http: / /www.ictp. t riosto.it/"pii b_off

TC/97/150

United Nations Educational Scientific and CnlJ.nral Organization

In tern at.ion al Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSTCS

SELF-ORGANIZING HEIGHT-ARROW MODEL:NUMERICAL AND ANALYTICAL RESULTS

Robert R. Shcherbakov1

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,141980 Dubna, lUissia2

andInternational Centre for Theoretical Physics, Trieste. Italy.

ABSTRACT

The recently introduced self-organizing height-arrow (HA) model is numerically in-vest.igated on l.lie square latt.ice and HTIM.1 ylicnlly on the Bet.he latt.ice. The concentrationof occupied sites and crit.ical exponents of distributions of avalanches are evaluat.ed forl.vvo sliglilly differ<;nt versions of t.Tie TVK)<]<;1. T'IK; obtain<;d expon<;nJ.K for (lislribnlioriH ofavalanches by rna.HS. area, dnralion and appropriate fractal dinienHionH ar<; close l.o I.TIONC;

for l.lie "BTW inod<;l, vvliicli sngg<;sl.s that t.Tie HA TVIO<]<;1 Tx;longK l.o l.}i<; same universalil.yclass. Tor comparison, the concentration of occupied sites in the 11A model is calculatedexactly on the Bet he lattice of coordination number q = 1 as well.

MTRAMARE TRIESTE

September 1997

1On leave of absence; from: Theoretical Department., Yerevan Physics Tnst.itnte, 375036Yerevan, Armenia.

-E-mail: shchertitlisunl.jinr.ru

1 Introduction.

The study of different cellular automata, which exhibit Self-Organized Criticality (SOC) [l],has been a subject of great interest in recent years. These models serve as tractable limitsof real dynamic systems with many spatial degrees of freedom, in which one might hopeto gain understanding of possible mechanisms of SOC. Unfortunately, in most cases ourcurrent knowledge; of t.Tic: effects of SOC is ™l.}i<;r limit.ed.

T'}i<; main peculiarity of t.Tic: dynamic dissipative models which are N<;lf-<]riv<;n inl.o l.lieSOC st.al.e is t.Tie presence of l.lie povver-1 aw in dist.ribut.ions of qiianl.il.ieN such as avalanchemass, duration, et.c. TheN<; <lisl.ribiil.ionN ar<; characterized by a set. of exponents. One ofl.he moHJ. inl.rigning qnest.ions concerns l.he classes of universality of these models. Thereare several att.empt.s l.o shed light, on t.his proi)lem [2. 3]. Recently, Ben-Hur and Rihani [2]proposed a classification scheme for the 2d models both stochastic and deterministic.They found that the original Bak, Tang, Wiesenfeld (13TW) model [l] belongs to theuniversality class of undirected models, directed models form a separate class, and thetwo-state Manna model [i] belongs to the universality class of random relaxation models.Later on, JNakanishi and Sneppen [3] examined several id sandpile models and suggestedl.hal. I.IK; two-st.ate Manna model and rice pile model [5, 6] belong l.o the same universalityclass.

This classification scheme is based on l.he type of relaxations at each site of l.he lattice.Tn l.he RTW model the particles from l.he toppled sites are uniformly reclist.ribut.od amongits nearest neighbors, whereas in the two-state Manna model the set of'Tieighbors is chosenrandomly. It is possible to introduce more complicated dynamical rules of relaxations ateach site of the lattice that will depend on some period T where T is the number oftopplings. During this period after each toppling the redistributions of particles from thegiven site form a minimal nonperiodic sequence. The 13TW model can be considered withthe period T = 1. In the two-state Mannna model the sequence of topplings at each siteis stochastic without any periodicity, therefore, one can put T = oo for this model.

Tn this paper, we investigate l.he recently introduced self-organizing height-arrow (HA)model [7. S]. Tt combines features of the RTW model [9, 10] and self-organizing Eulerianwalkers model (EWM) [7. 11]. The model is a cellular automaton defined on any connectedundirected graph. Tn this model, each site of the graph can be occupied by a particle orcan be empty. Addition of the particle l.o the occupied Kite makes it unstable and causesits toppling. The site becomes empty and the particles are transferred to the nearestneighbors. The redistribution of particles from an unstable site is governed by the secondsite variable, an arrow. Each outgoing particle from the toppled site turns the arrow bythe prescribed rule and the new direction of the arrow determines the destination pointfor this particle.

Tn l.he HA model T is formed by the nonperiodic sequence of turns of l.he arrow ateach Kite of l.he lattice. For simplicity, it is convenient l.o assign l.he same period T forall sites of l.he lattice. Tims, one can define l.he HA model with increasingly complicateddynamical rules. These pseudo-random models Lend l.o t.he random ones for large T —> oo.

The goal of l.he paper is l.he st.ucly of t.he HA model wil.h T = 2. when two Lopplingsrestore initial direction of the arrow. The model evolves at long times into the steadystate which is identified with the SOC.1 state as distributions of dynamic characteristics ofthe model show a power-law behavior. The obtained exponents for two slightly differenttypes of this model are very close to those for the 13TW one. which suggests that the 11A

2

model is in the universality class of undirected models. We also study the main staticcharacteristic of the model, the time averaged density of occupied sites. This quantity isobtained miiTKTica.ily on t.Tic: squa.re lat.tice a.nd exactly on t.Tic: "Bel.lie lat.tice of coordinationnumber q = 4.

The paper is organized as follows. The definition of the model wil.h two slightlydifferent, types of evolulion rules is presented in l.he next. section. Sex;. 3 is devoted to l.henumerical investigation of l.he HA model on the squa.re lat.tice. Exl.rapolal.ing resulls fromfinite size lattices we estimate the concentration of occupied sites in the model. The criticalpower-law exponents and scaling relations among them are defined in the framework offinite-size scaling analysis. We present results for the values of these exponents in theSOC state for distributions of various quantities of the 11A model. Then, in Sec. -1, weexactly calculate the concentration of occupied sites on the Bet he lattice of coordinationnumber q = 4. Discussion a.n<] conclusions are presented in Sex;. 5.

2 The self-organizing height-arrow model on thesquare lattice.

The 11A model we are going to investigate is denned as follows. To each site i of thetwo-dimensional L x L square lattice is assigned a height variable z,; G {0.1....}- andan arrow directed north, east, south or west from i. We start with an arbitrary initialconfiguration of heights and arrows on the lattice, initially, we drop a particle on therandomly chosen sit.e i. The succeeding evolulion of l.he system is delermined by l.hefollowing rules. We increa.se l.he height, variable at. l.he site i by 1, z- —'r z.-t -\-1. Tf the sit.e iis already occupied by the particle, it topples (2.,: —y z.-t — 2). To redislribule l.he pa.rliclesfrom the loppled Kite i among ils nearest neighbors, we turn l.he arrow t.wice accordinglo l.he prescribed rules. For l.he given period T = 2, t.here are only l.wo non-equivalentsequences of turns of l.he arrow at l.he given sit.e which preserve l.he model from beingdirected. Hereafter, these sequences of turns will be distinguished as JN-E-S-W-N and-N-S-W-E-JN. After each turn the new direction of the arrow points to the neighbor sitesto which we will transfer particles at the next time step. This process continues untila stable configuration is reached. The sequence of topplings of unstable sites forms anavalanche which propagates through the lattice. After an avalanche ceases, we go on byadding a. new particle and so on.

A given configuration of the model is a. Net of directions of a.rrows and heighls. Thelota.1 number of them is S1''*1'. During l.he evolution of l.he syslem the arrow at. any sitemight, be only in l.wo posilions due lo the fact l.hat. the two subsequent lopplings of l.hesite rest.ore the initial position of the arrow. Therefore, l.he set. of configurations of l.hemodel falls into 2 L x L equivalent classes which are determined by initial configurations ofarrows.

Starting from a certain configuration of arrows and an arbitrary configuration of oc-cupied sites, the model evolves through transient states into a dynamic attract or whichis critical. This attractor is identified with the SOC.1 state as different dynamic charac-leristics of the model show power-law l.ails in l.heir dislribulions. The model being inl.he SOC Htat.e pa.sses from one allowed st.able configuration lo another by avalanche dy-namics. This critical sta.le has been investigated in detail by Priezzhev [8]. He definedoperators corresponding to a.ddilion of a. particle at a. randomly chosen sit.e and showed

3

that they commute with each other. The algebra of these operators is used to calculatethe number of allowed configurations of a given class in the SOC state. This number isshown t.o be equal t.o t.Tic: determinant, of the discrete La.pl aci an matrix A of t.Tic: squarelattice. To check the given configuration l.o be allowed in the SOC state t.Tic: modificationof the bni'Tiing algorithm was also introduced [8].

3 Numerical results.

TTI order to investigate the static:, properties and avalanche dynamics in the HA model, wehave made numerical simulations with high statistics. We consider square lattices of sizeL X L with open boundary condition and L ranging from 100 up to 600. The HA modelhas been studied for two different types of dynamics (N-E-S-W-N and N-S-W-E-N) ofturns of arrows and various initial conditions.

Starting from an arbitrary distribution of occupied sites and certain initial directionsof arrows, the finite system evolves into a stationary state. In this state we have measuredthe time averaged density {p(z = 1)} and critical exponents for distributions of avalanchesby mass f.s), area (a) and duration (£). The mass .s is defined as a total number of topplingsin an avalanche whereas the: area, a is defined as a. number of distinct sites visited by anavalanche. The: sinnilta.Tieous topplings of different sites in a.n avalanche: at a. given timeis considered a. single: time step. The duration / is the number of this type of ste:p. Eora. more: detailed description of the: structure of an avalanche: it is also useful to define a,linear extent (diameter) of the: avalanche cluster via. a. radius of gyration (r). We: alsomeasured the corresponding fractal dimensions %ry, where x.y = -{s.a.t.r}.

As is shown in Tig. 1, the steady state is reached by the model after about 100 000avalanches on the square lattice of the linear size L = 600 for the system which is ini-tially empty and with a random initial distribution of arrows. In this simulation, wewere recording the averaged density {p{z = 1)} of occupied sites at each time when theavalanche ceases.

As ha.s been mentioned in Sec. 2, the number of configurations of the model falls into2'-'xl' classes depending on the: initial con figurations of arrows. TTI OUT simulations of the:HA model with t.Tic: N-E-S-W-N dynamics we started from random initial configurationsof arrows. Whereas, for the N-S-W-E-N dynamics the arrows were initially directed onlyeast or south. The later case wa.s chosen to simulate the: scattering of particles at. eachtoppling by 180° angle.

Fig. 2 displays the results of simulations for the time averaged density (p(z = 1)} inthe stationary state. They slightly depend on the lattice size L and are well describedby the equation (p(z = 1))/, = pc + cL~x. The numerical extrapolation of the L —> oolimit gives the values for the averaged density: -pr, = lini/.._j.o&{pfx; = 1))/, = 0.721 ± 0.001(N-E-S-W-N dynamics) and pc = 0.755 ±0.001 (N-S-W-E-N dynamics). These values arc:a. little higher in comparison with the stochastic two-state: Manna model [4] (see Table; 1).

Table 1: The lime averaged density pc of occupied silos for t.Tie HA model on llie Kqua.rolall.ice with I wo slightly different, types of dynamics and on llie Rellie lall.ice is comparedwith the value for the two-slate Manna, model. The uncertainly of the mimerica.l dala isabout ±0.001.

Density

Pr.

1.0

LA!l

.72111A6

0.755

Mode11 Ac

1

666Manna0.683

[4]

a N-E-S-W-N dynamicsh .N-S-W-E-N dynamicsc Bethe lattice

The form of avalanches in the 11A model has a layered structure. A typical avalancheis shown in Tig. 3 where the number of relaxations in each site is marked by different scalesof gray color. The sites with the same number of relaxations form a layer or shell. Wehave observed that layers group in pairs, in each pair a larger layer is a connected clusterwith holes whereas a smaller one is a disconnected cluster without holes. Therefore, inthe avalanche cluster there are very few holes only near the surface in the first layer whereeach Kite topples once.

Tn FigK. 4, wo present, the directly measured diKJ.ribiiJ.ions of avalanches by inasK ,s,aroa a and dnra.lion /. in a double; logarilTimio plot, for llie N-E-S-W-N dynamics and llielaltice of size L = 600. Those distributions display a. power-law behavior up lo a. cor laincutoff which depends on the system size L. Since our simulations are limited by llielattices of finite size we ought to apply the finite-size scaling analysis [12. 13] assumingthe distribution functions scale with the lattice size L

P{x,L) = L-^Ux-L-^), (3.1)

where f.r(xL~'J-1') is a universal scaling function, x stands for s, a, t or r, and 3.r and ux arecritical exponents which describe the scaling of the distribution function. The finite-sizescaling ansatz (3.1) can be rewritten in the following form [1-1]:

P(x, L) = x-3x/l/xfT(x • L-"1). (3.2)

Let us suppose that distribution functions in the thermodynamic limit (L —V -so) showpure power-law behavior for large enough stochastic variables (s.a.t.r)

P(x) - x~T* , x > 1 , (3.3)

where r.T. x G {•?, a,t, r\ are critical exponents. This conjecture is mainly supported bycomputer simulations and different, heurislic argiiTii(;nts[14]. T'}i<;re[ore. comparing (3.2)and (3.3) wo got the scalimj relations among those exponents

r. = £. (3.4)

From the fact that (s) ~ L2 in the undirected BTW-type models [9]. one can get anadditional scaling relation [1-1]

iyj2-Ts) = 2. (3.5)

If we also assume that the stochastic variables s.a.t.r scale against each other, the ap-propriate fractal dimensions */„.„ can be defined via the following relations [15]:

.s ~ «7f i l . a ~ Pat ,s ~ f7ff , a — rlar . (3.6)

s ~ r"'~"' , t ~ T7tr

where %ry = 7~i1. The set of exponents {r,:, 7;r,y} are not independent and scaling relations

have the form [15. 1-1]

Ixy

F r e m i (3.7) one ; e;an fine! l.he; s i m p l e ; e;xpre;sKie>nK

We; h a v e ; 10 n n k n e n v n expoTieml.K alt.e)ge;l.he;r. n a m e l y TX a.nel 7^..^ = 7 " ' , w h e r e ; :r,y £

{c;., ,s. /.. r } . bill. l.}i<;re C^XIKIH o n l y 6 lin<;ri.rly iTi<l<;p<;Ti<l<;Tii. s c a l i n g r<;lri.l.ioTis (3.7) a m o n g l .hem.

AddiJ.ioTiril s c a l i n g r<;lri.l.ioTis Crin Tx; obl.aiTi<;d from l.he Kp<;cific s l . n i d . n r e rni<l evo lu l io i i of

an avalanche and depend on the given model. The compactness of an avalanche clustergives us 7ar = 2 [16. H].

Thus, estimating only three critical exponents from the numerical data, we can cal-culate all the others using the scaling relations Eqs. (3.7). Having calculated more thanthree exponents we are able to check these relations as well. The accurate determinationof t.he T:H exponents is A more difficult. Uisk l.lm.n t.he 7 :K due l.o I.IK; KJ.rid. <]<;p<;Ti<]<;iice of

t.he r ' s on I.IK; syKJ.em siz.e L.

To r<;dnc<; t.he fluct.uat.ioiiH of t.he <la.l.a. w<; inLegrriLed each diKJ.ribnJ.ioTi ov<;r bin lengths .

T h e cxponenl.s 7^.^. ;r, y = {,s. a. I,}, ar<; mea.Kur<;d froin I.IK; slopes of I.IK; KJ.ra.ighJ. parl.K of

t.he; correspond ing graph K (Figs. 5). T h e obta ined values are shown in Table; 2.

Plotting integrated distributions P(t,L) • LPt versus t • L~1'1 on a double logarithmicscale, as is shown in Fig. 6 for different lattice sizes L, we obtained from finite-size scalinganalysis that the best data collapse corresponds to 3f = 1.78 ± 0.05. ut = 1 .'36 ± 0.05(Fig. 7). The scaling relation for the critical exponents (3.1) gives the value rt = 1.31 ±0.05.

Next., we; use; t.he; Tiie;a.snre;el valne;s of rt, ~jsl. and 7S!l t.e> e;st.iiiia.l.e; t.he; whole; set. ofexpoTieml.K UKing l.he; se;aling relat.ie)iiK, Eqs. (3.7). These; valne;s are; prese;nl.e;el in Table; 2.

The; simulat.ie>riK for t.he; HA nie)ele;l wil.h N-S-W-F-N elynamies wit.hin a. small une:e;r-t.ainl.y give; t.he; Kaine; valne;s for t.he; e;ril.ie;al e;xpe)ne;nt.s.

Table 2: The crit.ical exponent.s for the 2ci HA model evaluated in our work (first column)are compared with t.TioKe for the RTW and l.wo-Htale Manna models. The second column isthe critical exponents for the BTVV model obtained from numerical simulations, whereasin the third column we show exact values of the exponents for the 13TW model basedon the scaling relations (3.7), (3.8) and -7.,,. = rr + 1 [16]. Comparison of the criticalexponents of the 11A and 13TW models evaluated from numerical simulations shows thatthe 11A model belongs to the universality class of the BTVV model. The uncertainty oft.Tic: numerical dal.a for the HA model is about ±0.05.

Exponent.rs

n

Is*

1ST

Izt

i (IT

llT

HA

"1.18̂1.21b

1.31l / l l b

1.111.682.23b

1.512'"

1.33b

RTW

1.20 [17]1.22 [IS]

1.32 [17]l/12b

1.06 [2]1.64 [17]2.16b

1.52b

T1.32 [2]

Mod<::1

RTW554 ~~7 _

| =5

2b

± =

H _

2C

r>4 ~~

11111

21

1

a

.2 [19]

.25 [19]4 b

.5b

.25b

.5b

.6b

.25 [16]

Manna

1.30 [4]1.37b

1.50 [4]1.75b

1.23 [2]1.67 [4]2/19b

1.35 [2]2.01b

1.49 [2]

a Exact result,h The value of the exponent is obtained from the scaling relations (3.7) and (3.8)." From the compactness of an avalanche cluster [16].

4 The height-arrow model on the Bethe lattice.

In this section, we present exact analytical calculations for the averaged density of occu-pied sites in the 11A model on the Bethe lattice of coordination number q = A. The Bethelattice is defined through a (Jayk tree well known in graph theory which is a connectedgraph with no closed circuits of edges. Then, the Bethe lattice is an infinite Cayle treehomogeneous in the sense t.Tiat. all except. t.Tie outer vertices have the Hame coordinationnumber q [20].

Following Dliar [21]. we approach t.Tie problem by dividing t.Tie allowed configurationsof t.Tie HA model in t.Tie SOC stale int.o t.wo types: slromjly allowed and weakly allowed andconstructing t.Tie recurrent relations for t.Tie rat.io of these configurations on the branchesof the Bethe lattice. Using this ratio in the thermodynamic limit, we obtain the densityof occupied sites in the 11A model.

First, let us briefly describe the procedure of construction of the Cayle tree. Likemany tree-like structures, the Cayle tree of k generations of coordination number q canbe constructed by attaching q /vth-generation branches to a central site, as is shown inFig. S. In l.urn. every Mb-generation branch is conHtrucl.ed by connecting q — 1 (k — 1 )th-general.ion branches to a new root, and so on [20]. This properly allows us t.o build

7

recursion relations for the number of allowed configurations on the branches of the Cayletree.

The number of boundary NH.CN of the Ca.yle tree is comparable with interior ones.Hence. t.Tie ca.lcnlal.ion of the bulk properties in the l.Tiermodynaniic limit requires specialcare. Since; we are inl.<;reHJ.e<l in t.Tie Holul.ion on l.lie Bethe la.l.l.ic<;, we will take t.Tie resultfor th<; averaged densit.y of occupied silos calculated at. the cenlral Hil.e of the Cayle t.reeas t.Tie value for t.Tie "Bethe lall.ice.

The definition of the 11A model on this connected graph of coordination number q = Aremains unchanged. The only difference from the square lattice concerns the notationof directions of arrows and sequences of their turns. We will consider only sequentialclockwise turns by the right angle and denote the directions of arrows at each site simplyby {1,2,3,4}-

Lot C be an allowed configuration on the Hh-general ion branch Tj, with root, verlexa. Adding a verlex b to Tj,. one defines a subgraph T' = T;. U h. Tf t.Tie HubeonfignralionC on T' with z^ = 0 and an arrow directed up or right (Fig. 9) becomes forbidden, Cis called a weakly allowed (W) confignralion. otherwise it is called a. strongly allowed (S)one.

Now consider T;. wit.Ti a root, a t.Tiat. consisls of t.Tir<;<; (k — I )j.}i-g<;neratioTi branches

^'A--I ; ^k-\ an<^ -L'k-\ with roots «| . a-2 and a^. respectively (Tig. 10). Let Nw(^'k- n,f) an<^Ns(Tj.,n.'\) be t.Tie numbers of distinct W- and 5-typo configurations on Tj, with a givenheight za = n and dir<;ctioTi of t.Tie arrow at. t.Tie root, verlex a.

Let us also introduce

Nw(Tk)= Yl J2 Xw(Tk.n,r), (4.1)«.= {(),! }-r={t4}

Ns(Tk)= Yl E Ns(Tk,n,r).. (4.2)«.= {(),! }-r={t4}

wTier<; th<; first summation is over t.Tie values of the heights and the second one is over t.Tiedirections of t.Tie arrow. As has been already mentioned, the arrow at. each silo can lakeonly two directions.

Those numbers can be expressed in terms of t.Tie numbers of allowed subconfignralionson t.Tie t.hroo (k — 1 )lh-generalion branches Tj._\. T^J-^ and T^J-^:

V IT \ \^] V{2) vW _L v(U V^2^ \rM _L v(U v(2J \r(*) , v ' 1 ' v(2J \r(*) ,

. . (1) , , ( 2 ) , , ( 3 ) , . . ( 1 )

1F V S VT1' ~ r ^VFv ( l ) v ( 2 ) ,..(;^ ,..(1) , , (2 ) , , (3 ) ,..(1) v ( 2 ) v ( 3 ) v ( l ) . v

(3)

{2} V ( 3 ) 4-'} \:(1) \:(2) V ( 3 ) 4- 9 V ( 1 ) V ( 2 ) V t 3 ) 4- \:(1) V ( 2 ) V ( 3 )

when; N& = NJT^), a = W,S and i = 1,2,3.T̂ ;!. us define

A " = ^ . (4.5)

if we consider graphs T^ . rlfjt and ^£}, to be isomorphic. then Nil'l^ ) = Af (Ti-i) =Ar('T .̂L i) r<-Tid from (4.3) and (4.4) one oT)lains t.Tie following recursion relat ion:

X(Tk) = \(l+X(Tk_i)). (4.6)

With the initial condition X(TQ) = ^, this equation has a simple solution

= £ - £ 3 - ( * + 1 > . (4.7)

In the t her mo dynamic limit (7; —Y oo) the iterative sequence \X(Tk)} converges to a,stable point X"' = \ that characterizes the ratio of the weakly allowed configurations tothe strongly allowed ones in the SOC state.

Consider now a randomly chosen site O deep inside the Cayle tree (Fig. 11). Theproba.bilil.y P(\) of occupation of the sit.e O is

->total

where N(i) is the number of allowed configurations with z = 1 at the site O and A'totai =A'(0) + A;'(l) is the total number of allowed configurations on the Cayle tree. The numbersA'(0) and A'(T) can be expressed via the numbers of allowed configurations on the fourneighbor Mli-generation branches Tk , i = 1,2,3,4

;V(0) = 2[\+2X + X2]l[Ns(Tli)), (4.9)

1

;V(1) = 2[1 + 4.Y + 5 Y2 + 2A'3] J[ Ns(T{i}). (4.10)

F o r t.Tic sit.<;s f a r IVOTVI O K ; su r f aces in j.}i<; l .T ic r ivKxlynaTnic l i m i t , (k —Y co) w<; }iri.v<; .Y = ^ .

T I I U H , f ro i i i ( 4 . 9 ) Am] ( 4 . 1 0 ) w e o b l . a i n

The value for the concentration of occupied sites P(l) is in good qualitative agreementwith the numerical result obtained on the square lattice (see Table 1).

5 Conclusion.

We numerically studied t.Tie self-organizing }i<;iglil.-arrow (HA) model on the square Uil.t.iceand rinalyt.ic-ri.lly on OK; "Bel.he U>!l.ice. The dynamics of I.IK; model drives it. into I.IK; crit.ica.1at.l.ra.cl.or vvil.h Hpat.io-j.empora.l coinplexit.y. The obtained distributions of various dynamiccharacteristics show an explicit power law behavior which indicates long-range correlationsin the steady state of the system. To obtain the critical exponents of distributions ofdynamic quantities of the model in the SOC state, we applied the finite-size scalinganalysis. The values of these exponents are listed in Table 2 and compared with knownexponents of the 13TW model and two-state Manna model. Thus, we argue that the 11Amodel belongs t.o I.IK; universality class of iindirecl.ed models.

Furthermore, we investigated t.Tie averaged density of occupied sites pc in t.Tie SOCstale of I.IK; HA model. Tl. was also calculated exactly on t.Tie Bel.be lal.l.ice of coordinationnumber q = 4. The obtained resull.s are present.ed in Ta.ble 1.

Acknowledgments

1 would like to thank V.B. Priezzhev for valuable contributions and suggesting improve-ments. 1 thank N.S. Ananikian, 1). Dhar. E.Y. lvashkevich. 1). V. Ktitarev. VI.V. Papoyan,A.M. Povolotsky and B. Tadic for fruitful discussions.

1 am also grateful for the hospitality extended to me at the international Centre forTheoretical Physics. TriesU;, where; this work was completed.

Prti'l.iri.1 financial support by l.he Russian Foundation for Basic Research under grant,No. 97-01-01030 is acknowledged.

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[S] V.R. Pri<;zzhev: (1997), to be published.

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[10] V.R. Pri<;zzhev: J. Stal. Phys. 74, 955 (1994).

[11] R.R. Shcherbakov. Vl.V. Papoyan. and A.M. Povolotsky, Phys. Rev. E 55, 3686(1997).

[12] L.P. Kadanoff, SAL Nagel. L. Wu, and S. Zhou. Phys. Rev. A 39. 6521 (1989).

[13] M.JN. Barber, in Phase Transitions and Critical Phenomena, edited by C. Domb andJ.L. Lebowitz (Academic, London, 1983). Vol.8, p.111.

[11] K. Christensen and Z. Olami. Phys. Rev. E 48. 3361

[15] K. Christensen. ii .C. Eogedby. and 11.J. Jensen, J. Stat. Phys. 63 . 653 (1991).

[16] S.N. Majinndar and D. Dhar , Physica A 185 129 (1992).

[17] S.S. Manna, Physica A 179 : 249 (1991).

[IS] S.S. Manna, J. Stal. Phys. 59, 509 (1990).

10

[19] V.B. Priezzhev. D.V. Ktitarev. E.V. lvashkevich. Phys. Rev. Lett. 76, 2093 (1996).

[20] R.J. Baxter. Exactly Solved Models in Statistical Mechanics, (Academic Press., Lon-don. 1982).

[21] D. Dhar and S.N. Majumdar, J. Phys. A: Math. Gen. 23 4333

11

0.8 m i ii m i

0.7 -

0.6 -

0.5 -

0.4 -

0.3 -

0.2

0.1

0.0

0.720

0.715

i i i i i i i i i ; i i i i i i i t i I i i i i t i i i

100000 200000

Number of avalanches

300000

FIG. 1. A computer simulation of the HA model (N-E-S-W-N dynamics)on the square lattice of the linear size L=600 with open boundary conditions.The steady state is reached by the model after about 100000 avalanches.

12

A

II

0.722

0.720

0.718

0.716

0.714

0.712

_ i i i i 1 1 F T"~~ r 111 11*1 I I 1 -

r (a) i

nn

lnn

—

M

1 M

1

1 1

1

—" 1 1 1 1 1 1 1 1 1

11

11

11

1

-

TT

I r

w

1 1

1 1

1 1

1 1

11

II 1

1 1

1 1

1;

i i i * } i i t i E i -

0.000 0.005

1/L

0.010

FIG. 2. The dependence of the time averaged density of occupiedsites <p(z=i)> on the lattice size L. (a) The HA model with N-E-S-W-Ndynamics and random initial directions of arrows, (b) The same modelwith N-S-W-E-N dynamics and arrows initially directed east or south.The numerical extrapolation for the infinity lattice size (L —»™) gives(a) pc= 0.721 ±0.001 and (b) pc= 0.755 ±0.001.

A

II

V

0.755

0.754

0.753

0.752

0.751

0.750

0.749

0.748

• /

~r

I i

(b) 1

-—

r \I

t I

1 i

i

i I 1 I I

I I 1 I 1

—

I i 1 i i

i i

1 i

i1

I 1

1 11

t 1

I 1

— -

0.000 0.005

1/L

0.010

FIG. 2. (b)

FIG. 3: A typical form of an avalanche cluster of the HA model. The lattice size isL = 200. The avalanche cluster has a layered structure. The number of topplings in eachlayer is indicated in gray scale.

15

-1

-2

-3

60o

I-J

-5

-6

-7

„ ' • ' ' ' I • • ' • I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' '

I i i i i I i i i i I

0 1 2 3 4 5 6 7

L o g f s )> i o v

FIG. 4. Simulation results for distributions of avalanches by (a) mass,(b) area, and (c) duration of the HA model in the SOC state. The linearsize of the lattice is L-600. The number of avalanches for eachdistribution is 107,

16

I • • • • ' • • • I • • • l i • • • • " "

. . i i I i i i i I i , , r I . . . , I . . . . I . > i .-7

FIG.4. (b).

17

-1 r

-2

-3

-4

-5

-6

I I l \ I I i

(0)

-7I , , , , I . i i • I , • , i

1 2 3

Lo«io«

FIG.4.{c).

18

6

5

4

3

2

1

n

_, , , , I , ,

: (a)

1 1 I...L

-

- X

- • ' • • • 1 • •

. , 1 1 ,

/

, , i , ,

i i i i i

/

/

i

i i | i

•

.• J

Ar

, , i

i l l I I 1 tl 1

/ i

--

Log][)(a)

FIG. 5. Double-logarithmic plot of the dependence of the stochasticvariables {s, a, t} against each other for different lattice sizes.The distributions are integrated over bin lengths.

19

, , . , I , , , , I , , , , I , , , , I , , . , I

I I I I I I I I I I I I I I I I t I I I I f I I

FIG. 5.(b)

20

O,-1

D 1 I I I I I I I I I I I I I I [ I I I I I I I I

I I I I I I I I I I I I I I I I I I I I I I 1 I

FIG.5.(c)

21

o

-1 1

-2

-3

-4

-5

-6

-7

•' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' '

I I I I I I 1 I I I I I I I

FIG. 6. Double-logarithmic plot of the integrated distribution ofavalanches P(t,L) versus duration t for the five lattice sizesL-100, 200 500. Each distribution is averaged over 107 avalanches.

22

4

3

i i I i I r1 ' I ' ' ' ' I ' • • • I • ' • • I

o-1

-2

-4

i i I i i t i I i i i i I i i i i 1 r i r i

-2 -1

FIG. 7. Finite-size scaling plot for the integrated distributions P(t,L)-The data for different L collapse onto a single curve for p t = 1.78 andv t = 1.36.

23

Figure 8: Construction of the Cayle tree with q = 4 and k = 3 generations by attachingq = A Hh-generation branches to a central site. This procedure is explained in the text.

a

L

Figure 9: A fcth-generation branch Tthe rest of the subbranches of T&.

and vertex b form a subgraph T". The ovals denote

24

Figure 10: A feth-generation branch Tk consists of three nearest (k — l)th-generationbranches T ^ , T ^ \ and T^%

Figure 11: A site O with height z0 = n and a given direction of the arrow is located deepinside the lattice and surrounded by the four fcth-generation branches TJ: , TJ: , T^ ^ and

,(4)

25