Random walks on fractal structures and percolation clusters

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Random walks on fractal structures and percolation

clusters

R. Rammal, G. Toulouse

To cite this version:

R. Rammal, G. Toulouse. Random walks on fractal structures and percolation clusters. Journal de

Physique Lettres, Edp sciences, 1983, 44 (1), pp.13-22. �10.1051/jphyslet:0198300440101300�.

�jpa-00232136�

L-13

Random

walks

on

fractal

structures

and

_percolation

clusters

R. Rammal

_(*)

and G. Toulouse

Laboratoire de

_Physique

de l’Ecole Normale

_Supérieure,

24, rue _Lhomond, 75231 Paris Cedex _05, France

(Re~u le 28 octobre 1982, accepte le 10 novembre ₁₉₈₂₎

Résumé. 2014 La notion de dimension

_spectrale

d’une structure self-similaire _(fractale) est

_rappelée,

et sa _{valeur pour la famille des tamis} de

_Sierpinski

dérivée à

_partir

d’un argument d’échelle. On montre

que diverses

propriétés

de marche aléatoire telles que la

probabilité

de marches fermées et le nombre moyen de sites visités sont gouvernés par cette dimension

_spectrale.

On

_suggère

_{que le}

_{nombre SN}

de sites distincts visités

_pendant

une marche aléatoire de N _pas sur un amas infini au seuil _de

perco-lation varie

_{asymptotiquement comme SN ~ N2/3,}

en toute dimension. Abstract 2014 The notion of

spectral

dimensionality

of a self-similar _(fractal) structure is _recalled, and its value for the _family of

_Sierpinski

_gaskets derived via a

_scaling

_argument. Various random

walk

_properties

such as the

_probability

of closed walks and the mean number of visited sites are

shown to be

_governed

_by this

_spectral

dimension. It is

_suggested

that the

_{number SN}

of distinct sites visited

_during

an _N-step random walk on an infinite cluster at

_percolation

threshold varies

asympto-tically

as :

_{SN ~}

N2/3,

_{in any dimension.}

J.

_Physique

- LETTRES 44 (1983) L-1~3 - L-22 ler JANVIER 1983, Classification

Physics

Abstracts 05.40 - 72.90 - 64.70 1. Introduction. -

It is now

recognized

that there are _{many self-similar} structures

_(named

fractals

_by

B. Mandelbrot

_[1])

in nature and _{many ways} to model them. A random walk in free space or on a

_periodic

lattice,

a linear or branched

_polymer,

a

_percolation

cluster at threshold are

just

a few well-known

examples

of such realizations. When

_considering

the vast

_category

of these self-similar

_geometrical

spaces, it is

tempting (by

one turn of the

_crank)

to

_regard

them as basic

spaces and to

_study

some

general physical

effects thereon.

Among

past steps

in that

direction,

one can

_quote

the

_study

of random walks on

_percolation

clusters

_[2]

_(the

_problem

of« the ant in the

_labyrinth

_»),

the

_study

of

_phase

transitions and

_transport

properties

on various fractals such as

_{Sierpinski gaskets [3, 4],}

the

_study

of the

_density

of states on a

fractal and the definition of a « fracton »

_{dimensionality}

related to diffusion

_[5].

This _paper

_attempts

to

_present,

in a self-contained manner, some

_elementary

physics

on

self-similar structures. One

_important

thread runs

_through

the discussion : standard _{Euclidean spaces}

have translation _symmetry, while self-similar spaces have dilation

symmetry.

As a _consequence,

whereas Euclidean _spaces are well characterized

_by

one _space

_{dimension : d,}

_{self-similar spaces}

require

the definition

_{of (at}

_least)

three dimensions :

d,

the dimension of the

_embedding

Euclidean

(*) Permanent address : CNRS-CRTBT, B.P. _166, 38042 Grenoble, France.

PHYSIQUE -

space ; d,

the fractal

_(Hausdorff)

dimension

_{[1] ; d,}

the

spectral (fracton)

dimension

[5].

For Eucli-dean spaces, these three dimensions are

_equal.

This can be construed as an « accidental »

degene-racy and it is of interest to sort out the different

qualifications

of the

_{dimensionality,}

in some of the

many formulae where it enters. In this sense, the

study

of

physics

on self-similar _{spaces is} not

_only

an

_exploration

of a new

_world,

it also allows for a

_{deeper understanding}

of the

_physics

on more

traditionally

trodden spaces.

2. The

_spectral

_{dimensionality 3.}

On a

periodic

Euclidian

lattice,

of

dimensionality d,

the

density

of states

_p(~),

for excitations of

_frequency

c~, can be derived from a

knowledge

of the

dis-persion

relation

_w(k),

where k is a

wave-number,

and from a

counting

in

k-space (reciprocal

space) :

Therefore the

_argument

_hinges

on the translation invariance of the

lattice,

via a Fourier space

analysis

of the modes and the definition of a wave-number.

In the

_vicinity

of the lower

_edge

of the

_spectrum

_(e.g.

for elastic

vibrations),

the

_dispersion

relation becomes a

simple

_{power law :}

and the

_{low-frequency density}

of states also :

It turns out that a

_simple

law behaviour may also be found on self-similar structures. In this

case, the existence of the power law stems from a

_scaling

_argument,

which is more

general

and

deeper

than the

_previous

one, restricted to translation-invariant media.

2.1 SCALING ARGUMENT. - To be

_specific,

let us consider an elastic structure

_consisting

of

N identical units of mass _m, linked with

_springs

of

_strength

K. The

microscopic frequency

scale is

then set

_by

_c~

=

K/m.

If the _structure is

self-similar,

N is related _to

L,

the

length

scale of the

struc-ture

_by

This relation defines the fractal

dimensionality d

of the structure.

It

_implies

_that,

under a scale transformation

_by

a factor

b,

the number N varies

according

to

and

_{correspondingly}

the

_density

of states

_{(per elementary}

_unit)

scales

_according

to :

If,

under this scale

transformation,

the mode

_frequency

has also a

scaling

behaviour :

L-15 RANDOM WALKS ON PERCOLATION CLUSTERS

and

_{using (6) :}

Taking

b =

c~l~a

in the

r.h.s.,

_{one gets a}

power law :

from which one defines the

_{spectral dimensionality}

d-

as

d-

=

d ja.

On

_regular

Euclidean

lattices,

(5)

holds with d =

d,

the

Euclidean

dimension,

and

(2)

implies (7)

with a = 1. Thus the

spectral

dimension

d

is

equal

_to

d, (10)

reducing

_to

(3),

_as it should.

In order to _{prove that} a

scaling

relation like

(7)

_may indeed hold on self-similar structures, an

example

is

_given

below.

2.2 EXAMPLE: SIERPINSKI GASKETS. -

The

_{Sierpinski gaskets}

form a

_family

of self-similar

structures, which can be built in any Euclidean dimension

_(see

_Fig.

_1,

for d =

2)

and which have been

_already

elicited for studies

_[3-5],

because

_they

lend themselves

_particularly

_conveniently

to

scaling approaches.

_{g pp} Their fractal dimension is

_easily

_y found :

d

In

(d + 1 )

ln 2

’

Starting

from the

_equations

of motion for elastic vibrations

_(one

_equation

for each

_site),

one

eliminates the

_{amplitudes corresponding}

to the sites located at

_mid-points

of

_triangle

_edges

at the

lowest scale. This decimation

_procedure

leads to a reduced set of

_{equations, describing}

the same

physics

on a

_gasket

scaled

_(down)

_by

a factor b = 2. It turns out that the decimation introduces no new

_coupling

constants and leads to a

simple

renormalization of the

frequency (appendix).

Thus

the ratio

_W2/(02

ls

_{simply replaced by}

_{2 - ~} d + 3 .~

a2 co 2 -)

,2

2 2)

co co co0

In the notations

_{of equation (7),}

,_, ’"

and at low

_{frequency c~o ~}

_1,

one obtains

Fig. 1. - The two-dimensional

(d = ₂₎

_Sierpinski

_gasket. The iteration process _making the _structure

self-similar should be

_{pursued indefinitely.}

The

_figure

shows the structure after four _stages of iteration from

a relation which is cast in the form of

_equation

₍₇₎

with :

Thus,

exponent a

is obtained :

i

and the

_spectral

dimension of a

_{Sierpinski gasket}

in dimension d is :

This result _may be derived

_differently.

We have chosen this derivation for its

_simplicity.

The

_density

of states on a

_gasket

is

actually

a

highly

singular

function of

frequency [6]

and the

low

_frequency

behaviour described

_{by equation (10)}

is a smoothed

_expression,

to be used in

integrals,

such as

those

involved in the

_computation

_{of thermodynamic quantities}

at low

tempera-tures.

2.3

_INEQUALITIES

BETWEEN DIMENSIONS. - It is

easy to check that the three dimensions

(d,

d, d),

calculated above for the

_{family of Sierpinski gaskets,}

are ordered as follows :

The

_equalities

hold for d =

1,

which is the

baby

of the

family.

When

d increases, d

grows

mono-tonically

and is

_unbounded,

whereas

d

_approaches

its _upper bound

2=2.

The double

_inequality

₍₁₄₎

_appears to hold

_{generally (there}

is no

_{counter-example}

that we know

of).

It would be

_useful,

for various reasons which will become clearer

_later,

to know a

_family

of

fractals,

as

_simple

as the

_gaskets,

where

d

would not be restricted to values smaller than 2.

3. Random walks. - Classical diffusion and random walks in Euclidean

spaces are a well established field of

_physics.

We direct the reader to a recent review

_[7]

for references.

Several

_properties

will be

_examined,

_starting

with the fundamental one : the

_mean-squared

displacement

from the

_origin

at time t

_(or

after N

_steps

on a

_network).

An _exponent v is defined from the

_asymptotic

behaviour :

In a Euclidean space, of

dimension d,

it is well known that :

independent

of the

_{dimensionality.}

_Indeed,

the law

₍₁₅₎

is obtained as solution of a diffusion

equation,

which is

_analogous

to the wave

_equation

for

_vibrations,

_except

for the

_replacement

of a

second

_by

a first time derivative.

_Therefore,

for

_simple

dimensional reasons, the

counterpart

of

equation (2)

is

_{equation (15)}

with the value

₍₁₆₎

for the

_exponent.

From the

_arguments

_developed

in section

_2,

if on a fractal structure

_{equation (7)}

_holds,

then it

follows

_similarly

that :

L-17 RANDOM WALKS ON PERCOLATION CLUSTERS

3.1 SELF-AVOIDING WALKS. -

Expression (17)

implies

that the fractal

_{dimensionality}

of a random walk

_(on

a fractal

_structure)

is

_dR

= -

=

2 d

Therefore two such walks will

_generically

v _d

cross each other

_(or

one walk will

_self-cross)

if and

_only

if 2 dR >

d.

This

_condition,

which reduce

to

d

_4,

determines when the excluded volume

restriction,

in a

self-avoiding

walk,

is relevant.

It is worth

_noting

that this condition

_{depends only}

on the

_spectral

dimension.

3.2 CLOSED WALKS AND SCALING LAWS. - The

_probability

of return to the

_origin

after N

steps,

i.e. the

_probability

of closed walks of

_length

N,

is called

_Po.

On a Euclidean

lattice,

it is

easy to show

_[8]

that :

On

fractals,

such as the

_{Sierpinski gaskets,}

for which

_{(17) holds,}

the

_{generalization}

of

(18)

is :

More

_generally,

it is

_possible

to define a set of critical

_exponents,

in

_analogy

with the Gaussian model of

_phase

transitions

_[8].

Thus

_exponent

_y is defined from the total number of random walks of

_length

N and found

_equal

to

1,

for random walks on fractals as well as on Euclidean lattices.

The other

_exponents

can be derived from v and _y,

_using

the standard

_scaling

laws of the

theory

of

phase

transitions. It is

_interesting

to note that one of these

scaling

laws,

the

Josephson scaling

law :

involves

_explicitly

a _space

_{dimensionality.}

On a fractal structure, it is the fractal

dimensionality d

which enters

_(20).

This choice is

consistent

with

₍₁₉₎

since the

_{general expression}

for

_Po

is

_{[8] :}

3. 3 NUMBER OF VISITED SITES. - The

average number

SN

of distinct sites visited

during

an

N-step

random walk is a

quantity

which has been studied for

_thirthy

_years

[7]

on Euclidean

lattices and which is of interest for various

_problems

_(diffusion

in the _presence of _traps, for

ins-tance).

It will be

_helpful

to recall some established results for the

_asymptotic

behaviour of

_SN

on

Euclidean lattices :

where

_C,

C’ denote

_{lattice-dependent}

numerical factors.

The

_probability

of eventual return to the

_origin,

F = 1 -

E,

is

_equal

to

_unity

_{for d 2,}

whereas

0Flfor~>2.

A

_question

arises : which _{dimension governs the behaviour of}

_SN

on a fractal lattice ? We _argue

that it is the

_spectral

dimension

d.

Explicitly,

we claim that :

... ~ ~

For Euclidean

_{lattices, d}

_{= d,}

and

₍₂₄₎

is consistent with

_(22).

Let us introduce the notion of the

number of accessible sites

_EN,

in a random walk of N

_steps.

The

sites,

which are

_effectively

acces-sible,

are those located within a radius

_RN

of the

_origin:

Note that

_Po,

the

_probability

of return to the

_origin

after N

_steps,

_given

in

_(19),

can be recast as

Po ~

(EN) -1,

which says that the walker may find

itself,

after N

steps,

anywhere

within the

acces-sible cloud.

The number of visited sites

_SN

cannot exceed either the number of accessible sites

_EN

or the

number

of

_steps

N. Formula

(24)

simply

states that the

_asymptotic

behaviour

_{of SN}

is

_{governed by}

the lower of these two upper bounds. It appears as the most natural

_{generalization}

of formula

_(23).

We have

_proved

its

_validity

for

_{Sierpinski gaskets, using}

a decimation

procedure analogous

to

that of the

_appendix

for the

_generating

function

_[7]

of the walk.

3.4 LOCALIZATION. - For Euclidean

_lattices,

P. B. Allen

_[91

has discovered an

interesting

analogy

between classical diffusion and quantum localization.

_Transposing

his results to fractal lattices and

_using

_(24),

one reaches the

_prediction

that the

#-function

defined in

[10],

where

_g(L)

is the conductance of a

_sample

of size L should

_tend,

_{for g large,}

toward the limit

value :

But the conductance of a

_{Sierpinski gasket}

of size L _{is easy} to calculate

_[3]

and found to scale as :

where

has

_precisely

the value

_predicted

in

_(27).

This lends further

_support

to the idea that

formulae

₍₂₄₎

and

_{(27) might}

hold

_generally.

For fractals

with d

>

_2,

a

mobility edge,

for electrons

in a random

_potential,

is

_predicted

to occur with a localization

_length

_{exponent vL}

=

( ~L) -’.

4.

_Application

to the

_{percolation problem}

- The

percolation problem

is also a well established

field of

_physics

and we direct the reader to

_[11]

for references.

At a

_percolation

_threshold,

the infinite cluster is a fractal

_object.

_Therefore,

the

_concepts

of the

previous

sections can be

_brought

to bear on the

physics

in the

_vicinity

of a

_percolation

threshold

and this seems, at the

present

time,

to be

_perhaps

the

_{major application.}

L-19 RANDOM WALKS ON PERCOLATION CLUSTERS

be

_expressed

in terms of the Euclidean dimension d and of critical

_exponents

of the

_percolation

transition

_([3p,

_vp, indexed

_by

_{p for}

_percolation,

are defined below in

_equations

₍₃₀₎

and

(34)) :

In order to calculate the

_spectral

dimension

_d,

we shall use a

_simple

crossover _argument.

Slightly

above the

_percolation

threshold

_(Ap

>

_0),

there is one dominant

_length

scale,

the

corre-lation

_length

_{~p :}

-For excursions smaller than

_~p,

the diffusion law is

_expected

to be of the form of

_equation

₍₁₅₎

whereas,

for

_larger

excursions,

a standard power law is recovered :

The Einstein relation between the

_{conductivity a(p)}

and the diffusion constant

_{D( p)}

can be written as :

where

_{P( p)}

is the infinite cluster

_density.

In the

_vicinity

of the

_percolation

threshold :

(the

exponent t

should not be confused with

time)

and

implying :

Expressing

the

_equivalence

of

_{equations (15)}

and

₍₃₁₎

for R -

_~p,

we obtain :

which allows for a determination of the random walk _exponent v :

and of a characteristic time scale :

The

_{spectral dimensionality d}

of the

_percolating

cluster is then derived from

_{equations (17)}

and

₍₃₇₎

as :

This

result,

already

present in the work of E. Shender

_[12],

has been discussed in its

_{generality by}

S. Alexander and R. Orbach

[5].

In

_addition,

_they

have made the

_interesting

observation that

d,

as

Now formula

_(24),

_plus

the

observation d

_{= 3,}

lead to a

_simple

universal

_prediction

for the mean number of distinct sites visited

during

an

_N-step

random walk on a

_percolating

cluster,

in _any dimension.

Such a remarkable

prediction,

which lends itself

particularly

well to numerical

_check,

calls for a

physical explanation

which we now _present.

Consider the

_quantity

_dSN/dN

which is the

_probability

of access to a fresh

_site,

after N

_steps.

The claim is

_that,

on a

percolating

cluster which is

just marginally

infinite,

this

_probability

of

escape is of the order of the relative fluctuation in the number of accessible

sites,

due to the random

nature of the

_{percolation problem}

itself :

(41)

implies (40).

Another _way of

_phrasing

the same _argument consists in

_defining

the « _open frontier » of the random walk as the

_product

of the

_probability

of access to a fresh site

_dS~/dN,

by

the number of accessible sites after N

_steps

_2~.

It is then

_argued

that this open frontier is

marginally equal

to the fluctuation in the number of accessible

sites,

due to the random perco-lation _process.

Clearly,

this is

_physical

intuition,

not a demonstration. Is it exact or

_just

_a, somewhat

accidental-ly,

very

good prediction,

like the

_Flory

value for the

_{polymer exponents ?}

Here is a valuable

challenge

for future

_study,

since

₍₄₀₎

via

₍₃₉₎

_implies

a relation

_(the

existence of which has been

long questioned)

between

_dynamic

_(t)

and static

_{(!3p, vp)}

_percolation

_exponents.

Finally,

the situation above the upper critical dimension

de

= 6 for

_{percolation requires}

discussion.

For d > 6,

it is

_{generally accepted}

that

_!3p

=

1,

vp =

21 I t

=

3,

independent

of d.

Therefore the random walk

_exponent

v is

_expected

to stick to the

_{value v 6}

_according

to

equation (37).

However it remains an _open

question

whether the dimensions d

and d

are constant

or linear functions of

_d,

for d > 6.

Coming

now to _transport

_properties

on the

_percolating

_cluster,

we

_{firstly recognize}

that the value of

_{#L given}

in

₍₂₇₎

can be recast,

using

(39),

as

Since

_{#L permits}

a fine measure of the

_degree

of

localisation,

it is

_interesting

to _{compare its value}

for the

_percolating

clusters and for the

_{Sierpinski gaskets,}

as a function of the

_embedding

Eucli-dean

_{dimensionality.}

This is shown on

_figure

2.

Whilst,

for

_{Sierpinski gaskets, #L}

increases

_{monotonically}

with

d,

figure

2 shows

that,

for

percolating

clusters,

it

is

_always

smaller

_(except

for d =

1)

and furthermore it varies

essentially

in the

_opposite

_direction,

_{#L being}

a

_decreasing

function

of d, for 2 d 6.

This demonstrates

clearly

that the

_{Sierpinski gaskets}

cannot be

_accepted

as realistic

quantitative

models for

per-colating

clusters; instead,

their usefulness lies at a

_qualitative

level. The fact that

_PL,

for

per-colating

clusters,

remains

_always

far from its critical value

_f3L

= 0 _suggests that the localization

in a random

_potential

will be

_always

_strong,

an observation which is consistent with recent

results

_{[ 13] suggesting}

that the _{threshold for appearance of extended states,} in the

_quantum

percolation problem,

lies not at _p~, the

_percolation

_threshold,

but

_higher.

5. Conclusion. - The main content of this

paper is

expressed

in our

_equation

_(40),

which

sheds a new

_light

on the old

percolation problem.

We have

_presented

evidence that this

_simple

universal law is close to the truth. Further

_study

_{is necessary} to decide whether it is exact or not.

L-21 RANDOM WALKS ON PERCOLATION CLUSTERS

Fig. 2. - The

exponent

f3L’

defined in _{equation (28),} characterizes the size

_dependence

of the conductance. It is

_plotted

here as a function of the Euclidean dimension dfor three families of structures. Curve _(1), dashed : Euclidean lattice. Curve _{(2) :}

_{Sierpinski gaskets.}

Curve _{(3) :}

_percolating

clusters. For this last curve, the

tangent at d = 6 _can be obtained from _{an s} = 6 - d

expansion : ~L

° = - 2

+ 5 s

_[using

_equation

₍₄₂₎ 21

and reference

_{[ 11 ]].}

Acknowledgments.

- Discussions with R. Orbach and J. Vannimenus

are

_gratefully

acknow-ledged.

Appendix :

Recursion

_equations

for lattice vibrations on a

_{Sierpinski gasket.}

- We derive first

the recursion

_equations

for the two-dimensional

(d

=

2)

Sierpinski gasket (Fig.

1).

Identical

masses m are located at the sites of the

_lattice,

_{joined by springs}

of

_strength

K. Let us denote

Fig.

3. -

Principle of the decimation _procedure, used in the

_appendix,

to derive the recursion relations for elastic vibrations on a two-dimensional

_Sierpinski

_gasket.

the reduced

_squared

_frequency

and

_{

_{X j}

_{e‘~’‘ }}

the

_eigenstates

associated with a mode of

fre-quency w.

The set of

_equations

of motion for the

_mid-points

of the lower left

_triangle

_(in

_Fig.

₃₎

is :

where 6 denotes a

_neighbouring

site of _x,

_(resp.

_X2,

_X3).

The

_{corresponding equation}

for

_X

1

(or

Z1 )

is :

Using

(A .1 ),

one extracts _{x, , X2, X3} as functions of

_{{ Xi }}

and the same for

_{{ zi ~}

as functions

of

_{{ Zl ~. Inserting}

these values in

_{(A. 2),}

one obtains a new

_equation

where

_X

_t is a function of

X2, X3

and

_{Z2, Z3.}

This relation can be cast in the form of the first

_{equation (A .1 ),}

with the renormalization :

For

_general

value

of

the dimension

d,

one obtains after a rather cumbersome

algebra

the

simple

generalization :

References

[1] MANDELBROT, B. B., Fractals : Form, Chance and Dimension _(Freeman, San _Francisco) 1977.

[2]

DE GENNES, P. _G., La Recherche 7 ₍₁₉₇₆₎ 919 ;

MITESCU, C. _{D., OTTAVI, H., ROUSSENQ, J.,} AIP

_Conf.

Proc. 40 _{(1979) 377 ;}

STRALEY, J. _P., J.

_Phys.

C 13

₍₁₉₈₀₎

2991.

[3]

GEFEN, Y., AHARONY, A., MANDELBROT, B. _{B., KIRKPATRICK, S.,}

_Phys.

Rev. Lett. 47 (1981) 1771.

[4]

RAMMAL, R., TOULOUSE, G.,

Phys.

Rev. Lett. 49 ₍₁₉₈₂₎ 1194.

[5]

ALEXANDER, S., ORBACH, R., J.

_Physique

Lett. 43 ₍₁₉₈₂₎ L-625.

[6] RAMMAL, R., to be

_published.

[7]

MONTROLL,

E. W., WEST, B. J., in Fluctuation Phenomena, Eds. E. W. Montroll, J. L. Lebowitz

(North-Holland) 1979.

[8] See for instance TOULOUSE, G. and PFEUTY, P., Introduction au _{groupe de renormalisation (PUG)}

1975.

[9]

ALLEN, P. B., J.

_Phys.

C 13

₍₁₉₈₀₎

L-667.

[10]

ABRAHAMS, E., ANDERSON, P. W., LICCIARDELLO, D. C. and RAMAKRISHNAN, T. V.,

Phys.

Rev. Lett. 42

(1979) 673.

[11]

STAUFFER, D.,

Phys.

Reports 54 ₍₁₉₇₉₎ 1.

[12] SHENDER, E. _F., J.

_Phys.

C 9 ₍₁₉₇₆₎ L-309.