Series and Monte Carlo studies of 2 and 3 dimensions for axial hyperscaling in directed percolation

Physica A 189 (1992) 43-59 North-Holland

PHYSICA

Series and Monte Carlo studies of 2 and 3 dimensions for axial hyperscaling in directed percolation J.A.M.S.

Duarte Institut fiir Theoretische Physik, Universitiit zu K6ln, W-5000 Cologne 41, Germany

Received 15 May 1992

The relationship between the exponent %, characterizing the divergence, at criticality, of the cluster mass measured on the anisotropic axis of directed percolation, and the susceptibility exponent y, while well established in two dimensions, both by series and simulation work, has been doubtful for a decade for three and higher dimensions. In the present paper we furnish extensive simulations for the body centered cubic site problem that focus on that quantity, and study further the ascending moments of the number of sites on the axis. These numerical results are highly compatible with the scaling relation Y0= Y - Du±, where D = 2 is the transverse dimensionality. They also indicate the existence of a fixed gap exponent for the size moments of this quantity. Extensive series expansions in two dimensions are in acceptable agreement with the same picture (with D = 1, above). While direct determination of 3'o is still not satisfactory for some of the highly connected three-dimensional lattices we have adopted, the existence of a fixed gap exponent is confirmed in fairly reasonable agreement with the concurrent evidence of the BCC Monte Carlo studies.

1. Directed percolation and some of its current numerical issues T h e special b r a n c h of p e r c o l a t i o n t h e o r y that deals with its a n i s o t r o p i c v a r i a n t s , o b t a i n e d t h r o u g h the o r i e n t a t i o n of o n e or m o r e axes of the u n d e r l y ing lattice (see, e.g. ref. [1], ch. 6) can n o w be said to h a v e m a t u r e d : t h e r e is a m o r e careful c o n c e r n over highly precise estimates d e s c r i b i n g its overall b e h a v i o u r , as well as, f r o m h u m b l y l i m i t e d b e g i n n i n g s , a g e n e r a l i z e d r e f e r e n c e to it w h e n e v e r the p r o b l e m involves some sharp f o r b i d d a n c e o n the prog r e s s i o n a l o n g specific d i r e c t i o n s o n a net. P r o b l e m s o n which these c o n d i t i o n s a p p e a r as a basic g r o u n d l i n e i n c l u d e a n u m b e r of ecological e p i d e m i c s , for which p e r c o l a t i o n p r o v i d e s a n e u t r a l d e s c r i p t i v e u n d e r l a y o n which d y n a m i c a l s p r e a d i n g (of a disease, a n u n c o n t r o l l e d fire or a parasite species) is allowed to occur [1-4]. B u t also c o r r o s i o n , t r i g g e r e d d e p i n n i n g [5], surface d e p o s i t i o n in various guises [6, 7] a n d catalytic 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved

44

J . A . M . S . Duarte / Axial hyperscaling in directed percolation

processes implying the poisoning of a surface by a given chemical element [8-10] share in the characteristics (both qualitative and numerical) of the directed percolation phase transition. A number of more technical issues has nevertheless been studied by various researchers, and in recent years the universality with respect to variable coordination number [11], the determination of expansions for the percolation probability above the threshold [12], using corner matrix techniques, the successful development of a Dyson-type equation [11, 13] for most exact series expansions and the study of the stability of the cyclomatic index with regard to the type of averaging (whether it is animal-like or percolation-like [14]) have added meaningful results to the present theoretical understanding of this phase transition, particularly in the area of exact theorems and series expansions. Most Monte Carlo work aims at more practical applied models (and in particular tries to break out of the universality class of directed percolation in search of more exotic transitions [15]) and the phenomenological renormalization of several years back has now ebbed away. Among the unsolved riddles of the 1980's lies the question of axial hyperscaling for directed percolation. First proposed by Essam and De'Bell [16] in 1983, it can be stated as a property specific to directed percolation that focusses on the existence of a susceptibility associated with the sites exactly on the anisotropic axis. The exponent for this susceptibility, %, should be connected to the normal mass susceptibility exponent 7 by the following equation: %= y-Dr±

,

(1)

whose geometrical meaning is that of a subtraction of as many transverse correlation exponents as the dimensionality of the transverse space, D = d - 1. This was argued through mean field arguments and some series expansions in two and three [16] dimensions. The behaviour was very different: in two dimensions, well behaved series and reasonably stable exponents confirmed the equation; in three dimensions no sound numerical values could back it. The later, Dyson-like expansions of Essam and coworkers [13] greatly improved series for the transverse and anisotropic (or time) axis correlation exponents, vI and vt and the mass susceptibility without any further developments on 3'o in two dimensions. Following a simulation of the axis quantities [17] we present here extensive series expansions for the moments of the number of sites on the axis (up to order 21) for three different lattices in two dimensions. When analysed these series yield a consistent picture that reconfirms eq. (1), shows that the average distance to the origin of sites on the axis at Pc is given by the time correlation and that a constant gap exists for the exponents that characterize the ascending

J.A.M.S. Duarte / Axial hyperscaling in directedpercolation

45

moments of the number of axis sites. In three dimensions we bring together the results of Monte Carlo simulations and some three-dimensional exact series expansions and find good numerical support for eq. (1), despite significant difficulties with the exact expansions.

1.1. Two-dimensional expansions We have used three lattices: the triangular site, alternated square site and partially directed cases. The partially directed has neighbours on the North, West and East, say of a site, and the alternated square on the North, NW and NE (no neighbours on its own level). The triangular lattice also has 3 neighbours to a site along the oriented axes passing through it. The data (in table I) is given for the moments of the axis sites according to the equation

m(i) = ~

aig(s, a, t) p S ( 1 - p ) ' ,

(2)

s-- l,a,t

where s denotes the number of sites, a the number of them on the axis and t the site perimeter, g(s, a, t) is the number of clusters with a given label s, a, t. We have also studied by Pad6 approximants the average distance to the origin of the a sites. The location of poles of the original series expansion (also on table I) and its derivative (a very common strategy) show, for the triangular lattice, compatibility with the (expected) fact that sites on that axis (also called the direction of maximum spread [4]) expand in spatial moments characterized by vt (fig. 1), the same time correlation exponents as for the cluster itself. The partially directed lattice is disappointing. Resorting to dividing, term by term, the series for the radius expansion by M(1), a procedure which eliminates the Pc dependence and recenters the singularity at 1, while adding a unity to the exponent (see, e.g. [18]) did not improve matters substantially. On the other hand, the divided series strategy works well for the gap exponent A0, which characterizes the fixed separation of the successive moments of eq. (2). The number of series is significant: division of the M(i + 1) by the M(i) term by term, plus the derivatives of such series, means 24 Pad6 sets of approximants. Not all of them work equally well: fig. 2 summarizes a rich combination of the most valuable ones, and these back, unmistakably, the existence of a gap exponent for the axis moments, whose estimate certainly includes the scaling conjecture (marked on the figure) that this exponent A0 is related to the others by A0 = To + / 3 ,

(3)

with /3 the percolation probability exponent. Notice, once again, that the

46

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

Table I Partially directed square axis moments s

M(1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

M(2)

1 1 1 1 3 5 7 I0 20 27 52 60 155 116 472 130 1590 -535 5988 -5708 25192

M(3)

1 3 5 7 15 31 55 94 178 301 558 900 1757 2584 5454 7210 17040 18291 55480 38308 192320

M(4)

1 7 19 37 75 167 349 688 1358 2583 5014 9240 17783 31400 61012 103888 204696 328565 683034 993088 2293864

M(5)

1 15 65 175 399 931 2155 4762 10150 21013 43338 86844 174365 337024 667686 1255702 2467044 4487259 8879260 15469420 31404104

1 31 211 781 2163 5495 13717 33160 76790 171567 376462 806880 1708415 3527216 7278292 14601880 29581560 57736805 115745538 219584752 439564792

Triangular lattice axis moments s

M(1)

M(2)

M(3)

M(4)

1

1

1

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1 1 3 5 6 14 25 32 70 116 176 337 535 976 1479 2759 4825 6647 14848 21192

3 5 13 29 50 108 227 394 790 1546 2782 5257 9839 18242 32649 60797 111941 194901 367524 649698

7 19 51 137 306 698 1615 3380 7132 15152 30812 61945 125173 249148 484569 952697 1849897 3532451 6812986 12928392

M(5) 1

15 65 205 629 1706 4332 10895 25954 59698 136186 303118 657313 1416587 3013250 6283005 13056785 26814293 54476589 110101008 220416162

1

31 211 843 2945 9306 26714 73375 193292 486580 1193936 2864516 6683857 15340165 34677436 76965369 168884249 365949505 784127867 1663545298 3495804672

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

47

Table I (cont.) Alternated square site axis moments s

M(1)

M(2)

M(3)

1

1

1

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 3 4 10 15 37 51 137 168 525 476 2209 659 10804 -6779 65726

3 11 24 64 137 339 685 1629 3186 7425 13990 32807 58693 142722 232467 630318

7 33 106 328 891 2425 6015 15185 35724 85803 194498 451291 996869 2249248 4880341 10753970

Average radius expansion s 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Triangular site

Partial direc, square

1 2 7 18 33 78 174 316 666 1348 2506 4904 9377 17918 32528 62482 116840

1 2 3 8 19 34 61 124 224 420 711 1416 2183 4646 6289 15474 16085

p a r t i a l l y d i r e c t e d lattice is c o n s p i c u o u s l y a b s e n t f r o m the graph: in fact, the p a r t i a l l y d i r e c t e d family, while i n v o l v i n g axis sites at all levels, fails to p r o v i d e easy s e c o n d r o u t e c o n n e c t i o n s to t h e m . F o r the s e c o n d site o n the axis, for e x a m p l e , the shortest way r e q u i r e s a 2 site cluster, b u t the next a l t e r n a t i v e r e q u i r e s a s q u a r e (4 sites). T h e set of lattices w h e r e only o n e of spatial d i r e c t i o n s is o r i e n t e d is, in o u r e x p e r i e n c e , very n e g a t i v e l y affected a n d the e x p a n s i o n s are p o o r l y r e p r e s e n t a t i v e of the m a i n physical singularities.

48

J.A.M.S. Duarte / Axial hyperscaling in directed percolation 3

.

3

~

3.2 ¢

3.1

+~ 3 . 0 O*

0

~--. 2 . 9 2.8 o¢ 2.7

0

2

. .590

6

~ .594

~

.598

,02

values

Pc

Fig. 1. Pad4 approximant plot of the average radius series (first spatial moment of the axis sites). * denotes estimates from the series, ~ estimates from its derivative. On the vertical axis the sum % + v, is indicated by the horizontal lines (separated by the current uncertainty)•

collected pades 2 2 2 0 1 8 1 6

ol ~3

4 7~

1 2

0

1 0

t ~

8

¢~

o

0 0o

6 O

4

I

i

• 96

l

.98

reduced

J

i

1 oo

I

1.o2

I • 04

variable

Fig. 2. Pad6 pattern for divided series based on moments to the 5th order for the axis gap exponent. Symbols are: O for the 3rd/2nd moment of the triangular lattice, A for the 2 n d / l s t m o m e n t (triangular line), a for the 3rd/2nd moment on the alternated square 1, © for the 3rd/2nd m o m e n t derived alternated square series, * for the 4th/3rd moment triangular lattice, • for the 4th/3rd moment (triangular lattice), x for the 5th/4th moment derived series (also triangular lattice), y for the 2 n d / l s t moment derived series (triangular lattice).

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

49

2. Three-dimensional studies

2.1. Series expansions and their problems In three dimensions the situation is far more complicated: Essam and De'Bell [16] tried all of the usual lattices, but eq. (1) seemed to be either violated or not sufficiently illustrated by the chosen lattice topologies. The above mentioned authors presciently chose the second alternative, arguing that both the simple cubic and the body centered cubic variants, when taken to an order less than 13, do not sample the net effectively. In this light the extension to 18 terms for the simple cubic bond case [18], would be equally doomed. We have tried instead to develop series expansions that would include sites along the anisotropic axis on every transverse plane, or as many connections as possible to the nearest site along the direction of maximum spread. Two of these choices proved as disappointing as the earlier loose-packed cases, namely the partially directed cubic lattice to order 13 (with coordination number 5 with the square neighbours on the plane plus the successor site along the axis) and the 7-neighbour cubic site, including the first octant directions, their 3 face diagonals and the octant diagonal (to order 10). The series behaviour for the axis susceptibility is hindered, in both cases, by the lack of good and precise Pc estimates, while the study of divided series, which would map away this source of imprecision, is too scattered in singularity plots of Pad6 approximants. The value ranges and the reasons invoked by Essam and De'Bell ]16] apply equally to both of these cases. In fact, now that Grassberger [19] has added Pc estimates for three dimensional problems much sharper than the ones available to those authors in 1983, we have reexamined some series, biased at the central estimate value. On the simple cubic bond problem, for example 70-0.299 is very settled as an estimate, but in clear violation of eq. (1). The vogue launched by Grassberger's successful clutch of examples among the workers in related fields, notably the contact processes area ]8-10,15], which usually call this type "timedependent simulations", and the good series results in two dimensions obtained for the alternate square lattice have made us add the three-dimensional version, in which a site has 5 connections to the vertical successor on the upper square plane plus its 4 level neighbours, without any links in its own plane. Moderately demanding enumerations up to level 11 (about 100 times smaller than the triangular and partially directed data in the previous section) are presented in table II. The first use of this lattice, in a "time dependent" simulation context, was made by Noest [20] in 1986: unfortunately he chose to concentrate on the exponents. But in the context of the polynucleation growth model, D. Wolf and co-workers [6,7] have obtained the threshold to a perfectly

50

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

usable accuracy (Pc = 0.2723(3)) by several alternative methods. And one of the consequences of the optimism following [19] is that v, is more stable in three dimensions than it is in the two-dimensional version of the model (see e.g. [21]). When checked against the susceptibility and average perimeter data in Table II, the Pad6 approximant pattern is in very good agreement with 3' = 1.571(6),

(4)

as quoted in ref. [19]. Divided series place poles systematically below 1 and are unreliable. For the axis susceptibility, fig. 3 presents a combination of the pole plots from the series derivative with the biased estimates at 0.2723. It is difficult to draw conclusions based on the paucity of information below the threshold, with only two approximants. Values are markedly below what eq. (1) would give, around 0.06 to 0.10. Much better results are presented in fig. 4, Table II A l t e r n a t e cubic lattice s

M e a n size

Perimeter

1 2 3 4 5 6 7 8 9 10 11 12

1 5 25 107 455 1856 7570 29770 119190 457737 1822931 6840811

5 20 82 348 1401 5714 22200 89420 338547 1365194 5017880

2nd m o m e n t 1 15 145 1049 6679 38520 209468 1077558 5375224 25818015 122028949 559450541

Axis m o m e n t s s

1st m o m e n t

2nd m o m e n t

1 2 3 4 5 6 7 8 9 10 11

1 1 5 3 43 36 450 130 5522 -311 67107

1 3 17 29 191 380 2476 4090 32412 49695 423477

J. A. M.S. Duarte .25

Axial hyperscaling in directed percolation

,

I

r

I

i

I

I

'

~

I

'

I x '

_

×

-

.20

51

×

.15 x

.10

O

.05 0 -.05

2168"

i

- . 10. 2 6 4

I

,

I

i

I

J

• 272

I

I

I

i

.276

-

• 280

Fig. 3. Pad4 approximant plot for the first axial moment on table II: crosses denote location of poles for the derivative series, B denotes the estimation using the central value of ref. [6], whose uncertainty is indicated by the vertical lines. 1.2

'

'

'

'

I

'

'

'

I

'

'

'

'

I

'

'

~

I

I

}

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I

i

'

1.0

0

<3 +

t~

.8 .6

0

.4 .Z

0

1

I

• 265

i

I

I

]

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• 270

I

I

[

I

I

275

.280

threshold Fig. 4. Pad4 approximant plot combining the estimates from the second moment axial series in table II (symbol t ) , its first derivative (symbol * ) and biased estimates at the central Monte Carlo estimate of ref. [6] (like in fig. 3).

where the second moment of the axis (or spine distribution) shows a closer behaviour between the basic series and its first derivative. Biased values at the threshold are also indicated• Globally, the evidence is compatible with 70 + A0 = 0.75(15) and the pole locations of the various derived series are not conflicting.

(5)

52 2.2.

J , A . M . S . Duarte I Axial hyperscaling in directed percolation Monte

C a r l o s i m u l a t i o n s o n the b o d y c e n t e r e d c u b i c lattice

The best word for the series results of the previous section is, nevertheless, disappointing. We believe that Essam and coworkers [16] were right and that the generally poor behaviour of for example the partially directed cubic net can only be overcome with massive extensions. The best candidate for this is, however, the alternate cubic site problem but several terms will be required before the asymptotic regime is settled. Against this rationale, we have undertaken simulations on another lattice, the bcc site problem, whose critical value was put at 0.34450(12) by Grassberger [19], very close to the central series estimate of ten years ago, 0.344 [16]. Series were clearly not a solution to the problem due to the sublattice alternation that effectively reduces by a half the number of possible axis sites at each level. We started on a million to 1.5 million samples scanning of the interval 0.34460 to 0.34440 by 0.00005 intervals, up to 960 time steps along the axis. As in a previous two-dimensional paper [17] we studied several quantities, namely the probability of a connection to the origin P(t) ~ t -~ ,

(6)

the total number of sites up to order t M(t) ~ t ~ ,

(7)

the z exponent, given by the average transverse radius R~ ~ t ~ ,

(8)

as well as the gap exponent for the transition a = 3' + / 3 ,

(9)

the q'0 exponent, defined from the number of sites on the anisotropic axis on surviving clusters to time t, *0 =

(10)

and an equivalent gap exponent for this quantity zl0, normalized by v~. The transverse linear dimensions were 100, 200 and 300. For the slopes of the corresponding log-log plot the patterns for &, 4/, z, and 00 are given in figs. 5-8, respectively. Concentrating on the upper part of the interval according to

J. A . M . S . Duarte / A x i a l hyperscaling in directed percolation

53

indications in these figures we generated further samples at linear dimension 300 and 400, taking the times to 1400 and 1600 time steps and using a scanning interval of 0.00002. Longer times (we used open boundary conditions and

-

36

38

~...."~

i

:~ i

i

I

40

C¢3 4z

44

46 48 0

.05

. I0

inverse

.15

.20

time

Fig. 5. Successive slopes for the log-log decay in the number of surviving clusters with time (eq. (6)) in the critical region, are plotted against the inverse time (divided by 8). Sample size 300 times 300 to 960 time steps, averaged over 1.25 million trials. From the top, the lines are as follows: solid line, 0.34460; dashed line, 0.34455; solid line, 0.34450; dotted line, 0.34445.

C) O 00

1

2 4

1

22

i l l l l l l l i i l l l l l l

1 20

C~ 0 I

18

16 \

114

i 0

i

i

I .05

I

i

t

I .10

inverse

i

i

i

I .15

i

i

I

< I •

20

time

Fig. 6. Successive slopes for the log-log plot of M ( t ) (eq. 7) are plotted as in fig. 5 against the inverse time and for the same size and number of trials•

54

J.A.M.S. Duarte / Axial hyperscaling in directed percolation 1

148

1

146

1

144

I I F I I i l I I ] I I I I I I I I

C:) 1 142 C) U3 1 140 O~ O) 1 138 0

1.136 1.134 1.132 1.130 1.128

I

I

I

i

i

i

i

I

.05

0

I

I

I

~

I

I

.10.

I

I

I

'

.15

inverse

.EO

Lime

Fig. 7. Successive slopes (plotted as in fig. 5) for z (from eq. (8)) for the same size and number of trials.

.09

,,

,,

i , , ,

f

i , , , ,

I

]

,

,

,

I

i

i

i

i

.08 0 C~ .07 5q ~9 0 09

04

"l

0

i

i

i

I

i

.05

i

J

i

I

.10

inverse

i

i

i

i

.15

.20

time

Fig. 8. Successive slopes (plotted as in fig. 5) for qJ0 plotted against the inverse time.

controlled the n u m b e r of times a cluster hit the border to be no m o r e than 1 or 2 in a million samples) required 400 or higher in linear dimension. The n u m b e r of samples varied between 1.6 million for 0.34458 and 7 million for 0.34456 (for t = 1400) taken roughly in 0.4 million sections. Less extensive simulations up to t = 1600 (e.g. 4 million for 0.34456) were also done. Bearing in mind the

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

55

inevitable fluctuations, the results of the probability scanning are all in reasonable agreement. They can be described as follows: (i) For the susceptibility exponent ~0, our values are more in agreement with 1.222 than with a lower value, 1.214 [19]. The discrepancy is however, hardly serious. We find little deviation from the linear dependence in t -1 for the present case. (ii) No matter how many iterations were spent on the z exponent this is systematically extremely ill behaved and with large fluctuations. In the region 0.34456(2) it is in agreement with z = 1.133(2), well within the estimates z = 1.134(4) or z = 1.134(13), from either Monte Carlo or biased series [19]. (iii) The problem of scaling corrections come clearly to the fore in the decay exponent 4~, which is, together with the mass axis exponent and the A estimator the most pronouncedly curved. The expected behaviour must account for the fact that the probability of survival is P ( t ) ~ t *(1 + A t -W + "

").

(11)

A value of w = 0.6 does, in fact, correct most of the curvature in the plot. From the central Pc = 0.34456 the value leads to 0.455(7) in good agreement with ref. [19]. (iv) A very wide variation for 00 is apparent in fig. 8; the results are quite different from the old series apparent values quoted in the previous section, and much more in line with the scaling eq. (1). Errors are, however, likely to be significant because the exponent is very small and the curvature is even stronger than for the decay exponent. In fig. 11 we plot both the decay exponent and ~0 against a stretched scale and the axis exponent is recognisably more curved. (v) Another exponent that shows appreciable curvature is the gap exponent A. Results from the four ascending moments of the total mass on clusters disappearing at exactly time t, as described in ref. [17], are divided and plotted against a stretched scale in fig. 9. This is in fact one of the most stable exponents, barely sensitive to variations of the probability within the accepted interval. Fig. 9 also embodies, of course, scaling, since it clearly shows the constancy of distance between ascending moments of the cluster size distribution. (vi) When the corresponding moments for the number of sites exactly on the axis are studied, a very strong oscillation appears, although scaling, in the form of a constant gap is indisputable: the presumable value of the exponent (which should be given by Ao/u,, see ref. [17]) evolves slowly and then starts to acquire a superimposed fluctuation after times 500, roughly. Most of the evolution is done in this regime, therefore, and we tried, through rolling averages, to detect

56

J . A . M . S . Duarte / Axial hyperscaling in directed percolation

po = 0 . 3 4 4 5 6 1.7

i

I

I

i

I

i

I

~

I

i

I

% .~.., 1.6 5Q (D 1.5 0

E 1.4

1.3

i

I

i

.1

0

.2

I

.3

.4

l , / t °'6 Fig. 9. Successive slopes for the log-log ratio of moments, against a stretched scale, from eq. (7). Dashed lines for the third/second moment, solid line for the second/first moment, dotted lines for the fourth/third moment; 300 times 300 by 1600 time steps (time scale divided by 4); 4 million samples.

axis gap e x p o n e n t

°°I

75

,

I

'

i

L

i

'

I

i

I

i

]

I • 01

i

I .02

I

I .03

I

[ .04

i

L .05

I

I ,06

65

0 0 CO

6O

(2) N

55 50

4 5 _-i 4O

0

I .07

inverse time Fig. 10. Successive slopes for the log-log ratio of moments on the axial direction (plot as fig. 9, sample statistics, 7 million trials): solid line: fourth/third moment; dashed line: third/second m o m e n t ; dotted line: second/first moment.

J.A.M.S. Duarte / Axial hyperscaling in directed percolation

57

compensated curvature f

I

I

i

i

i

I

~

I

I

I

I

I

i

i

I

-.4

0

I

I

/

[

I

I

_

--.5

o u3 -.6 @ o

-.7

-.8

--.9

t

0

I

t

t

I

.05

~ t

I

I

[

.10

I

I

I

I

[

I

.15 t-0.6

I

I

I

I

I

.20

Fig. 11. Successive slopes on a stretched scale of the exponents for cluster decay (upper line) and the mass on the axis in surviving clusters (multiplied by a factor 10); 300 times 300 samples, extending over 1400 time steps at p = 0.34456.

any steady trend in its m e a n value. Most of the time the three m o m e n t ratios oscillate unanimously, so that any averaging of them would be useless. For the 0.34456 and 0.34452 values both the t = 1400 and t = 1600 values show a perceptible increase in the mean. We estimate a value of Ao/v ~ = 0.535(15) (fig. 10). 2.3. Results and discussion

T h e simplest value to discuss is the gap exponent for the mass m o m e n t s A/vt, given in fig. 9. A conservative range, taking into account the evolution of the values around the probable central Pc estimate leads to A = 2.160(15), which c o m p a r e s very well with the 2.1595 to 2.162 range obtained from either hyperscaling for the /3 exponent or direct calculation from the q5 exponent (/3 = 4~v,) [18,19]. Using central values throughout, and both correlation exponents in [19],/3 would be, then, around 0.589 and, logically, ~b = 0.458, a value that should be c o m p a r e d with the estimate in this p a p e r (section 2.2). T h e other gap exponent, one of the main reasons for the present study, although stable in the scaling region, should be c o m p a r e d with the central value resulting from equations 1 and 3 leading to a0/B = 0 . 5 4 5 ,

(12)

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J.A.M.S. Duarte / Axial hyperscaling in directed percolation

if the conjecture (eq. (3)) that works well in two dimensions is to be extended to d = 3. This is still plausible, though a bit high, from fig. 10. Combining older, coarser values of Reggeon field theory [22], or of the older series expansion analyses (as referred to in [18]) the central value could come down to 0.527. This seems too low for the evolution of the rolling averages: on the other hand, it is difficult to gauge the extrapolation of these slowly increasing averages in the range Pc = 0.34456(2) and whether it will reach the region 0.54 . . . . For the series values the estimate A o --0.702 (see fig. 4) is quite plausible. Finally, the three-dimensional data on the mass axis show a fan-like distribution of slope estimates (fig. 8) that is repeated when the interval 0.344500.34458 is studied. The value evolution breaks down between 0.34456 and 0.34454, by turning away abruptly at higher time steps. In fig. 11, where the exponent is c o m p a r e d with ~b, the central value arising f r o m the combination of eqs. 1 and 10 would be 0.0878, a plausible estimate w h e n the full curvature is compensated. This strengthens the case for a central estimate of Pc =0.34456(3), and q'0 should cover an interval 0.073-0.090 asymmetric about 0.087, our best central guess. When one turns to fig. 3, the dismal p e r f o r m a n c e of the series estimates is clear: the central Y0 value should be centered around 0.113 although its sensitivity to fluctuations in Y and v± is clear from the fact that by combining a lower value of y with the much higher (0.737 as against 0.729 in ref. [19]) v± from Reggeon field theory [22], its value can be brought down to 0.091.

Acknowledgments This research was funded by J N I C T through project C E N 379-90.

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