand, these results are in very good agreement with those obtained for the 2DBlume-Capel model in the absence of a crystal field (D ¼ 0): z ¼ 2:106ð2Þ from theratio F 2 and z ¼ 2:16ð2Þ from the fourth-order Binder’s cumulant [13] (for D ¼ 0; the2D Blume-Capel model is similar to the automaton studied in this work, in the sensethat both models have the same symmetry of the 2D spin-1 Ising model). Suchresults show that the exponent F2 yields an underestimated value of z whencompared with the fourth-order Binder’s cumulant. We may speculate some reasons

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for this underestimating. Firstly, by using the ratio F2; the critical exponent z isobtained from linear fits of log–log graphs of F2 against t; whereas the fourth-orderBinder’s cumulant demands a collapse between curves obtained from two distinctlattice sizes. Thus, we believe that the exponent z obtained from such collapse carrieslarger errors due to finite size scaling corrections. Secondly, the ratio F 2 turns out tobe more sensitive to the proximity of multicritical points. For example, under-estimated values of z were also obtained from the ratio F 2 along the critical line andclose to the tricritial point of the 2D Blume-Capel model. On the other hand, farfrom the tricritical point, the dymanic critical exponent z tends towards its known2D Ising model value [13]. A similar behavior was also observed when applying theratio F2 for the 2D Ising model with nearest- and next-nearest-neighbor competinginteractions: The critical exponent z tends towards its 2D Ising value only whenevaluated sufficiently far from the disorder point [11]. Finally, we may also speculatethat dynamic universality is much more subtle than static universality. Thus, thediscrepancy observed between values of z; obtained from the ratio F2 and from theBinder’s cumulant, may be reflecting such subtlety. On the other hand, we cannotcompletely discard the possibility that the ratio F2 itself presents its own subtlety, inthe sense that it may demand corrections close to multicritical points, and that thesecorrections become more and more irrelevant far from such points. However, despitethe speculative discussion above, a precise explanation for such underestimatedvalues of z obviously requires further detailed investigation.In the next section we obtain the global persistence exponent from the collapse

method, which depends on the value of z; and directly from Eq. (20), which does notdepend on the value of z: As we shall see in the following, the results for yg obtainedfrom these two approaches are in very good agreement with each other.

4.3. Critical dynamic exponent yg

By using Eqs. (21)–(23) given in Section 3.2, we have performed short-time MonteCarlo simulations in order to apply the collapse method of Section 3.2 and obtainthe dynamic critical exponent yg: According to the description of the method, thedynamic exponent z is to be known beforehand. Thus, we have used the valuez ¼ 2:097 obtained in the previous section. Monte Carlo runs were made up to 1000MCS for Ns ¼ 40 000 samples in square lattices with linear dimensions 20, 40 and 80.The collapse method was applied for pairs of linear dimensions ðL1;L2Þ ¼ ð20; 40Þand ðL1;L2Þ ¼ ð40; 80Þ; from which we have obtained yg ¼ 0:24ð2Þ and yg ¼ 0:25ð2Þ;respectively. In Fig. 4 we show the collapse of the curves obtained for L ¼ 40 andL ¼ 80 with yg ¼ 0:25:Finally, we have obtained the dynamic exponent yg directly from the power law

predicted in Eq. (20). To this end, we have performed short-time Monte Carlosimulations to obtain the quantity PðtÞ; from which we have constructed log–logcurves of PðtÞ versus t; as depicted in Fig. 5. From Eq. (20), the exponent yg wasobtained directly from the slope of such curves. Monte Carlo simulations wereperformed for square lattices with linear dimension L ¼ 80 and Ns ¼ 40 000 samples,where each sample began from the initial state m0 ¼ 0:005: Error bars were estimated

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-3.0 -2.5 -2.0 -1.5 -1.00

1

2

3

4

5

p = 0.3r = 0.194

log( t / Lz)

θg = 0.25 L = 40

L = 80

Lθ g

z P(

t)

Fig. 4. Collapse obtained from the scaling relation Eq. (21), corresponding to square lattices with linear

dimensions ðL1;L2 ¼ 40; 80Þ and yg ¼ 0:25:

1 2 3 4 5 6 7

-1.2

-1.5

-1.8

-2.1

-2.4

-0.9

m0 = 0.005

r = 0.194p = 0.3

lnt

1n P

(t)

Fig. 5. Typical decay of the probability PðtÞ (straight line in a log–log plot), obtained from short-time

Monte Carlo simulations of Ns ¼ 40 000 samples with initial state m0 ¼ 0:005:


from the Monte Carlo runs performed for Nb ¼ 5 sets of samples. In Table 3 wepresent the results obtained for the exponent yg and the corresponding goodness offit Q for three different time intervals. From Table 3, the best estimate for the globalpersistence exponent is yg ¼ 0:247ð4Þ; corresponding to Q ¼ 0:97: This result is invery good agreement with the estimates yg ¼ 0:24ð2Þ and yg ¼ 0:25ð2Þ obtained fromthe collapse method.

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Table 3

Dynamic critical exponent yg and the corresponding value of Q obtained for different time intervals

Time interval yg Q

½90; 300� 0:247ð4Þ 0:97½90; 400� 0:242ð3Þ 0:86½100; 500� 0:238ð6Þ 0:95


5. Conclusions

In this work we have revisited the probabilistic cellular automaton proposed byTome and Drugowich [4] on the basis of a previous cellular automaton devisedby Brass et al. [1]. From short-time Monte Carlo simulations, we have obtainedthe dynamic critical exponent z ¼ 2:097ð8Þ by using a recent technique thatmixes different initial conditions [8]. Such result is slightly smaller than the valuez ¼ 2:17ð2Þ obtained from the fourth-order Binder’s cumulant [7].We have also performed short-time Monte Carlo simulations in order to obtain

estimates for the dynamic critical exponent yg; the global persistence exponent, byusing two distinct approaches: The collapse method described by Eqs. (21)–(23), anddirectly from the power-law scaling given by Eq. (20). From the collapse method, byusing square lattices with linear dimensions ðL1;L2Þ ¼ ð20; 40Þ and ðL1;L2Þ ¼ð40; 80Þ; we have obtained yg ¼ 0:24ð2Þ and yg ¼ 0:25ð2Þ; respectively. Directly fromEq. (20) we have obtained yg ¼ 0:247ð4Þ; in very good agreement with the estimatesyielded from the collapse method.

Acknowledgements

N. Alves Jr. acknowledges financial support from Brazilian agencies FAPESP andCAPES. R. da Silva thanks GPPD of the Institute of Informatics of FederalUniversity of Rio Grande do Sul (UFRGS), for computational resources.

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