The perimeter in site directed percolation. Mean perimeter expansions

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The perimeter in site directed percolation. Mean perimeter expansions

J.A.M.S. Duarte, H.J. Ruskin

To cite this version:

J.A.M.S. Duarte, H.J. Ruskin. The perimeter in site directed percolation. Mean perimeter expansions.

Journal de Physique, 1986, 47 (6), pp.943-946. �10.1051/jphys:01986004706094300�. �jpa-00210290�

The perimeter in site directed percolation. ^Mean perimeter expansions

J. A. M. S. Duarte (*)

Solid State Physics, Imperial College, London SW7 2AZ, ^U.K.

and H. J. Ruskin (+)

School of Quantitative Methods, ^N.I.H.E. Glasnevin, ^Dublin 9, Republic ^{of Ireland}

(Reçu le 18 octobre 1985, accepte ^{le 29} janvier 1986)

Résumé.

Nous analysons ^{des séries} pour le périmètre moyen ^et la taille _moyenne ^en percolation dirigée, ^séries

dont nous avons obtenu des termes nouveaux. Nous obtenons des estimations plus précises pour le seuil de perco- lation critique pc dans le ^cas des réseaux triangulaires ^et cubiques. ^{Dans le cas du} problème ^de percolation ^{de site}

avec seconds proches ^{voisins sur le réseau carré, nous trouvons pc}

^{0,4965 ± 0,002.}

Abstract.

Exact expansions ^for susceptibility

^{like mean} perimeter series for directed percolation and extended

mean size series are analysed, ^{on two and three} dimensional lattices. The critical threshold pc estimates ^are refined for the triangular ^and simple cubic lattices. On the _square next-nearest-neighbour ^site problem pc is estimated as

pc

0.4965 ± 0.002.

Classification

Physics ^Abstracts

05.50

Introduction.

Studying ^the critical behaviour ^of the moments of the cluster size distribution is one of the standard _{ways of}

establishing ^{a set of critical} exponents ^{for the} percola-

tion transition in _any dimension. Consider directed site percolation, where clusters spread ^{from a} given origin along preferred orientations of the axes, between sites that are occupied ^with probability p. In this

special ^case it is clear that the various moments M ‘( p)

can be expanded ^{in low} density ^series (small ^values ofp)

from a knowledge ^{of the} total numbers of configura-

tions of finite clusters starting ^{from the} origin (g sf)

and we have

where s is the size of the cluster, and ^t (the ^{other essen-}

tial characteristic of the cluster) ^{is the} perimeter count,

(*) ^On sabbatical leave from : Laborat6rio de Fisica, Faculdade de Ciencias, Universidade do Porto, ⁴⁰⁰⁰ Porto, Portugal.

(+) ^{Work carried out in} part ^at Trinity College ^Dublin 2, Eire.

i.e. the number of bounding lattice sites along ^the

allowed orientation of the axes (see Fig. ^{1. A of}

Ref. [10]). The gst ^{are crucial to the} determination of such exact series expansions. ^{In this paper we shall}

concentrate on the analyses ^{of first moment} expan- sions (i

^{1 in} Eq. (1)). ^This gives the average number of sites connected to the origin, ^{or its « mean cluster}

size » S(p), ^{at a} given p. This quantity diverges ^near

the percolation ^threshold Pc with the leading ^critical exponent y

However, knowledge ^{of the} gst also enables alterna- tive calculations of ^an equivalent ^« susceptibility >>

like quantity, ^{the mean} perimeter T(p) by taking

Consider the critical behaviour ^{of the} correspond- ing low-density ^series expansions. ^{From the sum rule}

for the occupation ^{of the} origin,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004706094300

944

Fig. ^1.

^Pad6 plot ^{of y versus Pc} for the square site problem

from the mean size (2022) ^{and mean} perimeter ^series (+) [i/j ] - j degree ^{of the} denominator, i degree ^{of the numera-}

tor (the ^{mean size series are not} identified for easier reading).

one obtains by taking derivatives w.r.t. p,

and, clearly, ^near Pc :

Expansions ^for S(p) ^{have been} published ^{and ana-} lysed ^for y in dimensions 2 to 5 (the upper critical dimension for directed percolation) [ l, 2]. They ^follow

as a by-product ^{from a} knowledge ^{of the} gst histo- grams given ⁱⁿ [3] and in the reference [10]. ^From equation (6) T(p) ^{must have the same} leading ’ diver-

gence associated with the exponent y. Therefore t, the perimeter ^{in the} percolation ^{sense, is somehow a}

measure of the cluster size and the sums in equation (1) (for i

1) ^and equation (3) ^are dominated, ^for large

s, by ^{the same value of the} perimeter ^as shown in the reference [10]. ^In fact, ^at fixed size s the _average peri-

meter value is proportional ^to the size with coeffi- cient (1 - Pc)/Pc :

and these contributions dominate in equation (6)

the summations on s near PC.

We reencounter in this context a situation known from undirected percolation ^and by ^{now well-docu-}

mented [4, 5]. ^{The use of} alternative measures of cluster size was explored in reference [6] (and ^{also in} [7]) ^to

obtain alternative susceptibility ^series.

Series expansions ^for T ( p) ^{are of a} comparable length ^to those for S(p), when these are obtained from the gst

they ^can alternatively ^{be obtained} by ^a

combination of transfer-matrix and graph-theoretical

methods (see [1]). ^{Note that} expanding ^{the gst} (up ^to

order s) through ^{order s + 2, then} (conf. Eq. (4)),

and

where

so that, using equations (9) (or Eq. (5)), ^{2 more terms}

can be added to S(p) ^{and one more term to} T( p), ^if

gs+2 (the total number of clusters with s + 2 sites) ^is

known. This is particularly ^{useful for the directed} percolation ^site problems ^{where exact} expressions

are known for square, triangular, simple ^cubic (to

order 17) [8] ^and body-centred ^cubic (to ^order 42)

animals.

We have used this additional data to obtain expres- sions for T( p) and ^for S(p). ^{These are} given ^{in tables I}

and II and we describe the results of this numerical

study ^{in the} following ^section.

1.1 PADÉSlUDIESOF Pc ^AND y.

Strong non-physical singularities make ratio methods of the usual type,

or even refined modifications [9] almost useless for the present series. We have therefore formed dlog ^Pad6 approximants ^{to them and} figures ^{1 and 2} present ^the situation for the _square and simple ^cubic lattices for

S(p) ^and bi! T(p). ^The approximants for the square

mean size _are, of _course, found in [ 1 ] ^{and the mean} perimeter pole locations fall on a smooth line almost

super-imposed ^on them. Both the _y value in 2 dimen-

sions, y

2.269 ± 0.018 and the _Pc estimates from

series, ^0.7055 ± ^{0.0005 are in} agreement ^{with the}

present alternative evidence from T(p).

Table I.

Mean size series for ^{the various lattices} ((*)

^added coeffcients). On the b.c.c. all

173.737 and _a12

514.988 (earlier ^{terms in} [2]).

Fig. ^2.

^Pad6 plot of y ^versus p, for the simple ^{cubic site} problem ^{from the mean size} (2022) ^{and mean} perimeter ^series (+) [ilj] - j degree ^{of the} denominator, ⁱ degree ^{of the}

numerator.

Table II.

Mean perimeter ^series for ^{the various lattices.}

946

For the square next-nearest-neighbour ^{lattice the}

much shorter series we have obtained are in excellent agreement ^{with the value for y. The} poles ^{for both} S(p) ^and bit T( p) ^{fall once} again nearly ^{on the same}

line and we estimate

for this problem.

On the triangular lattice the pole patterns ^{tend to}

concentrate around values of both y ^and Pc higher

than the existing estimates. Also, ^the lines for the two

alternative susceptibilities ^{remain somewhat} apart.

We refine the current estimate for _Pc to

on the basis of the new longer ^series.

Of the 2 three-dimensional lattices, it is the simple

cubic which exhibits the most difficult features. i) ^More

than 13 terms are required ^{for the} singularities ^to

exceed the central estimates for Pc ^or y (see Fig. 2) ^but higher ^order approximants then become concentrated

on this higher region. ii) ^{A more serious} gap is _appa- rent between both lines of singularities (for S(p) ^and bit T(p)) ^{than was found to be the case for the trian-} gular lattice, although ^{the common} artificial exten- sion obtained by forming dlog ^{Pad6s to 1 +} T(p) brings ^the perimeter ^{and mean size} singularities ^closer.

The mean size singularities concentrate around _{/?c =} 0.4355 whereas perimeter singularities ^favour (with

the present extension) 0.4345. We estimate

from the present evidence. On the b.c.c. lattice the series are only marginally longer than those in [2]

and the singularities ^for T(p) ^{are less} regularly ^distri-

buted than those for the s.c. case. The _Pc value of 0.344 ± 0.004 in [2] ^{is still} adequate.

2. Summary.

The singularity analyses ^{for mean} perimeter ^expan-

sions in two and three-dimensional directed percola-

tion were found to be in agreement ^{with a} susceptibi- lity ^{- like} divergence ^{at the} critical threshold. Based

on the extended perimeter polynomials ^of [3] ^{and a} preceding paper [10], series for the mean size have also been extended (by ^a significant ^{number of terms}

for the triangular ^and simple cubic site problems).

The leading divergence ^{in both} types is, ^{of course,} the same.

Acknowledgments.

We are indebted to D. Dhar for information (on ^the

b.c.c. numbers) ^{J. Adler for} discussions and to the B.

Council and the Gulbenkian Foundation for funding.

One of us (J. ^{A. M. S.} D.) thanks N. Rivier and the Solid State Group ^{at I.C.} for their kind hospitality.

References

[1] ^DE BELL, K., ESSAM, ^J. W., ^J. Phys. ^{A 16} (1983) ^385.

[2] ^DE BELL, K., ESSAM, ^J. W., ^J. Phys. ^{A 16} (1983) ^3553.

[3] DUARTE, ^{J. A. M.} S., Portugal. Phys. ¹⁵ (1984) ^119.

[4] STAUFFER, D., Phys. Rep. ⁵⁴ (1979) ^1.

[5] STAUFFER, D., Introduction to Percolation Theory (London, ^ed. Taylor ^and Francis) ^1985.

[6] DUARTE, ^{J. A. M.} S., RUSKIN, ^H. J., Zeit. Phys. ^{B 46} (1982) ^225.

[7] PRIVMAN, V., VAGNER, ^I. D., ^Zeit. Phys. ^{B 50} (1983) ^353.

[8] DHAR, D., Phys. Rev. Lett. 51 (1983) ^853.

[9] PEARCE, ^C. J., ^Adv. Phys. ²⁷ (1978) ^89.

[10] DUARTE, J. A. M. S., ^J. Physique (Paris) ^submitted

for publication.