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Site and bond percolation distributions : a survey of perimeters for all values of p
J.A.M.S. Duarte
To cite this version:
J.A.M.S. Duarte. Site and bond percolation distributions : a survey of perimeters for all values of p.
Journal de Physique, 1979, 40 (9), pp.845-851. �10.1051/jphys:01979004009084500�. �jpa-00209169�
Site and bond percolation distributions :
a survey of perimeters for all values of p
J. A. M. S. Duarte (*)
Wheatstone Physics Laboratory, King’s College London Strand WC2R 2LS, U.K.
and Blackett Physics Laboratory, Imperial College London, Prince Consort Road, S.W. 7, U.K.
(Reçu le 29 mars 1979, révisé le 16 mai 1979, accepté le 21 mai 1979)
Résumé. 2014 On présente une étude de deux types de percolation pour différents réseaux et à plusieurs dimensions.
On montre qu’au seuil de percolation, les lois d’échelle sont suivies et que l’universalité des exposants critiques est
vérifiée. L’analyse asymptotique des coefficients confirme les hypothèses prévoyant l’évolution de petits amas
dans la région critique.
En outre, on étudie le régime non critique pour un certain nombre de réseaux particuliers. Les résultats ainsi obtenus permettent de comparer les modèles contradictoires.
Abstract. - This paper presents a comprehensive survey of site and bond percolation distributions. Agreement
with the scaling picture and universality for both types of percolation is found within the usual uncertainty limits.
Asymptotic analysis of coefficients also supports existing predictions for the high density region.
In addition, the non-critical region is studied for a variety of lattices. The results obtained enable a comparative analysis of the existing conflicting proposals for that region.
Classification
Physics Ahstract,s
05.50
1. Introduction. - Considerable numerical and theoretical efforts have recently been devoted to the
problem of perimeter distributions for percolation.
Reference should be made to Essam [1] for a general
review of the problem and to Reich and Leath [2]
and Stauffer [3] for a recension of the current ideas
on the perimeter distributions in connection with the critical behaviour of random mixtures.
Briefly, in site (bond) percolation a site (bond)
is considered occupied with probability p and vacant
with probability 1 - p, and in the non-critical region
the probability p can alternatively be expressed as
the sum of probabilities of a site (bond) in finite
clusters (de Gennes et al. [4]) :
where n is the size of the site (bond) cluster, b the
number of boundary sites (bonds) that ensure its
isolation on a lattice and gnb the number per site
(bond) of different configurations with a given pair (n, b).
For fixed large n the histogram gn6 is conveniently represented by a continuous distribution with a
variable a = b/n, the perimeter-to-size ratio.
The onset of criticality is evidently located at the point where the higher moments of the cluster size distribution start to diverge
The reasons for this behaviour where first seeked
by Leath in the limiting shape of gnb (Reich and
Leath [2]). It is now known that the continuous
analog of gnb, gn(a) shrinks to a b-function in a, located at some ao» On the other hand, the total number of clusters is supermultiplicative (Klarner [5])
and if the property is extended to the 9,,(a), Leath [2]
From an analogy with the Bethe lattice later made sounder (Reich and Leath [2], Domb [6]), the sug-
gestion of Stauffer [7] for ac was
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004009084500
846
implying for all lattices
To obtain the required non-classical scaling expo- nents the shift in mean value (indeed in mode) became
the physically important quantity and for the mean
value at p = p,,, Stauffer [7] proposed
with Q = 1 /J, and L1 the gap exponent (Essam et al. [8]).
In the critical region, Kunz and Souillard [9] have provided evidence for the non-analiticity of the ( n’ > and obtained for p > Pc a linear dependence
of the form (6) for ( bln > :
almost simultaneously published with equivalent suggestions by Stauffer [3] and Hankey [10]. d is the
lattice dimensionality. The initial segment of the Bethe solution for gn(a) namely
was proposed as the asymptotic form of the distri- bution for real lattices.
The tests of Stauffer [3], Flammang [11] ] on exact
information of Sykes et al. [ 14] and Stoll and Domb [1 S]
Monte-Carlo experiments gave support to these
assumptions for two and three-dimensional site
percolation.
The situation appears less well established for the non-critical region where several conflicting limiting
forms for gn(a) have been proposed.
With the recent completion of the bond problem expansions for all dimensionalities on the hypercubic
system, Gaunt and Ruskin [16] a comprehensive
numerical survey became possible for both types of
percolation. An earlier report for cluster statistics at p = p,,, and p = 0 for two dimensional site per- colation has been published, Duarte [17]. By its nature, it excluded the variations with dimensionality, studies
in the high density region and tests of the universality hypothesis for both types of percolation.
Moreover, in the non-critical region, where a
tabulation of values for each lattice is unavoidable, the mutually rejecting proposals remain without inspection.
For that purpose, an extensive range of distributions has been analysed. Section 2 describes the numerical
techniques used for the critical and non-critical
regions, at Pc and p = 0, and their efficiency. Section 3
discusses the numerical evidence obtained in the
light of the current ideas on the problem.
2. Numerical analyses. - i) High density region. -
For this region Kunz and Souillard’s result implies
that the first-order-correction to linear dependence
satisfies a power law of the type n-’Id (eq. (7)). This
should easily be tested by standard extrapolation (Gaunt and Guttmann [18]) (1), from exact sequences.
(Sykes et al. [14], Gaunt et al. [20]) : these have been
scanned at p = 0.700, p = 0.800, and for d > 2 p = 0.600 and p = 0.500. Figure 1 gives estimates by the ratio method of - l/d for the simple quadratic
site problem. Table 1 gives estimates for the diamond
lattice site problem. Results of equivalent significance
can be given for the upper dimensionalities (site
problem) in acceptable agreement with Padé studies.
Bond results for d > 2 are generally poorer (Gaunt
and Ruskin [16]). As a rule, the convergence is quite noisy and clearly associated with the existence of
subsidiary modes of some relevance in the compact end of the histograms gnb.
Fig. 1. - - 1 /d estimates by the ratio method (crosses) and linear intercepts on them (dashed line) are plotted against lin (the inverse site content) for the simple quadratic site problem (ref. [14]) at
p = 0.700 (see section 2 (i) and eq. (7)).
Table I. - - l/d estimates by the ratio method and
linear intercepts on them for the diamond site problem
at p = 0.500 (section 2 (i)).
(’ ) Pearce [ 19].
ii) p = p,,. At the onset of criticality, the power law dependence of the first order correction varies as
ni ld- 1 (eq. (5)). Precise - literally conflicting pre- cise - estimates for this exponent exist : see section 3 below. Certainly the estimate in Duarte [17] for the triangular site problem must be considered as excep-
tionally confident, from the smooth behaviour of
the data.
The method is, of course, not reliable in the low
density side because of the usual uncertainty limits
for It is therefore interesting to analyse the results obtained for bond percolation where a number of
exact critical probabilities are known (Essam [1]).
Figure 2 gives ratio estimates and linear intercepts
for the simple quadratic bond problem (Gaunt
and Ruskin [16] and references therein). A reasonable
estimate is - 0.62 ± 0.02, in good agreement with the earlier estimates and the scaling hypothesis (Essam et al. [8]).
For d > 3 the most reliable method seems to be the detection of the correction by Padé approximants (hopefully powerful enough). It has been found difficult to estimate with precision non overlapping
intervals for d = 3, 4, 5 between (as required by the scaling hypothesis for d ) - 0.60 and - 0.50. The existing scatter even for d = 3 is commendable
(compare Flammang [11], with Kirkpatrick [12],
Essam et al. [8] or the prediction of Amit [13]). No
estimate will be added here.
iii) p p,. As could be expected, close to p,,, in the low-density region, the correction term transition is not sharp. Nevertheless, for values sufficiently
distant from Pc it is possible to estimate with some
confidence the mean value limit. For the simple quadratic site problem by the methods already employed in Duarte [17] and described in detail in (iv) below, it is possible to estimate a sequence
0.96 ± 0.02 (p = 0.400) , 1.065 ±0.015 (p = 0.250) ,
Fig. 2. - Ratio method estimates for Q - 1 (crosses) and linear
intercepts on them (dashed line) are plotted against 1 /n (the inverse bond content) for the simple quadratic bond problem at Pc (see
section 2 (ii) and eq. 6).
when looking at the limiting value of bfn for non-zero,
p below Pc’ in superb agreement with Stauffer [21 ].
This problem is particularly interesting since not only Stoll and Domb [15] have recently covered by
Monte-Carlo studies the close vicinity of Pc (like Leath before them in a somewhat narrower range)
but also Stauffer [21] ] has covered the whole of the
low-density region.
The parallel analyses for the other lattices can
therefore be accepted with confidence. The region
of inapplicability is found to vary between Pc - 0.20 and Pc - 0.15 in two dimensions, narrowing in
three and higher dimensions (see section 3 below for
a discussion). Lists of values for other lattices are available on request (Duarte, unpublished).
iv) For p = 0, the pure animal distribution is obtained. Adoption of (1 - p)/p for the coefficient is impossible and the least stringent assumption is clearly that ( bln > - A + Bn’*-’, which can be fitted by triads of successive values. For the significant exponent 0-* - 1, table Il lists estimates for various Table II. - Estimates for the coefficient 0’* - 1, by triad fitting (see section 2 (iv)),from refs. [14,16] and [20].
848
lattices, problems and dimensionalities. A general pattern is evident although convergence gets predic- tably poorer for the higher dimensionalities. It is
tempting to conclude that the first correction for real lattices will be Bethe-like. (For Bethe lattices B is either 2 (site problem) or z (bond problem).) The
sole exceptions were the bond problem distributions (for d > 3) and full clarification for these will cer-
tainly require far more than the present availability.
To obtain estimates for A, linear intercepts from the successive ( bln > seemed therefore in order : the
resulting monotonically decreasing sequences and the sequences from triad fitting (mostly monotonically increasing) provided estimate ranges for each problem.
In order to make preciser estimates for central values,
two more methods were used : a) Neville tables which showed remarkably small drift in several cases
(see table III for an example) and b) a regularity
criterium whereby A was varied in the determined range and Q* - 1 estimated by the ratio method ; the more likely value for A being that which resulted in a smoother sequence for Q* - 1.
Table IV lists estimates for A in the usual lattices.
Table III. - Neville table for the mean value, in
a = b/n, of the exact animal expansions for the trian- gular site problem (ref. [14]).
Table IV. - Mean value location for animals on
various lattices and problems ( from references cited
in the text).
The value there given for the simple quadratic site problem is within 0.2 % of the Monte-Carlo central estimate of Stauffer [21].
3. The data presented and the existing models. - i) p > p,,,. In the high-density region early circulation of the Kunz and Souillard results made for the
widespread acceptance of the Bethe-lattice solution
as valid between a = 0 and a = a,,,, for all lattices.
Two points seem interesting in connection with exact expansions for low and intermediate size.
(a) The asymptotic behaviour of the coefficients gnb is taken to be equal to that of the combina- torial coefficients
n + b
) (Reich and Leath [2],n
Schwartz [22]), and this implies that for fixed n they
are asymptotically normal (Harper [23]), i.e. verify
a central limit theorem. Neglecting the corrections
arising from the difference after Gc’ the leading linear dependence of not only the variance but also of the upper semi-invariants (third and fourth cumulants, for example, see Kendall and Stuart, section 3 [24])
should be known and submitted to numerical tests.
The difficulty in checking the variance evolution even
from Monte-Carlo experiments has been mentioned in the literature (Leath and Reich [25]) and for the
same quantity it has been found that series analyses
do not enable sound predictions of the first-order corrections.
Despite these difficulties, formulation of a central limit theorem gives a reliable qualitative indication
on the behaviour of the distribution function cor-
responding to gnb(1 - p)b. Clearly, straightforward
substitution of the normal law for gn6 is simply
illicit and inevitably fails to give the correct p-depen-
dence of even the mode location (as remarked by
Duarte [ 17]).
(b) However, conclusions about ’the rather more
stringent local convergence condition have to be
guardedly taken. In fact, for two dimensional site
percolation the gn6 are unimodal with a highly regular
compact end, but for the upper site dimensionalities and bond percolation in general, the condition is
simply not verified. A brief survey of published
(and unpublished) sets of data shows that such anomalies are very much lattice dependent and a graph-theoretical analysis is bound to be detailed.
The way in which these irregularities will tie in with a local limit theorem according to the equivalent suggestion of Harper [23], for other doubly-indexed
sequences is therefore a matter for speculation.
So far, two points only can be assumed on the Bethe- lattice curve : a = 0 and a = ac’ The latter stems from the condition for non-analiticity of the moments at
the percolation threshold ; the former implies that
the most compact configurations for fixed site (bond)
content are non-supermultiplicative. Compact is taken
here in the sense of with the minimum site (bond) perimeter. The Kunz and Souillard derivation, for example, established the non-analiticity for p > pc of the analogue of the free energy for percolation solely on the basis of the contributions of a set of compact configurations which are nevertheless non-
extremal (the squares for the simple quadratic pro-
blem). By straightforward inversion of the so-called
j-polynomials in Sykes et al. [26], such extremal solutions are known through order 30 for the trian-
gular site problem. And, by noticing further that the site perimeter and bond perimeter of the space-types contributing to these extremal solutions are exactly corresponding they can be written through order 37, using Sykes et al. [27]. Now, Duarte and Marques [28].
have analysed the structure of the isoperimetric
solutions on this lattice and it is clear from their discussion that since the lattice constant of such solutions is fixed (except for the new value introduced
at each stage of the nesting of extremal solutions
like e.g. that for site content 38), the growth parameter
(Klarner [5]) will effectively be 1, as required. This-
rather convenient correspondence between site and bond perimeter solutions is not carried through to
other lattices, as wrongly assumed in the literature
on cyclomatic numbers (see e.g. Domb [29]) and the analysis of Duarte and Marques cannot be used for the site percolation problem on the other lattices.
Nevertheless the pattern remains very much the
same in spite of the considerable rearrangement of lattice information between both perimeter classifi-
cations.
ii) At p pr, the value for Q - 1 in Duarte [17]
(a - 1 = - 0.619 ± 0.003) differs from the Monte- Carlo results of Leath and Reich [25] (with two determinations of
The very last indirect estimate of Reynolds et al. [30]
is centred outside both estimate ranges, overlapping marginally with them. If a value of around - 0.612
was indeed to be adopted, the triangular site estimates would need a terminal pattern slightly different
from that shown in the presently available data.
However, the direct estimate of Essam et al. [8] is,
of course, sufficiently imprecise to encompass almost all of the proposed ranges and it is perhaps rather
more significant that for bond percolation the agree-
ment with the scaling picture is quite consistent
with in the larger uncertainty limits for J - 1. In the low density region close to Pc’ the detected scaling
correction could be attributable to Leath’s initial
assumption of the continuity of the Bethe-lattice solution beyond ac (Reich and Leath [2]). The proposal,
which is not without theoretical difficulties implies
that the mode location is 1 - P in the low-density
p Y
region : the Monte-Carlo data give, in fact, values somewhat lower, as found by the same authors in
their careful numerical reassessment (Leath and
Reich [25]) and within the equivalent range of concen-
trations exact expansions can be used to determine the correction
to 1 p p
for the limiting mean value.A missing constant term is easily indicated by the
estimates.
iii) p ;:-, 0. Confrontation with the need to force down the Bethe lattice curve (8) after ac gave rise,
in naturalibus, to a number of assumptions in varying degrees of analytical clarification. By chronological
order of publication, they are :
(a) Leath’s, latter clarified in Reich and Leath [2].
Essentially a continuation of the Bethe-lattice curve
up to some value ai where a departure occurs. a; being
not necessarily ac but satisfying the condition ai ao where g-l(À) = ao (À the growth parameter for the
problem, Klarner [5]).
(b) Domb [6], assuming a departure at ac with a
correction term to log (g(a» of the form
and including a modal shift for the animals giving
rise to a confluent singularity of the generating
function for their total number and non-classical
scaling.
(c) Hankey [10], assuming a departure at a,, with
a discontinuity in the derivative of the real lattice
curve such that Ge is the mode for animals.
(d) Stauffer [3], assuming, like Domb, a correction
term C(a) but with an exponent directly responsible
for scaling :
(e) Reich and Leath [2] on Domb [6]. By scaling requirements, the former authors correct Domb’s C(a) to :
and find it impossible to distinguish between (a) and (e) within the available concentration range.
Since results of a significance equivalent to the
Kunz and Souillard theorem [9] are probably out
of the question for the low density side of the animal distribution, the most realistic point to investigate
seems to be which of the present assumptions comes
closer to the true distribution. Leath’s original assumption, for example, should perhaps not be
stretched beyond the author’s intentions, while by comparison with the mode Monte-Carlo estimates for 0 > p > p,,, Stauffer found acceptable agreement with his own version for the simple quadratic site
(2) But see Hoshen et al. [31] Monte-Carlo test, closer to - 0.62.
