The clustering problem : some Monte Carlo results

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The clustering problem : some Monte Carlo results

D.H. Fremlin

To cite this version:

D.H. Fremlin. The clustering problem : some Monte Carlo results. Journal de Physique, 1976, 37

(7-8), pp.813-817. �10.1051/jphys:01976003707-8081300�. �jpa-00208477�

(Reçu

^le

5 janvier 1976,

^{révisé le 1 er mars}

1976, accepté

^{le 2 mars}

1976)

Résumé. ²⁰¹⁴ Je

présente

^une série de résultats statistiques ^{sur la} distribution de la taille des amas

formés par les ^atomes répartis aléatoirement dans un milieu continu de deux ou trois dimensions.

Si la portée des interactions est r0, les densités critiques

(auxquelles apparaissent

^{les amas}

infinis)

^sont probablement ^de 2,7 ± 0,1

atomes/sphère

de rayon r0 (en ³ dimensions) ^et 4,4 ^± 0,2

atomes/disque

de rayon r0 (en ² dimensions).

Abstract. ²⁰¹⁴ I present ^a series of statistical results on the distribution of the size (i.e. the ^{number of} atoms) of clusters formed by ^{atoms scattered}

randomly

^{in a} continuous medium of two or three dimensions. If the range ^at which they interact is _r0, the critical densities (at which infinite clusters

appear)

^are

probably

^{of 2.7 ± 0.1}

atoms /sphere

^{of radius} r0 (in ³ dimensions) ^{and 4.4 ± 0.2} atoms/disc

of

radius r0

(in ² dimensions).

1. Introduction. ^- In

[8]

^and

[9]

^the

following problem

^was introduced.

Suppose

^{that atoms are}

randomly

distributed in 2- or 3-dimensional _space

according

^{to a Poisson}

point

process.

Suppose

^that

atoms are linked

by

^some interaction when

they

^are

within a

distance ro

of each other. What is the _pro-

bability

that there will be an infinite cluster of linked atoms ? The answer

plainly depends

^{on the}

density

^W

of the atomic cloud. It is known

(see ^[10]

^{for a}

general

review of this and allied

problems)

^that

^there

^{is a}

critical

density We

^such

that,

^{for W}

Wc,

^the

proba- bility

^that any infinite cluster exists is _zero, while for W >

Wc,

^the

probability

^{that a}

given

^atom

belongs

^to

an infinite cluster is non-zero, and the

probability

that some infinite cluster exists is 1.

The

problem

^{that I wish to address here} is that of

estimating WC.

^The

proof

^that

We

^exists

yields

^weak

upper and lower bounds for it

(see [10], § 7),

^{but these}

bounds are

widely separated,

and the theoretical

problems

associated with

narrowing

^them

usefully

^are

daunting.

^{In this} paper I offer statistical

evidence,

based on computer simulation of random

clusters,

which I believe enables us to locate the critical values with a certain

degree

of confidence.

I should like to

acknowledge

my

gratitude

^to

Pr. M.

Gordon,

^for

bringing

^this

problem

^to my

notice;

^{to Dr. D. D.}

Freund,

^for

helpful discussions ;

and to Mr. C.

Bowman,

^for

making special

^efforts

to

provide

^the

computing

facilities I needed. I am also

grateful

^to the referees for their

helpful

^comments.

2. Method. ^- I

employed

^{a PDP-10} computer

(at

^the

University

^of

Essex)

^to generate ^{and count}

clusters, using

^a

pseudo-random-number

generator.

The method was

essentially

^{that of}

[3],

^{with some}

refinements in the

programming.

^{For each}

cluster,

^an

atom was fixed at the

origin,

^{and the}

sphere

^{of radius} ro around it

searched,

^{i. e.}

points

^were

generated

^{in this}

sphere

^{in such a} way ^{as to} simulate a Poisson

point

process,

using

^the distribution functions described in

[9],

part

4,

^and

[8],

part ^{3. The atoms}

found

^{at this}

stage ^were listed and the

ro-sphere

around each was

searched

again.

^{In order to eliminate}

double-counting,

each atom found must be checked to determine whether it lies in a volume

already ^searched;

^i.e. whether it is within a distance _ro of some atom whose

neighbour-

hood has

already

^been

investigated.

^{To reduce the time}

required

for these checks

(which

appear ^at first

sight

to

require 0(n2) operations,

^{where n is the} size of the

cluster),

^{I used} scatter-storage

(hash-coding)

^{to list the}

coordinates of the ^atoms found.

(The

^{method is}

described under the name open type

addressing

system in

[7].)

In my programs the coordinates of each atom were stored in an address

keyed by

^the

integer

parts of the coordinates

(modulo ro) ;

^thus

only

^{9 or}

27 areas of the store had to be referred to while

checking

^{each new} atom, and the

computing

^time

required

^to generate ^a cluster of n elements became

0(n).

^The

later,

^more

refined,

versions of the _program

ran in about 140 K of store,

handling

^clusters up ^to 25 K in

size,

and took about 20 ms per ^atom.

Following [3]

^and

[6],

^{I also in most of the work}

counted the

generation

^{sizes. Here the O’th}

generation

consists of the

starting

atom; ^{the first}

generation,

those atoms within

ro of

^the

starting

atom; ^{the second}

generation,

^{those atoms within} ro of an atom in the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003707-8081300

814

FIG. 1 a. ^- Histograms of cluster sizes found for 3-dimensional clusters, for values of W between 2.60 and 2.85. The bars at the bottom correspond ^to clusters that reached the computational limit of 25 001 atoms. A modified logarithmic scale is used for the vertical axis. The numbers of clusters on which the histograms ^are

based are as follows :

FIG. lb. ^- Histograms ^{of cluster sizes found for} 2-dimensional clusters, for values of W between 4.0 and 4.6. The numbers of clusters

on which the histograms ^{are based are as follows :}

first

generation (other

than those in the O’th and first

generations);

^{and so on.} The information

yielded

^is

summarized in

figure

^2.

Two caveats must be made here. As with _any

computer-generated numbers,

the coordinates of the atoms must be distributed not

continuously

^{but on a}

lattice. In my programs, the

grain

of the lattice was

2-11 _ro

(in

^three

dimensions;

^2-15 ro in ^two dimen-

sions).

^{This seems to}

correspond

^{to an error in}

ro of

one or two

parts

ⁱⁿ

211,

^{or an error in W of three to six}

parts ⁱⁿ

211;

^{I am} confident that the real error from this cause is less.

Secondly,

^{there is}

always

^{room for}

doubt

concerning

random-number

generators;

my

own is a home-made version which has been checked

(in

^a

simple-minded fashion)

^and seems, in

particular,

to have no

periodicity

^{of order 2 or 3}

(see [1]).

3.

Theory.

^{- In the}

interpretation

^{of the results}

it will be

helpful

^{to know the}

following

^{facts. Let}

Gk

⁼

Gk( W)

^{be the mean} size of the k’th

generation

of a cluster

generated

^with

density parameter

^W.

FIG. 2a. ^- Growth curves for mean generation ^{sizes in 3-dimen-} sional clusters, for values of W between 2.60 and 2.80. A square-root scale is used for the vertical axis. Each point plotted ^{is based on at}

least 20 clusters.

FIG. 2b. ^- Growth curves for mean generation ^{sizes in 2-dimen-} sional clusters, for values of W between 4.0 and 4.6. Each point

plotted ^{is based on at least 20 clusters.}

Proof.

^{- For a}

rigorous proof

^{I think the}

theory

^of

disintegration

^{of measures} is needed.

However,

^I believe that the

following

argument ^is

essentially

sound.

Imagine

that the clusters are

generated

^{as in}

the computer program,

generation by generation,

so

that,

^{at the n’th} _stage, ^{we have a}

region V

^{that has}

already

^been

searched,

^{and a} finite number of atoms of the n’th

generation,

^at

points

ai, ^..., are For the rest of the

growth

^{of the}

cluster,

^{it is}

only

^the

region V (from

^{which atoms are} henceforth

excluded)

^{and the}

points

a1, ^..., ar that are

significant ;

the distributions of the atoms of earlier

generations

withiri v is imma- terial. Write

for the

expected

total number of atoms in the _genera- tions n +

1,

..., n + m,

given

^{that the atoms of the}

For the effect of

excluding

^the

region V

^{from our}

further

searching

^{must be to} reduce the number of descendants.

And E ^Gk

^is

^by

definition

Q(Ø ; 0),

I -k-m

which

by

^the

homogeneity

^{of the}

underlying

^Poisson

process is

equal

^to

6(0 ? a)

^for any ^a.

Thus we see that

So, given

that the n’th

generation

^{has r} atoms, ^the

expectation

^of the number of atoms in the

generations

n +

1,

..., n + ^m is not greater

than r E ^{Gk- (The}

I -k-m

argument ^above

implicitly

^{assumes that}

r > 0 ;

^but

obviously

^{the last} statement at least is true also for

r ⁼

0.) Summing

^{over all the}

possible

^{numbers of}

atoms in the n’th

generation,

where

p(n, r)

^{is the}

probability

that the n’th

generation

contains ^exactly

^{r atoms.}

Corollary.

^-

If,

^{for some} n,

Gn(W) 1,

^{then the}

expected

total size of the

cluster,

is

finite,

^{and W}

Wc’

00

Proof.

^-

Clearly y ^{Gk( W)}

^{is the}

^expected

^total

k=0

size of the cluster. If

Gn(W) 1,

then for any ^m > 0

we have

so that

and now W W’ ,

Wr.

Remark. ^- It is clear from this

corollary

^that

G(W)

is finite if there is some n for which

Gn(W) 1;

^{in this}

case, of _course, the

Gk’s

^decrease

exponentially

^after

a certain

point (see

^the

graphs

ⁱⁿ

Fig. 2).

It appears to be unknown whether

G(W)

^is finite for _every

W

Wc.

4. Results. ^-

Following [3]

^and

[6],

^{I use the}

density parameter W

^{to mean the} average number of atoms in a

sphere (in

³

dimensions)

^{or a disc}

(in

2

dimensions)

^{of radius} ro, where _ro is the distance _up to which the atoms interact. Thus each atom is on

average connected to W others.

In

figure

^{1 I} present

histograms

of the cluster sizes found. Since the computer is limited in store, the bar at the bottom of each column collects

indiscriminately

all clusters of sizes greater ^{than 25 000.}

Naturally,

^this

bar

lengthens

^{as W} increases. The

phenomenon

^that

gives

^{us some}

hope

^{that we have}

actually passed through

^a critical value is the dramatic fall in the

proportion

of clusters of size 1 000-25 000. I suggest that this is

because,

^{for W} >

Wc,

^the

larger

^{a cluster}

becomes,

the better is its chance of

becoming

^an

infinite cluster.

Unfortunately,

^{I have no}

proof

^that

the visible advance and retreat of

the finite part

^{of the}

histogram

^{- that}

referring

^to clusters of size _up to 25 000 ^- is not an artefact of the

logarithmic

^{scale I}

am

using,

^or of the actual limit 25 000. Another _way of

expressing

^these

problems

^{is set out in table I. Here I}

give

^the average sizes of clusters of size less than or

equal

^{to 25 000. These}

give

^dramatic

peaks

^around

W ⁼ 2.7 in 3 dimensions and W ⁼ 4.4 in 2 dimen- sions. But if we repeat ^the exercise with a limit of 2 500 instead of 25

000,

^the

picture changes;

^the

peak

is less dramatic ^- as is natural ^- but also there is a

distinct shift downwards ⁱⁿ the value of W at which the

peak

^{occurs in} 2’dimensions. The statistical ^errors in the tabulated values are enormous, but the real

problem

^{is to find some}

argument

^{to assure us that} there would not be a

corresponding

^{rise in the} appa- rent critical value if we were to

repeat

^{this work with}

a

larger

^{limit on} the cluster size. In view of

figure 1, I

find it difficult to

believe

that

Wc

^can

really

^be greater than 2.85 in 3

dimensions,

^{but I have to admit that this}

method still

gives

^no

really

^solid

grounds

^for

believing

that I have found _upper bounds for

Wc.

Concerning

^lower

bounds,

the situation is much

816

TABLE I

(a) Average

^sizes

of

3-dimensional clusters

of

^{size not} greater than no,

for

no ⁼ 25 000 and 2 500 and values

of

^W between 2.6 and 2. 85. The _ranges

given

^are the estimated standard deviations

of

the numbers

they follow.

The numbers in brackets are the numbers

of

^{trials in each case.}

(b) Average

^sizes

of

2-dimensional clusters

of

^{size not} greater than no,

for

no ⁼ 25 000 and 2 500 and values

of

^{W between 4.0 and 4.6}

(*) ^The discrepancy between these figures and those above them is due to the inclusion in the lower figures ^of data from clusters

generated ^{with a limit set between 2 500 and 25 000.}

happier. Following [3]

^and

[6],

^I present ⁱⁿ

figure

²

curves of mean

generation

^size

plotted against

gene- ration numbers for selected values of W. As remarked in

paragraph 3,

^we know that if

Gk,

^{the mean size of}

the k’th

generation,

^{is ever less than}

1,

^{then W}

Wc.

Subject

^{to the}

possibility

of statistical _error,

therefore,

the curves here show that

Wc >

^{2.65 in 3 dimensions}

and

We

> 4.2 in 2 dimensions.

(Fortunately,

^{we have}

upper bounds for

Gk (see [3], § 3.2),

^{so that the}

probability

^{of a}

spurious

^{result is at least}

estimable.)

However,

^we

again

^{have no} way of

being

^{sure that}

even the

steepest

^curves

plotted

^{here are not}

going eventually

^to

peak

^out and decline towards zero.

So

concerning

upper bounds for

We

this method is

even less

satisfactory

^{than the}

histogram approach.

Finally,

^I

give

^three

drawings

of actual two-dimen- sional clusters

(Fig. 3).

^{There has been a certain amount}

. of

speculation concerning the frontiers

of these clusters

(see

e.g.

[5]),

^{and it seems worth while}

having

^further

pictures

^to

supplement

^{those of}

[2].

5. Conclusions. ^-

Subject

^{to the}

possibility

^of

statistical _errors, the

probability

of which is estimable and can be reduced

by repetition

of the above calcu-

lations,

^{the curves in}

figure

^{2 show that}

Wc >

^{2.65 in}

3

dimensions,

^and

Wc >

^{4.2 in 2} dimensions. The

histograms

^of

figure

¹ and the statistics of table I suggest ^that

We

^{2.8 in 3} dimensions and

Wc

^4.6

in 2

dimensions,

but there are deficiencies in the method which mean that without some further theore- tical advance no amount of

repetition

of the calcula- tions can

really

establish these

figures.

FIG. 3. ^- Three two-dimensional clusters. The atoms are drawn

as circles of diameter _ro, so that two atoms interact if the correspond- ing ^circles overlap. The solid circle in each picture ^{is the} starting

atom. Those atoms from which the cluster is, ^or may be, still growing

are marked with crosses. a) ^{W =} 4.3; ^{4 397} atoms, complete.

b) ^{W =} 4.4; ⁷ 000 atoms, still growing. c) ^{W =} 4.5 ; 6 203 atoms,

complete.

ture of clusters, ^J. Physique ³⁴ (1973) ^341-344. percolation theory, ^Adv. Phys. ²⁰ (1971) ^325-357.