Anomalous scaling of moments in a random resistor networks

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Anomalous scaling of moments in a random resistor networks

G.G. Batrouni, A. Hansen, M. Nelkin

To cite this version:

G.G. Batrouni, A. Hansen, M. Nelkin. Anomalous scaling of moments in a random resistor networks.

Journal de Physique, 1987, 48 (5), pp.771-779. �10.1051/jphys:01987004805077100�. �jpa-00210497�

Anomalous scaling ^{of moments in}

^a

random resistor networks

G. G. Batrouni

(1,*),

^{A. Hansen}

(2,**)

^{and M. Nelkin}

(+)

(*) ^Newman Laboratory of Nuclear Studies, ^Cornell University, ^{Ithaca, New York 14853, U.S.A.}

(**) Department ^of Physics, ^Cornell University, ^{Ithaca, New York} 14853, ^U.S.A.

(+) ^{School of} Applied ^and Engineering Physics, ^Cornell University, Ithaca, ^{New York} 14853, ^U.S.A.

(Reçu le 18 dgcembre 1986, accepté ^{le 26} janvier 1987)

Résumé. ²⁰¹⁴ Nous considérons un réseau carré de résistances aléatoires au seuil de percolation ^{de lien}

pc = 1/2. Nous calculons la distribution de ^{courant sur} l’amas infini naissant au moyen d’une méthode de

gradient conjugué accélérée par transformation de Fourier. Nous calculons le n-ième ^moment de cette distribution à la fois dans l’ensemble à courant constant et à voltage ^{constant sur} des réseaux de taille variant

jusqu’à ^{256 x} 256, ^et examinons le comportement d’échelle du moment en fonction de la taille. Une analyse

conventionnelle de nos données produit ^des exposants critiques ^{en accord} avec ceux obtenus dans des

publications antérieures. Les auteurs précédents ^avaient supposé que ^ces exposants ^{sont déterminés} théoriquement dans la limite de n grand par les liens simplement connectés. Ceci correspond à ^une hypothèse implicite ^{sur l’ordre} des limites pour lequel ^{nous ne trouvons aucuene} justification théorique. ^Nous réanalysons ^{nos données en} soustrayant la contribution des liens simplement ^{connectés avant} de calculer les moments. Dans la limite du réseau infini, ^{ceci ne} peut pas faire de différence mais, ^{sur nos réseaux} finis, ^les exposants critiques apparents ^sont fortement modifiés. Le problème que ^nous posons ^ne peut pas être résolu

numériquement. Nous examinons brièvement comment il pourrait être résolu théoriquement.

Abstract. ²⁰¹⁴ We consider a random resistor network on a square lattice ^at the bond percolation ^threshold

pc = 1/2. We calculate the ^current distribution on the incipient infinite cluster using ^a Fourier accelerated

conjugate gradient ^{method. We} compute the ^{n-th moment} of this distribution in both the constant current and constant voltage ensembles for lattices up ^to 256 x 256 in size, and examine how these moments scale with lattice size. When analysed in the conventional way, ^our scaling exponents agree with published ^results.

Previous authors have assumed that these exponents ^are theoretically determined in the limit of large n by ^the singly connected bonds. This makes an implicit assumption about the order of limits for which we find no

theoretical justification. ^We reanalyse ^{our data} by subtracting the contribution of the singly connected bonds before calculating ^{the moments. In} the limit of infinite lattice size, ^{this can make no} difference, ^{but for our}

finite lattices, ^the apparent scaling exponents ^are strongly affected. The dilemma that we pose ^{can not} be resolved numerically. We discuss briefly ^{how it} might be studied theoretically.

Classification

Physics ^Abstracts

05.40 ^- 64.60

1. Introduction.

The

study

of fractal

objects

characterized

by

^an

infinite number of

scaling

exponents ^{is of consider-}

able current interest. This

subject

^was first studied in the context of

fully developed

turbulence

[1],

^and

has led to some interesting ^recent

speculations

ⁱⁿ

that context

[2, 3].

^{For more tractable}

problems ranging

^from

dynamical

systems

[4]

^{to diffusion}

(’)Present

^{address :} Department ^of Physics, ^Boston University, ^Boston, Massachusetts 02215, ^U.S.A.

(2 )

Present address : Groupe ^de Physique ^des Solides,

Ecole Normale Supérieure, 24, ^rue Lhomond, 75231 Paris Cedex 05, ^France.

limited aggregation

[5],

^to

problems

ⁱⁿ

polymer physics [6],

^the

subject

^{has had much recent}

activity.

Perhaps ^{the most}

straightforward application

^{is the}

distribution of currents among the bonds on the backbone of a random resistor network at the

percolation

^threshold

[7, 8].

^To be definite we

consider a constant current ensemble where a unit current flows

through

the backbone in each realization. An

important

^{role is}

played

by ^the

singly

connected bonds which carry all of the ^current. The number of these bonds has been shown by Coniglio

[9]

^{to scale as the}

1 / v

^{power of the} lattice size. In two dimensions, ^the correlation

length

exponent ^{v is} believed to be exactly ^4/3

[10].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805077100

772

The moments of the current distribution are

expected ^{to scale as} powers of the lattice size for

sufficiently large ^lattices. We define the

scaling

exponent qn

by

where (in>

is the n-th moment of the current distribution.

The exponent qo is

just

^the fractal dimension of the backbone

[11],

^the exponent q2 is the resistance exponent ^{and the} exponent q4 is related to the

amplitude

^{of the}

1 / f

^{noise in a}

percolating

system

[7].

^{Since the current is}

always

^{less than or}

equal

^to

one, qn ^{must be a monotone}

non-increasing

^function

of n. In any finite lattice, ^{the moments for}

large n

will be dominated

by

^the

singly

connected bonds for which i = 1. This leads to the

assumption

^that

which is 3/4 in two dimensions.

Equation (1.2)

^is

strongly

supported

by

^all

existing

^{numerical results}

for finite lattices, ^{and is} supported

by Migdal-

Kadanoff renormalization _group calculations

[12],

and by ^series

expansions

^{near six} dimensions

[13].

The main theoretical contribution of the present paper is ^{to note that}

equation (1.2)

^{assumes that the}

limits n -+ oo and L - oo commute. The numerical support ^for

equation (1.2),

^{and the}

published

^theor-

etical arguments ^{in its} favour, ^both

implicitly

^take

the limit n -+ oo first. The definition of qn, however,

from

equation (1.1) clearly

^{takes the} limit L -+ oo

first.

Why might

^{this make a} difference ? Consider the n-th moment

(i n)

^{as a function of L for}

large

^but

finite n. The contribution to

(i n)

^{from the} singly

connected bonds with i _{= 1 grows} as the _{3/4 power} of L. We introduce the finite system

scaling

expo- nents defined

by

where 4 is a finite difference. _{For any} finite n we

expect Zn:> 3/4. Thus for

sufficiently large

^lattices

the contribution to

(i n)

^{from the}

singly

^connected

bonds will become

negligible compared

^{to the contri-}

bution from the currents i 1. If we define the

exponents qn

by

^the

limiting

process of

equation (1.1),

^{we see that} for any finite ^{n, no matter} how

large,

^the qn’s ^{are not} determined by ^the singly

connected bonds. Thus there is no obvious theoreti- cal reason to expect

equation (1.2)

^{to be valid.}

The exponents qn ^for large n ^are determined

by configurations

^{with currents} nearly

equal

^to

(but

^less

than)

^{one. These currents} appear ⁱⁿ bonds that are

connected in parallel ^{to clusters with} very

high

resistance ; ^an example ^{of such a bond is} given ⁱⁿ

figure

^{1 where it} is marked A. These

high

^resistance

Fig. ^{1. -} Several features in the current histograms ⁱⁿ figures ^{2 and 3 are} identified in this backbone. A is a bond

carrying ^{a current close to one. B is} perfectly ^balanced,

and thus carries no current. C are singly ^connected bonds,

or links. ^{D and E are} configurations responsible ^{for the} peaks marked D and E in the histograms.

clusters must

necessarily

^be large. Furthefinore, ^if

such a cluster is removed from the backbone, ^the corresponding ^{bond in} parallel ^{to it will become} singly connected. The appearance of ^currents close to one are therefore associated with very non-local

properties

^{of the backbone. We see no a} priori

reason why ^the distribution of these currents close to one, ^or

equivalently,

the distribution of the large

structures

causing

them, ^{should be}

simply

^{related to}

the number of

singly

connected bonds thereby

making equation (1.2)

^{correct. We return to this}

question

in section 4 since it is the essential theoreti- cal

point

^raised by ^the present ^work.

We have studied this

question numerically

^for

finite systems by

introducing

^{the « blob » moments} Bn ^defined

by

where NL ^{is the} average ^{number of}

singly

^connected

bonds _per realization. That means we subtracted from each moment the contribution of the lines which is

unity

^{for each}

singly

^connected bond ; ^thus

we teep

only

^{the slob} contribution. We then intro- duce the finite system

scaling

exponents

In the limit of an infinite lattice the exponent

z,,, y,,

and qn

^must agree for any finite ^n. Our numerical results for L between 16 and 256 show that for the larger lattices, ^{the fits to}

equation (1.3)

and

equation (1.5)

^are

approximately equally good,

but the exponents Zn

and yn

^are

numerically quite

different for

large

^{n. In}

particular

^our

numerically determined Yn

^{seem to}

approach

^a limiting ^value

near 1.0 and not 3/4, ^{as n becomes} large, ^{which the}

Zn form the same simulation

approach

^the

expected

limit 3/4.

In section 2, ^we _present ^our numerical results for the statistics of the current distribution. In section 3,

we give ^the detailed results for the scaling ^{of the}

moments and the ^« blob » moments. We conclude in section 4 with a brief discussion of the theoretical

significance

^{of our results.}

2. Numerical methods and results.

We study ^a two-dimensional _square lattice of size L with two types ^{of bonds.} They either ^{have a resist-}

ance R with

probability

p ^or infinite resistance with

probability (1- p ).

^{In all of our} calculations we

have taken p ^{= 1/2,} which is the threshold for bond

percolation in the infinite system ^{or finite} system with self

duality.

^{We look for an}

incipient

^infinite

cluster which carries current from one side of this lattice to the

opposite

^{side. We start from the lower}

left-hand _corner,

looking

^{for a} connected cluster

reaching

^across the lattice from the lower edge ^{to the}

upper edge. ^{If no} such cluster is found, ^{a new lattice}

is generated. ^{If such a} cluster is found, ^{we then start}

from the upper

right-hand

^corner

searching

^{for the}

first node on the upper edge ^{connected to this} incipient ^{infinite cluster. This node at} the upper

edge

of the lattice and the leftmost node at the lower

edge belonging

^{to the}

incipient

infinite cluster are then chosen as the two contact points between the cluster and an external current source. In our constant current ensemble, ^{a unit current flows}

through

^the incipient ^infinite cluster in each realization.

There are three classes of bonds in the cluster : those which will carry ^a current,

perfectly

^balanced

bonds and bonds

belonging

^to

dangling

^{ends. The}

first two classes of bonds form the backbone

[14].

^In figure ^{1 the bond marked B is}

perfectly

^balanced.

We

identify

^{the resistors}

belonging

^{to the backbone} by using ^{the «}

burning

^»

algorithm

^of Herrmann, Hong ^and

Stanley [15].

^{It is}

quite

costly ⁱⁿ computer time to

identify

the backbone at this

point

^{in the}

calculation since the computer ^{time needed} grows ^as the number of

loops

^{in the backbone. An} alternative way would be to solve the Kirchhoff

equations

^for

the entire cluster, ^{and then}

identify

^{the backbone}

from the currents that the resistors carry. As ^our iterative methods for

solving

^{the Kirchhoff}

equations

improved, ^{it turned out that this} alternative method would have been more efficient for the larger

lattices. We only discovered this after we had

completed the calculations

reported

^here.

To solve the Kirchhoff

equations

^{on the} backbone,

we used the conjugate gradient ^method for lattices of sizes 50 and 200, and the Fourier accelerated

conjugate

gradient method for lattices of size 16, 32, 64, 128 and 256. We have described this method in a recent publication

[16].

^{For the} largest lattices, ^the implementation of Fourier acceleration made the

conjugate gradient

^{method run}

approximately

5 times faster in cpu time. We generated ¹⁰⁰⁰

backbones for L = 16, 32, 50, ^and 64, ^{1 250 for}

L = 128, ^{350 for L =} 200, and 160 for L ⁼ 256. The currents were determined to within an accuracy of

10- 8,

which is the maximum obtainable

precision

^on

the FPS 264 we used. The error criterion we used is discussed in reference

[16].

^In figures ^{2 and 3 we}

plot histograms

^{of the current} distributions for L ⁼ 64 and L ⁼ 256. The statistical errors in these histo-

Fig. ^{2. -} Relative number of bonds carrying ^{a current} i, N (i,

L )/ E

^{N (i, L ), versus loglo} i for 1 000 lattices of

i

size L ⁼ 64. The leftmost bin in the histogram contains all bonds carrying ^a current i __ 10- 8.

Fig. ^{3. - A} histogram ^{of the same} quantities ^{as in} figure ^2,

but based on 160 lattices of size L ⁼ 256.

774

grams ^are comparable, and of the order of 2 % of the

height

of the last bin for the L ⁼ 64

histogram,

^and

4 % of the height ^{of the} corresponding ^{bin for the}

L ⁼ 256 histogram. Note the finite size effects at

certain currents. These are associated with the features marked C, ^{D and E in} figure ^{1. C denotes}

the « links » or

singly

connected bonds which _carry a

current i = 1. If one of these were cut, ^{no current} would flow in the backbone. D denotes a configura-

tion with four bonds, ^each

carrying

^{a current}

i = 1/2, ^{and E a}

configuration

with three bonds

carrying

^a current i = 1/4 and one bond

carrying

^a

current i = 3I4. These features are all more

import-

ant for L = 64 than for L ⁼ 256 as can be seen in

figures

2 and 3. A curious feature in these

histograms

is that the number of

perfectly

^balanced bonds, marked B, ^{relative to the total} number of bonds in the backbone seems

approximately equal

^{to 0.5 %,}

independent

^{of lattice size. In these}

histograms,

which are normalized _{to one,} the horizontal axis is

logarithmic

^{with ten} bins per ^octave. This was a

strategic ^{error on our} part. ^{Had we realized from the}

beginning

^{that we were}

going

^to

emphasize

^the

results for large ^{currents so} heavily, ^{we would have}

stored our current distributions in a different form.

Since we

computed

^{the moments} of the distribution

directly

^{from the} solution of Kirchhoff’s laws and not from the

histograms,

^the

principal

^{results of this}

paper ^{are not} affected.

3. Moments of the current distribution.

We computed ^{the n-th moment of the current}

distribution defined

by

where the sum over k _goes over all bonds on the backbone for each realization, ^{and the sum over r}

goes ^over each of the

NR

realizations. These sums were computed

directly

and stored for each realiza- tion for n ⁼ 0 to 9.5 in intervals of 0.5. The contribution to each of these moments from the

singly

^{connected bonds}

(links)

^{is the} average number of links _per realization. In table I we list our data for the cases n ⁼ 0, 2, 4, 6, ^{and 8. The column to the far}

right gives

^the average number ^of links _per realization. In each entry ^the top ^{line is the full n-th} moment, and the bottom line is the ^« blob ^» moment with the contribution from the links subtracted.

It is

easily

^seen that, ^over the range of L studied,

n ⁼ 0 is dominated by ^the blobs, n ⁼ 2 has compar- able contribution from links and blobs, ^and n = 4, 6,

and 8 are dominated

by

^{the links. On the other hand}

it is

equally

easily ^{seen that the blob} contributions to the

higher

^{moments are} growing faster than the link contributions with

increasing

system ^{size. This is a} natural consequence ^of the links

forming

^a

vanishing

subset of the backbone as L - _{oo ;} the blob contri-

Table 1. ^{- Shows the current} moments in > ^{and « blob » moments}

Bn

= in > -

NL for

^{n =} 0, 2, 4, 6, ^and 8, ^and

lattice sizes

^{L =} 16, 32,50,64, 128,200, and 256. The top entry ^{in each line is} the full moment, and the lower entry ^is the« blob » moment. The column to

the far

right ^is the number

oflinks, NLI for

each lattice size.

bution to the higher ^{moments has to} grow faster than the total moment with

increasing

^{L in order to}

« catch up with » the link contribution and eventually

dominate the moments. This leads to y,, -- zn

[17].

The

question

which of the finite size exponents y, ^or z, better approximate ^{the « true »} exponents

qn ^cannot be determined from this observation :

ignoring

^{all other} finite size effects than the presence of the links, ^{there are two} possible situations that are

possible : A)

^{The full moments scale with} z,, ⁼ qn, and the ^« blob » moments approach ^the

straight

lines of the full moments from below in

log-log plots

versus L.

B)

^The blob-moments scale with y,, ⁼ zn, and the full moments approach ^the straight

lines

of the ^« blob ^» moments from above in log-log

plots

versus L. Both of these scenarios obey ^the

inequality

above.

It would seem as this

point

^{that the} only way ^to

answer the

above-posed question

^is

by

determining

whether the full moments or the « blob » moments scale better for the lattice sizes that can be studied

numerically.

However, ^{the so-far}

ignored

^{effects of}

the boundaries will restrict the sizes of both the full moments and the ^« blob » moments for the smaller

lattices, ^thus

increasing

the values of _{both y,,} and z,,. Now, if scenario A is correct, ^we expect ^{that both}

the full moments and the « blob » moments to

approach

^the

asymptotic straight

^lines from below in

log-log plots

^versus L, ^{and with the « blob » mo-}

ments

having

^{more curvature than the full moments.}

If, ^{on the other} hand, ^{scenario B is} correct, ^we expect ^{that the « blob » moments will}

pick

up ^some

Fig. ^{4. - The Oth} current moment

(i °)

^{and the corre-}

sponding ^{« blob » moment} Bo =

(i °) - NL

^{as a function}

of lattice size. The straight ^{lines are based on} least squares fits. The fit to the full moment is based on all the data

points. ^{The fit to the} « blob » ^moments _neglects the L = 16 point.

curvature in these plots ^and

again approach

^the

asymptotic

lines from below. The full moments

however, ^will have the effect of the links and the

boundary

^effects

pulling

ⁱⁿ

opposite

directions, ^and

the net effect is that the full moments will reduce their curvature. Thus, ^it may appear that the log-log

plots

^{of the full moments are closer to}

straight

^lines

also in this scenario.

To determine the scaling exponents ^{z,, and}

y,,, ^we carry ^{out a} least squares ^{fit to}

equations (1.3)

and

(1.5) respectively.

^In

figures

^{4, 5 and 6 we} give

Fig. ^{5. -} The second current moment

i2 >

^{and the}

corresponding ^{« blob » moment} B2 ^{as a} function of lattice size.

Fig. ^{6. - The 8th} current moment

i 8>

^{and the corre-}

sponding ^{« blob » moment} B8 ^{as a} function of lattice size.

776

Table II. ^- Shows the scaling exponents Zn based on the

current moments ^i" ), ^{and the} scaling exponents _y.

based on the ^« blob » moments

Bn

= in ) -

NL

^for

n ⁼ 0, 2, 4, 6, and 8.

log-log plots

^whose

slopes

^{lead to those} exponents for n ⁼ 0, ^{2 and 8. In table II we}

give

^the

resulting

exponents ^{based on lattice sizes 32 to} 256, ^{and in}

figure

^{7 we}

plot

^these exponents ^{as a} function of n.

We see that the full moments give ^{better fits than the}

« blob » moments, but except ^{for 1} = 16, the differ-

ences are small. However, ^the

resulting

^effective scaling exponents yn and zn ^are very different.

The exponents zn appear

large compared

^to previous- ly published ^values

[7, 8].

^{We attribute this to our}

choice of boundary conditions ; ^{our backbones are}

connected to the current source

through only

^two

nodes on

opposite

edges ^{of the lattice,} while in other calculations the backbone has been connected

through

^bars along each of these edges. ^For

n ⁼ 0 the difference between yo and zo is small since the links do not play ^an important ^{role in the}

effective fractal dimension of the backbone, ^{even for}

small lattices. For n ⁼ 2, ^our exponent y2 ^{is consider-} ably

larger

than Z2, and this implies ^a substantial

Fig. ^{7. - The} scaling exponent z,, based on the full current

moment, i’>, ^{and the} scaling exponent yn ^{based on the}

« blob » moment Bn ⁼ in) - NL ^{as a} function of n.

change

in the resistance exponent

[18].

^{A similar}

change

would have been found by ^other authors,

had they

analysed

^their data in the way that we did., For larger ^{values of n,} all of the moments for all L up to L ⁼ 256 are dominated by ^the links, ^{but the}

contribution of the blobs is still

accurately computed.

In light ^{of the} previous discussion of the effect of the boundaries on the

scaling

behaviour of the full moments and the « blob » moments, ^{we find it} impossible ^to

distinguish

^{between scenario A or} B,

or

equivalently,

^determine

whether yn

^or zn ^are closer to the « true » exponents qn.

Finally

^{we consider the} distinction between our

constant current ensemble and the constant voltage

ensemble considered

by

other authors

[8].

^{Instead of}

holding

^{the current in each backbone} fixed, ^{we could} have

kept

^the

voltage

^{across the backbone fixed.}

The current in the links would then have been

proportional

^to the conductance of that backbone rather than

being

^{fixed at i = 1. The current in each}

of the other bonds would also have scaled pro-

portional

^{to the} conductance. This would smear out all of the finite size effects in the current distribution,

making

^the

corresponding histograms

^{of the currents}

distribution smooth. Especially, ^{there would be no}

trace of the distinctive

peak

^{due to} the links. In reference

[8],

large ^{lattices were} analysed ^{in a}

constant

voltage

ensemble with the scaling assump- tion

Under reasonable

assumptions

^of statistical

indepen-

dence, ^{it was} suggested in reference

[8]

^that

Since we stored all of the moments for each realization, ^{we could}

analyse

^{our data for the}

moments in a constant voltage ^{as well as a constant} current ensemble. We found

good

agreement ^with

equation (3.3).

The statistical

independence

assump- tions

leading

^to

equation (3.3)

^{are not obvious. The}

n-th current moment in the constant-voltage ^ensem-

ble can be written

as Gn in)

^{in the} constant-current

ensemble, where G is the conductance of the back- bones.

Equation (3.3)

^assumes that this moment scales

as G) n i n).

^To check this assumption, ^we

measured the correlation between the conductance and the number of links, ^{since we} _expect ^{the current} associated with the links to be more correlated to the conductance than the currents associated with the bonds making up the blobs. We found a correlation coefficient of

approximately

^{0.2, and we found that}

this correlation coefficient did not vary

appreciably

with lattice size.

4. Theoretical considerations.

When

analysing

^a

probability

distribution which contains an infinite number of

independent

scaling