The branching of real lattice trees as dilute polymers

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The branching of real lattice trees as dilute polymers

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

To cite this version:

J.A.M.S. Duarte, H.J. Ruskin. The branching of real lattice trees as dilute polymers. Journal de

Physique, 1981, 42 (12), pp.1585-1590. �10.1051/jphys:0198100420120158500�. �jpa-00209355�

1585

LE JOURNAL ^DE PHYSIQUE

The branching ^{of real lattice trees as dilute} polymers

J. A. M. S. Duarte

Solid State Group-Physics Department, Imperial College, Prince Consort Road, ^{London SW} 7, U.K.

and H. J. Ruskin

Metabolic Unit, ^Hunts House, Guy’s Hospital, ^Medical School, ^London SE,1, ^U.K.

(Reçu ^le 12 février 1981, accepté ^{le 31 août} 1981)

Résumé.

On montre dans cet article que le branchement ^en arbres sur un réseau réel est lié à la présence ^des exposants ^{de deux} préfacteurs différents.

Pour tous les réseaux où les arbres trivalents sont inscrits, ^cet exposant ^est estimé être de même nature que dans le cas des animaux. En outre, ^{on donne des} estimations du paramètre ^de croissance pour des réseaux à deux et trois dimensions.

Abstract.

In this paper, the branching of real lattice trees is shown to be related to the occurrence of two different

prefactor exponents. For all lattices where trivalent trees are embeddable, ^this exponent is estimated as being

animal-like in nature.

In addition, estimates for the growth parameter ^are given ^{for a} number of 2- and 3-dimensional lattices.

J. Physique ⁴² (1981) ^{1585-1590 DÉCEMBRE} 1981,

Classification Physics ^Abstracts

05.50

1. Introduction and _summary.

Studies of real lattice configurations ^{with no} loops ^have repeatedly appeared in the literature in connection with detailed field theoretical calculations of branched dilute poly-

mers (Lubensky [1], Lubensky and Isaacson [2]), ^and

also in a percolation ^context (Stephen ^{[3], Wu [4],} Straley [5]).

In the latter case, the fundamental physical question

of whether or not the configurational entropy ^and distribution of such clusters is sufficient to ensure

percolation (’tree percolation’) ^{has led to} completely contradictory conclusions thus far. A closely ^related topic is the calculation of the mean cyclomatic ^number

for normal percolation. Trees make no contribution to this quantity, which has been put ^{forward as a measure} of the degree of ramification of a cluster in a set of numerical studies (Domb [6], Cherry ^{and Domb} [7])

and it is a conclusion of such studies that the mean

cyclomatic ^{number in the} vicinity of the critical

probability p,,, is o very small ».

While the specihc weighting of cluster and boun-

dary ^sites (or bonds) required ⁱⁿ percolation ^{is not} directly ^{relevant to the} statistics of polymers ^{on a} lattice, it will be shown in 2 that the series expansions

for the mean cyclomatic ^number help ^to extend infor- mation on the total number of trees for both site and

bond configurations (i.e. strong ^{and weak} embeddings

for a given ^{size of} cluster).

Of interest also, is the correlation between site (and bond) valence and the existence of closed loops ^{in a} configuration.

The valence of a site (bond) ^{is the} number of site

(bond) neighbours belonging ^{to the same cluster} (Duarte and Ruskin [8]) ^{and the} theory ^{of free tree-}

like graphs with restricted valence has been the subject

of considerable combinatorial research (see e.g. Klar-

ner and Rivest [9], ^Klarner [10]). Further, ^Klarner

(in [10]) provides ^an expression for the number of

planted ^{bond trees on} 2-dimensional regular ^lattices.

Against ^this background of closely ^related problems,

we undertake in this paper ^a wide ranging ^investi- gation ^{on the} configurational entropy of ^{bond and} site lattice trees and report succintly (section 2) ^{on the} graph theoretical manipulations that lead to the

corresponding ^series expansions.

In 3 the standards methods of extrapolation ^are applied ^to the series expansions ^thus obtained, ^the

main conclusions listed and values for the tree growth parameter ^{and the} prefactor exponent estimated in 2-, 3-dimensions. The extension of the work to higher

dimensions on the simple hypercubic system ^{shall be}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420120158500

1586

considered in a companion paper (Ruskin ^and Duarte,

to be published).

2. Site and bond valence and the enumeration of lattice trees.

If lattice animals are taken as reali- zations of dilute polymers with each dimer embedded

on a lattice bond, sites with valence « v » will represent the embeddings ^of possible space-types corresponding

to ^« v-functional units » (Lubensky and Isaacson [2]).

Classification of sites by ^{their « valence »} (« degree »

or « valency » ⁱⁿ Cherry [11] ^and Cherry ^{and Domb} [7]) ^{in this way, leads to the} derivation of linkage ^rules

which relate the number of loops ^{and sites in a confi-} guration ^to such v-discriminations (in ^this instance, these refer equally ^{to both} strong and weak embed-

dings).

Trees.

i.e. polymers ^{with no closed} loops

can then be singled ^{out from an} exhaustive v-discri- mination simply by specifying ^the corresponding linkage ^rules

where sv is the number of v-functional units, s ^{and n} the number of sites and bonds in the polymer respec-

tively and where the summations run from 1 to z

(lattice coordination number). Equation (2.1c) is,

of course, the Euler equation ^{for one tree.}

For those configurations verifying (2.1c) ^{there are} topological limitations on the bond degree (measured by ^the pair of site valences at its ends, Cherry [11]) =

on the triangular lattice, ^for example, ^a (6,6)-bond

cannot belong ^{to a} tree-embedding in either the strong

or weak senses. On the _square lattice maximum degree

bonds can be found in bond trees only.

(2 .1 c) shows that ^tree identification requires ^no

more than a doubly-indexed site bond discrimination.

However, ^a consideration of both site and bound valences can be useful in the prediction ^of the asymp- totic behaviour of the total number of site trees on a

certain number of lattices, ^{as we shall} proceed ^to

show :

Let g (sv, nw, c) denote the weak embedding (1) ^of

a given space-type with sv ^v-sites, nw w-bonds (here,

a (i, j)-bond ^is considered as having ^a total valence i + j - 2), ^and cyclomatic ^{number c and let}

g (sv, nw,, c’) denote the strong embedding ^{on the} covering lattice of the corresponding space-type ⁱⁿ

the usual bond-to-site transformation, ^with sÛ,, ^v’- sites, nw, w’-bonds and cyclomatic number c’ :

(1) ^{For a detailed account of} graph-theoretical definitions _see, for example [18].

( = stands for bond-to-site transformation). Clearly,

the following linkage ^{rules are valid on the} original

lattice

and on its covering ^lattice

with the passage relations

From (2. 3), (2. 4), (2. ) ^an expression for the difference in cyclomatic ^number between the original ^{and trans-}

formed embeddings ^can be obtained (in ^{terms of the} original lattice) :

Equation (2.6) ^{shows that the} cyclomatic ^number only ^{remains constant in a} bond-to-site transfor- mation for those configurations ^which verify

i.e. for all polygons ^{and self} avoiding walks, ^thus

SAW’s are the only ^{kind of trees} which retain their

loop ^{structure on} transformation in this _way. Further,

since the transform of a given ^{SAW is} always ^a neigh- bour-avoiding ^walk (Watson [12]) the total number of site trees for the Kagomé, square covering ^and Kagomé matching lattices is just the total number of SAW embeddings (per bond) ^{on the} honeycomb,

square and triangular lattices, respectively. ^These

have been well studied and the corresponding ^gene-

Fig. ^1.

^{Site trees on :} a) ^the honeycomb lattice ; b) ^the kagomé lattice ; c) the archimedean (3, 3, 3, 4, 4) ^lattice.

rating ^{functiôns are known} through reasonably high

order (Watts [13] ^and references therein).

To determine the site trees for other lattices which

are not covering lattices of _any bond problem, ^the existing ^valence discriminations can be used (Cherry

and Domb [7], ^Duarte and Ruskin [8]). ^{For such}

lattices, the site valence _may always ^violate (2. 7), ^but

the extent to which it does so depends ^{on the} type ^of embedding ^{and the} connectivity of the lattice. For

triangular site trees, sU - 0 for all v > 4 and the same occurs for the archimedean (3, 3, 4, 3, 4) ^and (3, 3, 3, 4, 4)

lattices (Fig. lc), while maximum valence site trees can easily ^{be drawn on the} honeycomb and square lattices (Figs. la ^and 2a). ^On close-packing ^the latter,

trees can still be tetravalent on the square-matching lattice, ^{but then} only next-nearest-neighbour ^con-

nections can be involved (Fig. 2b), ^and inevitably,

when altemating tetrahedra ^are suppressed ^{to obtain}

the square covering lattice such tetravalent trees

collapse, ^{as was} shown before (Fig. 2c). ^Parallel

arguments ^are valid for the higher dimensions, ^where

site trees on loose-packed ^{lattices can} possess maxi-

mum valence sites.

For the determination the total number of site trees up ^{to a} reasonable order on those lattices for which valence discriminations were available (tri- angular, simple quadratic, simple cubic, body-centered

cubic and 4-D hypercubic lattices) equation (2.1c)

acted as a check on simple site-bond discriminations.

Additional information could also be found in bond

percolation distributions (see ^for example ^Blease,

Essam and Place [14] ^{for the} triangular ^bond problem).

The technique ^for deriving the tree-like embeddings

uses as a starting point ^the complete ^{set of} strong embeddings ^{on the same} lattice. On these, ^{the bonds} surrounding the cluster sites can be divided in 2 types

Fig. ^2.

^{Site trees on :} a) ^the simple quadratic lattice ; b) ^the

square matching lattice ; c) the square covering ^lattice.

internal bonds (belonging ^{to the} cluster), given by n

in equation (2. la) ^{and extemal} (or perimeter bonds)

so that in addition to (2.1a) ^and (2.1b) ^another

relation is required

The system ^of equations (2.1) ^and (2.8) only ^holds

for strong embeddings ^{but all weak} embeddings ^{with s}

sites can be obtained by deleting ^{internal bonds} (and adding ^{them to the bond} perimeter). By ^{this method} (called ^the yield ^factor technique, Blease et al. [14])

the process terminates with the generation ^{of all} weakly

embedded s-site trees derivable from a given ^section- graph. ^{Hence the n-bond tree} percolation polynomial

on a given lattice has length ^{c +} 1, where c is the maximum cyclomatic number of n + 1-site clusters

on the same lattice. Further, ^{the total number of} strongly embedded n + 1-site trees will _appear as

the maximum perimeter ^{lattice constant in those} polynomials. Using ^the triangular ^bond problem ^{as an} example, ^the data of Blease ^{et al.} [14] ^almost complete

the bond trees on the triangular ^lattice through ^{order 7}

and for completion ^of the total number of 8-bond

trees the mean cyclomatic ^{number can be used.}

The series expansion ^{for the} quantity ^{follows from}

and

1588

From (2 , 9b) the site valence must clearly ^be > 0

so that isolated sites are not considered. g stn is the lattice constant of clusters with n bonds, s sites, ^t perimeter ^bonds (per site). K( p) ^{is the mean number}

of clusters.

Equation (2. 9b) ^{acts as a sum rule for the} cyclo-

matic number discriminations and in conjunction

with the sum rule for the primary species in perco-

lation, ^it determines the number of 1-loop clusters of

perimeter 32, ^the only ^{non-tree like} configuration required for the 8-bond polynomial ^not given by

Blease et al. [14]. ^{The total number of trees can be}

found by ^direct subtraction of the loop configurations

from the total number of 8-bond clusters.

3. Asymptotic analyses.

^To determine the nature of the singularity ^{of the tree} generating ^{function we}

assumed algebraic singularities

Standard ratio and Padé techniques ^{were used. The} growth parameter ^was determined from the _sequences

and linear intercepts ^on them, ^{of the} type

For the prefactor exponent ^unbiassed sequences ^are of the form

and a value conjectured ^from (2.14) ^{is used to deter-}

mine biassed sequences four À

Tables 1 and II give the values for Tn (per site) ^{in 2}

and 3-dimensions, respectively.

Table Il.

Total number of ^{n-size trees} (per site)

in 3-dimensions.

Table 1.

Total number of ^{n-size trees} ( per site) ⁱⁿ 2-dimensions.

Table III.

Sequences of ^estimates for ^the prefactor

exponent according ^{to equation} (2.14) ^on 2-dimensional

regular ^lattices.

i) 2-dimensional regular ^lattices.

^{Table III} gives

the estimated sequences for 0 according ^to (2.14).

These _sequences are highly irregular, especially ^when compared ^{with the} corresponding sequences for the total number of animals on the same lattices. There is, however, ^a final section of regular estimates, reflecting

the settled behaviour of the linear intercepts (2.13).

The final value ouf0, ^{from linear} extrapolation ^{of those}

last few values in (2.14) ^is compatible ^{with 0 -} 1.0,

the animal-like estimate for the prefactor in 2-dimen-

sions, ^{and this} result finds support ^{in d} log ^Padé approximants ^{to the} derivative of T(x) ^{from the} early approximants.

ii) 2-dimensional semi-regular ^lattices.

^The Kagomé lattice site trees are, of course, SAW-like in nature and 0

1/3. ^{For the} archimedean

(3, 3, 3, 4, 4) ^and (3, 3, 4, 3, 4) lattices, ^the present extension of the expansion ^{has not} reached the smooth behaviour zone yet. ^The pattern ^{of d} log ^Padé singu-

larities of the derivative of T (x) is nevertheless _very similar to those found for simple quadratic ^and honeycomb ^{site trees, for} example : ^{there is an uneven} dispersion ^of pole-residue points ^{around 0}

^{+ 1.0,}

with the various pole ^{locations close to,} but compa- tible with, ^{the biassed} sequence values following (2.14).

iii) 2-dimensional high coordination lattices.

Con- vergence improves ^{for these lattices,} and the whole sequence of estimates for 0 can be used for linear

extrapolation (cf. ^Table IV). ^{There is once} again ^a

substantial indication that the limiting ^{value of 0}

could be the ^{same as} for lattice animals, ^{i.e. that the} singularity ^of T(x) ^is logarithmic. ^In fact, ^{for the} square

matching ^{site trees} the behaviour of the On ^{is more} regular ^{than the} sequence for the lattice animals

(published by Peters et al. [16] ^{to s} 11), ^{but the} comparison, ^{is in} général, ^however, unfavourable

to trees.

Table IV.

Sequences of ^estimates for ^the prefactor

exponent according ^to equation (2.14) ^on 2-dimensional lattices.

Overall estimate

1.00 + 0.05.

iv) 3-dimensional lattices.

With the sore exception

of the b.c.c. site trees, the prefactor ^{for trees is} clearly

animal-like in nature (cf. ^Table V).

Table V.

Sequences of ^estimates for ^the prefactor

exponent according ^to equation (2.14) ^on 3-dimensional lattices.

Overall estimate for 0 in 3-dimensions

1.55 + ^0.1.

On the simple ^cubic lattice, ^{where a} comparison

between site and bond trees is possible, ^a steady

difference between the On sequences is apparent, ^which closely parallels ^{that found for lattice} animals, ^and

seems to comply ^{with a} general pattern ^{in the} hyper-

cubic system. Noticeable discrepancies ^{of this} type

were also reported ⁱⁿ five-parameter ^{fits to the total}

number of animals (Guttman ^{and Gaunt} [15]).

Owing ^{to these} problems ^the final estimate range has been widened so as to encompass the individual estimates on table V.

The diamond lattice sequences show ^an incipient

evolution towards this final value, while the b.c.c.

results are most irregularly behaved, despite ^{the fact}

that sites of all possible ^{valences can be found in site}

trees.

For the growth parameter  ^{we have} considered the

sequences (2.13) ^and (2.15) ^{in all cases. For the}

triangular ^{site trees the} growth parameter is smaller

than the published estimate for site animals Gaunt

1590

Table VI.

Estimates for ^{the tree} growth parameter ^À

on 2- and 3-dimensional lattices.

et al. [13] with v 3, ^as could be expected. ^The comparatively larger uncertainties for the archime- dean (3, 3, 3, 4, 4) ^and (3, 3, 4, 3, 4) and the b.c.c.

lattices reflect the difficulties found in the extrapola-

tion analyses ^as mentioned in 3ii) ^and 3iv).

4. Conclusions.

The numerical evidence in 2 and 3 dimensions substantiates the assumption that,

for all lattices where trivalent trees are embeddable,

the total number of lattice trees leads to an animal-like

prefactor exponent in the usual fitting ^formula.

By ^{contrast all site trees on} covering lattices of bond lattice problems ^{have a} différent, ^SAW-like prefactor exponent.

For all lattices in the first category, the ratio of tree to animal growth parameters, ^while consistently large, is also found to increase systematically, ^{as one}

goes from site to bond enumerations on a given

lattice.

Acknowledgments.

^{We are indebted to N. Rivier}

and C. Pearce for discussions and correspondence.

The hospitality ^{of the Solid State} Group ^{at I.C. is} gratefully acknowledged by ^{one of us} (J.A.M.S.D.).

A substantial endowment towards computing ^costs

was given by the Natàlia Costa Matos Overseas Fund.

References

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