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### Submitted on 1 Jan 1976

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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 wassearched

### 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 was2-11 _{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;

myown 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. Writefor 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 much816

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 iseven 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.}