Rooted Spiral Site Trees on Triangular Lattice

HAL Id: jpa-00247160

https://hal.archives-ouvertes.fr/jpa-00247160

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Rooted Spiral Site Trees on Triangular Lattice

Sitangshu Santra

To cite this version:

Sitangshu Santra. Rooted Spiral Site Trees on Triangular Lattice. Journal de Physique I, EDP

Sciences, 1995, 5 (12), pp.1573-1576. �10.1051/jp1:1995218�. �jpa-00247160�

J.

Phys.

I France 5

(1995)

1573-1576 DECEMBER1995, PAGE 1573

Classification

Physics

Abstracts

05.20-y

05.40+j

64.60Cn

Rooted Spiral Site Trees

_on

Triangular Lattice

Sitangshu

Bikas Santra

Laboratoire de

Physique

et

Mécanique

des Milieux

Hétérogènes,

Ecole

Supérieure

de

Physique

et Chimie

Industrielles,

10

rue

Vauquebn,

75231 Paris Cedex 05, France

(Received

18

August

1995,

accepted

in final form 21 August

1995)

Abstract. ~Ve bave studied rooted

spinal

site trees embedded _{on a}

triangular

lattice in two dimensions

by

exact enumeration. We found that trie

scaling

function form for trie number of

trees and trie radius of gyration exponent v are different from trie _square lattice results.

Lattice animals have been much studied in literature _as lattice mortels of braiiched

polymers

in dilute

solution,

and

they

also descnbe the statistics of

large percolation

dusters below

percolation

threshold

il,2]. Spiral

lattice site animals has been defined

by

Li and Zhou 1985

[3],

Base and

Ray

1987

[4],

^Base et ai. 1988a

[Si,

^1988b _[fil ^{and Santra and Base} 1989 1?i and

belong

to a new

universality

Mass different from those of undirected and directed lattice animals. In

a

spiral

^animal the constraint is such that each _site of the duster _is attached to the

origin through

at least _one

spinal path.

In a

spiral path,

connection is either in the forward direction

or in _a

specific

rotational

direction,

_say dockwise. For rooted

spiral

trees

Ii.e.

animals without any

loop

with fixed

origin)

embedded _on the _square lattice it has been found that _a dimensional reduction

by

four _occurs in the

problem

_[Si. Another

interesting

result that has been obtained

is that

spiral

lattice animals and

spiral

lattice trees

belong

to different

universality classes,

_i e.

loops

have _a non-trivial effect _on

spiral

lattice animal statistics [fil. Santra and Bose 1989 [7]

have

conjectured

a relation 9~ ₌ 90 oc where 9~ is the animal number

exponent

^{for animals}

~vith _c

loops,

₉₀ is the tree number exponent for _trees and _{a m} 1.62. This result is different from the relation 9~

= 90 c which holds for both undirected and directed animals. In this _{paper we}

study

the

scaling properties

of rooted

spiral

lattice site trees embedded _{on a}

triangular

lattice

in two dimensions. Dur _atm is to obtain the

scahng

function form and the numencal values of the

exponents

for the number of trees and their radius of

gyration

and to compare them with the _square lattice results.

To enumerate the tree number

exponent

we have used _a

scaling

relation similar to the

spiral

self-avoiding

walk

[8,9]

since the

spiral

site trees on the

triangular

lattice should not have _any

branching

except at the ongin.

Figure

1 shows a

spinal

lattice site tree on the

triaiigular

lattice connected

by

_a solid line. If _one considers the site

represented by

_{an open} circle in

Figure

1 connected

by

a dashed line to the

previous

site then it will form _a

loop

which

implies

that there _can not be any

branching

after the _ongin. The relation _states that in the

asymptotic

©

Les Editions de

Physique

1995

1574 JOIJRNAL DE

PHYSIQUE

I N°12

P

Fig.

I. A rooted

spiral

lattice site tree _{on a}

triangular

lattice. The root is denoted

by

_{a cross.} Trie site

represented by (o)

is _a forbidden site.

n - cc

limit,

where _n is the number of _sites in the tree, ^tue total number of trees an goes as

~ ç~

~(n~)~-~ j~j

"

where 9 is the tree number

exportent.

^The radius of

gyration Rn

scales with the size _n asymp-

totically

_as

Rn

_c~ n~

(2)

where _v is the radius of

gyration

exponent.

Rn

is defined _as

Rn

=

ll~ rÎ/nl~/~l

₁₃₁

where _{r~ is} the distance of _a duster site from the centre of _mass of the duster and

(.

denotes the average over all _trees.

In order _to calculate the

exponents

^{for the rooted}

spiral

lattice site _trees embedded _{on a}

triangular lattice,

exact enumeration of duster

properties

has been

performed

_up to _n

= 25.

All the enumeration data

along

with the exact data for radius of

gyration

_are listed in Table 1.

To have _an estimate of the

exponent

^à in

equation (1)

_we have calculated _a

quantity (assuming equation (1)

to be

valid)

Xn

=

'Ogl~ ^~~

^{m An~~ +}

^'Ogli

~

141

n-2 n+2

~

where A

=

46(1- à) log(À)

and

~T = 2 à. In the

asymptotic

_{n - cc}

limit, log(Xn)

_{m -~i}

log(n)

and _we have estimated

~T from the

following

relation

~

~ÎÎÎÎÎ ÎÎOgÎÎ~Î~

^~~~

The diflerence between

log(Xn-2) (instead

of

log(Xn-i ))

^and

log(Xn)

has been taken to min-

imize the odd-even oscillation in the values of

~T. In

Figure

2 we have

plotted

~T versus

1In

to

show that

~T converges in the

asymptotic

^limit. Tue average of last four values gives _~T m

1.31,

so à _m 0.69. Thus tue

scaling

relation for _the

spiral

site trees embedded _on

triangular

lat- tice is diflerent from that of the _square lattice. In the square

lattice,

the

scahng

relation _is an c~

À"n~~

in the

asymptotic

_{n - cc}

limit,

the

exponent

^à = 1. It _can be _seen from [Si that

N°12 ROOTED SPIRAL SITE TREES ON TRIANGULAR LATTICE 1575

Table I. Exact data

for

rooted

spirai

iattice site trees on a

trianguiar

iattice.

n an

Rn

_{n an}

Rn

1 0.0000000 14 189036 2.6565927

2 6 0.5000000 15 366612 2.7875866

3 21 0.7758448 16 699742 2.9153740

4 62 1.1014329 17 1317162 3.0398781

5 168 1.2248601 18 2446422 3.1615829

6 420 1.4221328 19 4489524 3.2805438

7 1012 1.6011560 20 8144682 3.3970î91

8 2316 1.7732979 21 14620800 3.5112668

9 5154 1.9342901 22 25983884 3.6233259

10 11084 2.0900256 23 45748941 3.7333466

11 23283 2.2386266 24 79834998 3.8414857

12 47718 2.3826322 25 138158404 3.9478314

13 95906 2.5214937

4° ~~ ^{----~- -~}

35

Gy~~~ ~Ît/~)~Î

)

f

25

~~

_{oo 0.05 0.~°}

~~

In

Fig.

2. A

plot

of tlm exportent _{~y =} 2 6 _versus

1In.

they

have found

satisfactory

convergence for and 9

considering

à ₌ in the

scaling

relation

il)

for the square lattice. No convergence has been found for and 9

considering

^à ₌ 1 in

same relation for the

triangular

lattice _up to n = 25. The value of à is also diflerent from o-à

as in the _case of

spiral self-avoiding

walk.

The radius of

gyration

exponent v has been calculated from the local

slope

of the

log-log plot

of

equation (2)

and is denoted

by

_un. The

slope

_{un is given}

by

ÎogjRnj ÎogjRn-2j

~" "

Îogjnj Îogjn 2) i~)

In

Figure

3 _we have

plotted

_un

against 1In

for both the _square and

triangular

lattices. Ex-

trapolation

of last few data points to n _- cc has been

performed

for the

triangular

lattice

by employing

^standard numerical

procedures [loi.

The

intercept

on the _{y axis gives an} accurate

estimate of the

exportent

v. The hnear

extrapolants vj

= nvn

in -1)vn-i

and their _averages

v(

=

(vj

₊

vj-1)/2

should

approach

v as ~ - oe. The _crosses in

Figure

3 indicate the values of _v for square and

triangular

lattices. For square lattice the value of _v has beeii taken from [Si.

For

triangular

lattice _v

= 0.618 + 0.002 whereas for _square lattice the value _{is v =} 0.653 + 0.01.

It _seems that the exponent v for the

triangular

lattice _is diflerent from that of the square

lattice.

1576 JOURNAL DE

PHYSIQUE

I N°12

o-go

o.75

~~

>° o-m

/~

,, 0.65

0.60

0.00 0.05 0.10 0.15

1/n

Fig.

3. Plot of the radius of

gyration

exportent _{un versus}

1/n

for _square

(D)

and

triangular (.)

lattices. Trie _crosses denote trie bnear extrapolants.

To _sum up, ^exact enumeration for the numbers and the radius of

gyration

for rooted

spiral

lattice site trees embedded _{on a}

triangular

lattice has been

performed

_up to _{n =} 25 where _n

is the size of the tree. No convergence has been found for the ratio _an

tan-i

_versus

1In plot considering

usual lattice animal

scaling

form _{an c~}

À"n~~

_{in the}

asymptotic

n _- cc limit.

Instead,

an

exponent

^à m 0.69 defined in

equation iii

has been calculated. For lattice animal the value _is à

= whereas in the _case of

spiral self-avoiding

walk it is à

= o-à- Also the radius of

gyration exponent

v = 0.618 + 0.002 has been calculated and is diflerent from the square lattice value. So it can be conduded

that,

like

spiral self-avoiding walks,

rooted

spiral

lattice site trees exhibit _a breakdo~vn of

universality

^if we compare square and

triangular

lattices.

Acknowledgments

The author thanks Dietrich

Staufler, Stéphane Roux,

Hans Herrmann and Indrani Bose for

helpful

discussions.

References

Ill Lubensky

T.C. and Isaacson

J., Phys.

Reu. A 20

(1979)

2130.

[2] Stauffer D. and

Aharony A.,

Introduction to Percolation

Theory,

2nd

ed,

2nd print

(Tay.for

and Francis, London

1994).

[3] Li T.C. and Zhou Z-C-, ^J.

Phys.

A 18

(1985)

67.

[4] ^{Bose I. and}

Ray

P., Phys. Rev. B 35

(1987)

2071.

[Si ^Bose I., Ray P. and Dhar D., J.

Phys.

A 21

(1988)

L219.

[6] Bose

I., Ray

P. and

Mukhopadhyay S.,

J.

Phys.

A 21

(1988)

L979.

[7] Santra S.B. and Bose I., J.

Phys.

A 22

(1989)

5043.

[8] Blôte H-J-W- and Hilhorst H-J-, J. Phys. A 17

(1984)

Llll.

[9j ^Lin

K-Y-,

J.

Phys.

A 18

(1985)

L145.

[10] ^{Gaunt D. S. and} Guttmann A.J., Phase Transition and Critical

Phenomena,

Vol. 3, C. Domb

and M.S. Green Eds. (Ne~v York _:

Academic, 1974)

_p. 97.