Maghetization Distribution on Fractals and Percolation Lattices

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Maghetization Distribution on Fractals and Percolation Lattices

R. Mélin

To cite this version:

R. Mélin. Maghetization Distribution on Fractals and Percolation Lattices. Journal de Physique I,

EDP Sciences, 1996, 6 (6), pp.793-806. �10.1051/jp1:1996237�. �jpa-00247215�

Magnetization Distribution

_on

li~actals and Percolation Lattices

R. M41in

(*)

CRTBT-CNRS, BP 166X, 38042 Grenoble Cedex 9, France

(Received

29 December 1995, received in final form 4 March 1996, accepted 6 March 1996)

PACS.05.50.+q Lattice theory and statistics; Ising problems PACS.05.70.-a Thermodynamics

PACS.05.70.Ln Nonequilibrium thermodynamics, irreversible _processes

Abstract. We study the magnetization distribution of the Ising model _on two regular frac- tals

(a

hierarchical lattice, the regular simplex) and percolation clusters at the percolation thresh- old in _a two dimensional imbedding _space. In all these _cases, the only fixed point is T

= 0. In

the case of the two regular fractals, _we show that the magnetization distribution is _non trivial below T* _ci A*

In,

with _n the number of iterations, and A* related to the order of ramifica- tion. The _cross-over temperature ^{T* is} to be compared with the glass _cross-over temperature Tg m Ag

In.

An estimation of the ratio T*

/Tg

yields _an estimation of the order of ramification of bidimensional percolation clusters at the threshold

(C

= 2.3 +

0.2).

1. Introduction

We consider the

problem

of

Ising

spin systems on a

family

of fractals with _{a zero} temperature fixed

point.

More

especially,

_we consider the _case of _a hierarchical

lattice,

the

regular simplex (regular fractals),

and bidimensional

percolation

clusters at the

percolation

threshold. This

problem

has its

origin

in the

pioner

work of Mandelbrot

[I].

^The

problem

is also of _{an ex-}

perimental

interest because of the existence of random

magnets

^with a fractal structure

(see

for instance

[2]).

As _a

byproduct,

_we derive _{a measure} of the ramification order of

percolating

clusters. To do _{so, we compare to cross-over} temperatures: 1) ^the

glass

_cross-over temperature

Tg, 2)

^the

magnetization

distribution _cross-over temperature T*. The dominant behavior of

the

glass

cross-over temperature _was discovered in the 80's

[3,4].

^This quantity is related to the correlation

length

of the magnet. ^Below

Tg,

^the correlation

length

is

larger

than the system size,

leading

to _a linear

regime

of the

logarithm

of the relaxation time _{as a} function of the inverse temperature. Above

Tg,

^the correlation

length

is smaller than the linear size of the

cluster, leading

to a

parabolic

behavior for the

logarithm

of the relaxation time _{as a} function of the inverse temperature. ^{This behavior is well} established _{on a} numerical basis [5]. ^The

second _cross-over temperature T* _was not much studied in the past. ^Below T*

(which

is inverse

proportional

to the

logarithm

of the number of

sites),

the

magnetization

distribution is _non Gaussian. These _non Gaussian

magnetization

distributions also _{accur on} the

ferromagnetic Cayley

tree [6], ^{however with} a different

decay

with the system ^size. More

surprising

is the appearance of similar

magnetization

distributions in

magnetized

spin

glasses

_[7].

However,

in (* ^{Author for} correspondence

(e-mail: [email protected])

@ Les

(ditions

_{de Physique 1996}

z

a~

Z~

n=0 n=1

Fig. I. Construction ofthe hierarchical lattice. At the _{n =} 0 step, we start with two sites connected.

After _one step, ^{the lattice contains 6 sites. At} each step, the links _are transformed _as in the _n

= 0

-

n = I transformation.

real

spin glasses,

this _non Gaussian

magnetization

distribution is not

expected

to vanish in the

thermodynamic

limit _as it is the _{case on}

ferromagnetic

systems. ^{We first}

study

the _case of reg- ular fractals

(a

hierarchical lattice and the

regular simplex).

We _use the Swendsen

algorithm

to generate

equilibrium

states [8], ^{and thus} to calculate the

magnetization

distributions. In the _case of the

regular simplex,

it is

possible

to determine the

geometric origin

of the different local maxima of the

magnetization

distribution at low temperatures. ^In Section 4, we make _a

general

argument to estimate the

magnetization

distribution _cross-over scale, which is shown to be inverse

proportional

to the number of sites, and

proportional

to the ramification order C. In the _case of

percolating

clusters, we calculate the

magnetization

distribution at low tem-

perature.

Averaging

over the geometry, we can see the existence of finite size effects: the spin system ^is more

ferromagnetic

if _one lowers the number of

generations.

We also studied the correlation between different

equilibrium

_states, which

gives

us an other _way to estimate T*.

Finally,

_we calculate the ration Tg

/T*,

which

yields

_an estimation of the ramification order of bidimensional

percolation

clusters. We end up with _some

concluding

remarks.

2. A Hierarchical Lattice

The interest of hierarchical lattices is that the

thermodynamics

of the

Ising

model is

exactly

soluble _on these structures. We consider the _case of the hierarchical lattice built with the rules of

Figure

1. The coordination of _every site is bounded above, which is _not

always

the _case for hierarchical lattices

(for

instance for the hierarchical

diamond,

in which _case the coordination

diverges

in the

thermodynamic

limit

).

The number of sites of the hierarchical lattice of

Figure

is Nn ₌ 4.6"

IS

₊ 6

IS,

whereas the end _to end distance is Ln ₌

4"~l, leading

to the fractal dimension d

= In 6

/

In 4.

An

interesting

aspect ^of hierarchical lattices is that their

Yang

and Lee _{zero can} be calculated

easily

in the temperature

plane.

This _was done in [10] ^{in the} case of the hierarchical diamond.

The

Yang

and Lee _zeros set is _a Julia set. For comparison with the results of [10], we also

calculate the

Yang

and Lee _zeros of _our hierarchical lattice.

First,

_we need to derive recursion relations for the

partition

function.

2. I. RECURSION _{FOR THE} PARTITION FUNCTION. The

technique (standard

for hierarchical

lattices)

consists in

carrying

out the _{trace over} the the

spin

variables at the smallest

scales,

and to derive the renormalization of the temperature and

magnetic

field. The

partition

function of

the hierarchical lattice with _one

generation

is

zi(z, if)

₌

~j ePJia0~+1«0+a311ai+a2)1+«3z'ePHia0+...+a3), (1)

°oi%,°2,°3

where the spins _are

pictured

_on

Figure

1. A

straightforward

calculation leads to

Zi(L, L')

= 2 (e~~~ ^cosh

(flJ(L

₊

L')

₊

4flH)

₊

^e~~~~

cosh

flJ(L

+

L')

+ cosh

(flJ(L L')

₊

2flH)

₊ cosh

(flJ(L L') _{2flH) (2)}

+2 cosh

flJ(L L')

_{+ 2} cosh

(flJ(L

+

L')

₊

2flH)

The

partition

^function

Zi(Z, L')

_can be rewritten under the form

Zi(L, L')

=

~ife~~~~'e~~~~+~'l (3)

We have introduced here three parameters ~if,

I

_and

k

for three distinct

partition

functions

Z(+, +), Z(-, -)

and

Z(+, -)

=

Z(-, +).

The

partition

function _can thus be

consistently

be

brought

under the form

(3).

The identification of

(2)

^and

(3)

leads to

M

=

(zi(+,+)zi(+,-)~zi(-,-))~~~ ₍₄₎

~~~

p j

^M

(zi(+,+)zi(-, -))~/~

zi(+,-)

M ~~~

exp4pit

=

j~)~'~~. ⁽⁶⁾

In the _case of _{a zero}

magnetic field,

_we obtain

zn(J)

=

M~~~~zn-i(J), ₍₇₎

with

~if~

= 8

(cosh 6flJ

₊ 3 cosh 2flJ +

4) (cosh

4flJ + 2 cosh

2flJ

₊

1)

,

(8)

and

~~j cosh

6flJ

+ 3 cosh

2flJ

_{+ 4}

~ 2

(cosh 4flJ

₊ 2 cosh 2flJ + 1) ^~~~

The renormalization

equation (9)

^has

only

_one fixed

point:

J

= 0, so that the

spin

system is

always

disordered at any finite temperature ^{in the}

thermodynamic

limit.

2.2. DIGRESSION: YANG _AND LEE ZEROS IN THE TEMPERATURE PLANE. We _now cal-

culate the

Yang

and Lee _zeros in the temperature

plane.

This is not the main

subject

of this paper, but it is

interesting

to compare with

existing

results _on the hierarchical

diamond,

where the

Yang

and Lee _zero in the temperature

plane

form _a Julia set [10]. ^{We work in the}

plane

of the variable _z

=

exp2flJ

and invert the relation

(9),

which leads to _a

polynomial

^of

degree

6:

z~

2iz~

₊

(3 4i)z~

₊

4(2 I)z~

+

(3 4i)z~

2ix ₊

= 0,

(10)

If _z is a solution of

(10), 1/z

is _a solution too, _so that the set of _zeros of the

partition

function is invariant under the invei'sion _{z ~} l

/z.

In order to compute the set of _zeros of the

partition

function, _{we use} the methods of [10], ^{that is} we start from _a

given

point in the

complex plane,

we calculate the _zeros of

(9),

choose _one solution _among the six _zeros at random, ^{and reiterate.}

6

~

_9' ,/

_--'~~

2 J

--~

o

-6

0 0.5 1 1.5 2 2.5 3

Fig. 2. Set of _zeros of the partition function in the plane _z

= exp 2pJ We have plotted 39000 _zeros calculated by the procedure described in the text. 1000 _zeros have been eliminated.

0.04

0.03

0.02 ~_-.

_~d'

~, 0.01

,.

-"~~

o

-0.01 _.-.--.--,~

',.,_

'~~~_

-0.02 '~

-0.03

-0.04

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Fig. 3. Zoom of the _set of

zeros in the vicinity of the origin, for 19000 _zeros of the partition function.

The density of points corresponds to the density of _zeros. The density of _zeros vanishes in the vicinity of the origin.

The _{zeros are}

computed using

a

Laguerre's

method [11]. ^The

resulting

set is

independent

_on the initial value of _{z, as} far _{as one} eliminates the first _zeros. The result is

plotted

_on

Figure

2.

The set of _{zeros are} concentrated _on

lines,

which intersect the real _axis in the

vicinity

of the

point

_{z =} 0. A _zoom in the

vicinity

of _z

= 0 reveals that the

density

of _zeros vanishes _{as one}

approaches

the

origin (see Fig. 3). Contrary

to the _case of the hierarchical diamond, the Lee

and

Yang

_zeros are concentrated _on lines in the _z

=

exp2flJ plane,

whereas the Julia set of the hierarchical diamond is _a fractal.

2.3. GLASS CRoss-OVER TEMPERATURE. The

glass

temperature cross-over Tg ^{is related}

to the

decay

of the correlation function from _one end of the fractal to the other

[3,5,

_9].

Since the

Ising

model is soluble _on the hierarchical lattice, it is easy ^to get the behavior of the correlation

length

between the extremal sites _{as a} function of the number _n of iterations and the temperature. We call _x the dimensionless inverse temperature _{x =}

pJ.

In the low

temperature

regime,

_we

get

^from

equation (9)

_{xn+i Cf xn} In

2/2,

where _xn is the value of _x after _n renormalization steps. ^In the low temperature

regime,

_{xn+i ~bf}

2x(.

The limit between the

high

and low temperature

regimes corresponds

to the minimum _K of 2x~ _{x +} In

2/2.

We find K ci 0.59. The correlation between the _two extremal sites is

nothing

but

iii')

=

~

Z[p

₌ 0],

(ii)

Z[°) °P

where

Z[p]

c~

~j e~~n~~'e"~~', (12)

I,z,

so that the correlation is

simply (it')

₌

tanhpJn,

which decreases from unity to _{zero as} n

increases from I to +oJ. The _cross-over temperature Tg

corresponds

to _xn > K and _xn+i < K,

leading

to the dominant behavior of the

glass

cross-over temperature

~ n~~2

^~~~~

2.4. MAGNETIzATION DISTRIBUTION. The

magnetization

distribution becomes _non gaus- sian below _a temperature ^{T*. This} is due to the existence of

macroscopic

domains that _are

weakly

connected ot the rest of the structure. The temperature cross-over T* is evaluated in Section 4 in _{a more}

general

context, _so that _we do _not

reproduce

the argument ^{here in the} case

of the hierarchical lattice. The result in the _case of the hierarchical lattice is

~~

n In 6' ^~~~~

where _we have

kept only

the dominant behavior for

large

n. The results for the

magnetization

distribution _are

plotted

on

Figure

4 for

n = 4.

Using (14)

for _n

= 4, _we get T* ci 0.55, which is consistent with the data of

Figure

4. Notice

that,

in this _case, Tg < T*.

3.

Regular Simplex

The

regular simplex

recursion is

plotted

_on

Figure

5. The number of sites if Nn ₌ 3~, and the

corresponding

distance is Ln

= 2", so that the fractal dimension is d ₌ In

3/

In 2.

lo

T"0.5,0.6,0 7,08

8

IC

I 6

E

)

Ic

fl

% ₄

~

2

0

0 0 2 0.4 06 08

Normalized number of up spins

Fig. 4. Magnetization distribution for the hierarchical lattice with 4 generations. For clarity, the

plots have been shifted along the y axis. 750000 iterations of Swendsen algorithm _were carried out.

Z~

a~~

~

~23

~2 _{~12 ~13} 23

Fig. 5. Construction of the regular simplex.

3.I. RECURSION FOR THE PARTITION FUNCTION. The partition function in _{a zero exter-}

nal field _can be calculated

recursively by calculating

the trace _over the

spins

at the

deepest generation.

With the notations of

Figure

5, the partition function is

zjL~, L~,

L~ _{= exp}

(pJ(ii (a12

+ a13 +

L2(a21

+ _a23 +

L3 (°31

+ _°32

)) (IS)

exp

(flJ(a31a32

+ _a21a23 + _a12a13

(~~)

We look for Z under the form

z(Li, L~, L3)

₌

Mexp (flJ(LIL~

^{+ L~L~ +}

^LiL3 )) ⁽¹⁷⁾

The renormalized form of Z has two parameters

(N

and

I),

_{for two distinct}

_equations:

one

for

(ii, L2, L3)

E

Si,

and the other for

(ii, L2, L3)

E

52,

with

Si =

i(+,+,+),(-,-,-)1 (18)

52 =

1(+,+,-),(+,-,+),(-,+,+),(-,-,+),(-,+,-),(+,-,-)I (19)

The parameters M and

I

are

simply

given

by

Carryingoutthe trace

over

_thi

a

ariables

leads

_to

Z(+,

_+,

+)

= 27e~~~~ ₊

27eP~

₊

9e~P~

₊

e~~~ (22)

Z(+, +,

-)

=

3e~"~

+

19e~~~~

+ 33e~~ + _e~~~

(23)

In the

high

egime, the ^onstant

is

fl _ci

x~ and, in _the

low temperature

regime,

fl _ci

x - (In ₃₎ /2, where x =

The

between

_{the high}

and

low temperature _egime

orrelationunction

~~~

~~l~j ~K

~l

_~XP

_(Xn(~1~2

_{+ ~l ~3 +}

_~2~3)

_eXp

_(J1~i~2)

=

~~~~ _~ ~~

,

(25)

xi,z~,z~

e3xn ₊ 3e~xn

from what _we deduce the dominant behavior of the _cross-over temperature ^scale

~

nln3' ^~~~~

3.2. MAGNETIzATION DISTRIBUTION. The

magnetization

distribution is

plotted

_on

Fig-

ure 6 for various temperatures.

Again,

_we do _not

reproduce

the calculation of the

magnetization

cross-over temperature ^T* since _a

general

argument in

developed

in Section 4. The result for

the dominant behavior of T* in the limit of

large

n is

~

nln3 ~~~~

In the _{case n}

= 7

(which corresponds

to

Fig. 6),

_we have T* _ci 0.8, which is consistent with the data of

Figure

6. Notice also that, ⁱⁿ the low temperature

regime,

when the _non Gaussian

structure of the

magnetization

distribution is

fully developped,

it is

possible

_to

identify

the

origin

of the different local maxima of

P(m).

To do _{so, we carry} out a low temperature

expansion

of the

magnetization

distribution around the broken symmetry state m = I, _up to the order

p~,

where _p

=

e~P~ /(eP~

₊

e~P~)

_ci

^e~~P~

The result is

n-i

P(m)

₌

(I np~ Ap~)b(I m)

₊

3p~ ~j _b(m

₁₊

2.3P~") (28)

p=o

»-2 ~ ~

+p3 p=o £(3"~P 3)b(m

+

2.3P~")

+ 6

~j _{b(1 ~)}

₊

_o(p~

_).

₍₂₉₎

MeA

~

lo

T"0 7,08,_Q-Q,

8

Ic

I ₆

(

e

Ic

fl

I 4

~

~

2

0

0 02 0A 0.6 0.8

Normalized number of up spins

Fig. 6. Magnetization distribution for the regular simplex ^{with 7} generations. For clarity, the plots have been shifted along the _y axis. 750000 iterations of Swendsen algorithm _were carried out.

A is _a normalization constant and the set A is

A =

(M

_E

(I,..

,

[N/2]).

3k, 31,o I < k, M

= 3~ +

3~). (30)

We _can

recognize

_on

Figure

6 the contributions from _m

=

1/3,1/9,1/27

but also from _m

1/3

+

1/27,1/3

+

1/9,1/9

+

1/27.

4. General

Argument

for

Regular

Fractals with

a Zero Temperature ^Transition We first recall what

happens

on the

Cayley

tree [6], ^{where there exists also} a

glassy

cross-over,

with Tg c~

J/

In _n, where _n is the number of

generations

of the tree, and also _{a crossover} in the

magnetization

distribution, with T* _c~

J/

In _n. It is

possible

to derive exact recursion relations for the _average

magnetization,

to solve the recurence, and to obtain the temperature cross-over

T*. The _cross-over temperature is

interpreted

_as follows: below T*, there exists less than _one

broken bound _{on a}

path connecting

the _center of the tree to the leaves, which leads to the criterium for the

magnetization

distribution _cross-over temperature:

np ci1.

(31)

Taking

the

logarithm

leads to T* _c~

J/lnn.

In the _case of

fractals,

^there are not

bifurcations,

so that the criteriuni

corresponding

to

(31)

is

pCN»

_c~ 1,

(32)

which _means

that,

at the cross-over, there exists

only

C broken bounds _on the

graph,

where C is the number of bounds that need to be cut to isolate _a p

generations

^fractals in the bulk

3

2.5

~ 2 3

%

$

~ 15

$

$ + 8 6

05

0

2 3 4 5 6

Number of generations

Fig. 7. Inverse temperature _{cross-over as a} function of the number of generations for the hierarchical lattice. The _cross-over is such that the average magnetization is _some constant, the two extremal spin being frozen in the _up direction. The constant is taken equal to 0.02.

of _{a n}

generations

fractal. This is the ramification order. We thus obtain

T*

=

j ₍₃₃₎

In order to check this

equation,

_{we come} back to the _case of the hierarchical

lattice,

where _we obtained

recursively

the

partition

function in _an external

magnetic

field. We impose _a small

magnetic

field, and calculate the

partition

^function in the _presence of the small

magnetic field,

and deduce the _average

magnetization numerically

_as a function of the number of generations.

For _a

given

number of

generation,

we look for the temperature ^for which the

magnetization

is

a

given

constant. We thus obtain the _curve

fl*(n),

which is

expected

to be _a

straight

line from

equation (33).

This is indeed the case, as shown _on

Figure

7, which validates the

previous

heuristic argument.

5. Percolation Clusters at the Threshold

5. I. GLASS CRoss-OVER TEMPERATURE. Percolation clusters at the

percolation

threshold

were studied in the past as an

exemple

of critical

dynamics,

with _a

glassy-like dynamics

below Tg

[3,5,9].

The

glass

_cross-over temperature Tg ^{is evaluated in} [3] with the argument ^that

below

Tg,

^the correlation

length

_[12]

(T

_{= exp}

l~ ^{j~~ (34)}

TW.34, 0.42, 0.50,058, 0.66,0 74 9

8

7

C 6

I I

5 c c

°

fl 4

~

i~

3

2

1

0

0 0.2 0 4 0.6 0.8

Normalized number _{of up} spins

Fig. 8. Magnetization distribution for

a percolation cluster of size N

= 2660 sites

(this

cluster is

pictured _on Figure 9). The temperature is T ₌ 0.34, 0.42, 0.50, 0.58, 0.66, 0.74. 750000 iteration of the Lanczos algorithm _were carried out.

is lower than the linear size of the cluster. One gets

easily

_[3]

2fivp

Tg "

fi' (35)

where N is the number of sites of the

percolation

cluster.

5.2. MAGNETIzATION DISTRIBUTION. The

magnetization

distribution of _a

single

^cluster at low temperatures looks like the _one of

regular

fractals, apart from the fact that the localization of the local minima

depends

_on the geometry and is not easy to localize. For

instance,

_we

generated

_a

percolation

cluster at threshold in _a bidimensional

imbeding

space (p~ ₌

o.593).

The

corresponding

low temperature

magnetization

distribution is

plotted

_on

Figure

8 for _var- ious temperatures. We show _on

Figure

9 two

configurations belonging

to the local maximum with _a normalized number of _up

spins Nt IN

ci o.38. The

interpretation

of the

magnetization

distribution thus

depends crucially

_on the geometry. It is therefore

legitimate

to ask whether the local maxima structure

persists

after

averaging

_over the geometry. The _answer is 'no': _we

plotted

_on

Figure

lo the

magnetization

distribution

averaged

_over the geometry for different

temperatures. ^We observe _no local _maxima. If the temperature

increases,

the

weight

^of the

ferromagnetic

maximum

decreases,

whereas the tail of the distribution

increases,

which is _con- sistent with the fact that the

magnetization

distribution is Gaussian in the

high

temperature

regime.

Moreover, by averaging

_over the geometry, we can adress the question of finite size effects.

We

plotted

_on

Figure

11 the distribution of

magnetization

for two different _{sizes at} the _same temperature. We _see that the small clusters have _{a more}

pronounced ferromagnetic peak,

which is in agreement with the fact that T* decreases with the system size.

90 90

80 80

70 70

50 50

40 40

30 30

20 20

lo io

0 0

0 lo 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

Fig. 9. Two spin configurations in the peak Ni/N _cf 0.38.

iooo

T=.5 T=4 T=.3

ioo

Ic

I(

fi

8 lo ;."

=

fl fl

~

l

Q-1

0.9 091 0.92 0 93 094 0 95 0.96 0.97 0.98 0 99

Normalized number of_up spins

Fig. lo. Magnetization distribution averaged _over the geometry. ^{The clusters} are generated in _a 60 _X60 box, and 100000 steps ^{of Swendsen} algorithm _are carried out. For T

=

0A(0.3, 0.5), 106(200,

58) different geometries _were generated. Notice the semi-log scale.

5.3. CORRELATIONS. We wish to

analyze

the correlation between the different

spin config-

uration at _a

given

temperature. ^{To do} so, we introduce _n

replica

of the

spin

system. We call

q)"I

E

(0,1)

the value of the Potts variables in the

replica

_a

(1

< _a <

n),

and _we consider the

T=0.5 14

25x25 60x60 12

io

Ic

I

E 8

(

c

fl ₆

I

~

~

4

2

0

0 6 0.65 0 7 0.75 08 Q-W 0 9 0.95 1

Normalized number of up spins

Fig. ll. Magnetization distribution averaged _over the geometry. ^{The clusters} are generated in

a 60 X 60 box and _a 25 X 25 box, and 100000 steps of Swendsen algorithm _are carried out. The

temperature ^is T

= 0.25.

f~uowing

correlation:

Rp~iq, T)

=

( _{~~~_}

~~

l~ ql"'ql~~

l~~~

~=i laPi

for _a

given

geometry with _a

given magnetization (all

the

replica

have the _same

magnetization).

The variable _{q is} the _average

magnetization:

N

To,

_q

=

j ~j

_q)°1

₍₃₇₎

1=1

The correlation RN

(q, T)

in _a sector of

given magnetization

is

nothing

but _a kind of Edwards- Anderson order parameter in _a

subspace

of

given magnetization.

In the

high

temperature

regime, qf

is 0 with _a

probability 1/2

and I with _a

probability 1/2,

_so that the _average of

q)"~qj°~

^is

1/4,

_so that

RN(q, t)

₌

) ₍₃₈₎

q

if T _> T*. We _are interested in the variations of R for _a

given magnetization

_{as a} function of the temperature. These variations _are

plotted

_on

Figure

12 for _a

given

geometry. The

magnetization

is _q

=

Nt IN

_ci 0.38. The four different _{curves are} plotted for

neighboring magnetizations.

^As

expected,

R tends _to 0.38 above T*. The estimation of T* from this

technique

is coherent with the estimation from the

analysis

of the

magnetization

distribution

(see Fig. 8).

Nup=1753 +-

Nup=1754 -~-

Nup=1755 a

0.9 Nup=1756 x

Nup=1757 --

0.8

x~,,

0.7

06

o.5

04

0.3

015 0.2 0 25 0.3 0.35 0 4

T/2

Fig. 12. Variations of the correlation R _{as a} function of the temperature for the cluster of Figure 9.

The _average number _{of up} spins is chosen _to be around 0.38.

5.4. EVALUATION oF THE RAMIFICATION ORDER _oF PERCOLATING CLUSTERS. We know

from _two different methods that the _cross~over temperature T* for N

= 2660 sites is about T* _ci 0.6. From this data, _{we can} deduce the ramification order C,

which,

in the _case of

regular

fractals _was

interpreted

_as the number of links _to be cut to isolate _{a p}

generations

^fractal inside

a n

generations

fractal

(n

_>

p).

We deduce from

(33)

and

(35)

that

(

=

~. (39)

Upd

On the other hand, _we know that up ci 1.3 in _a bidimensional

imbedding

_{space [13],} and that d ₌ 1.78 + 0.02

(14].

With N

= 2660 sites, we evaluate Tg ci 0.6 from

(35). Using (39),

_we get ^C = 2.3 + 0.2. The

incertainty

_comes from the

incertainty

ⁱⁿ the location of the _cross-over

temperature T*. This result is in agreement ^with the fact that the order of ramification of

percolating

clusters is believed _to be finite and

superior

to two

(percoliting

clusters _are not

quasi one

dimensional)

_[15].

6, Conclusion

We have studied the

magnetization

distribution _on fractals with _{a zero} temperature transition.

Below _a temperature ^inverse

proportional

to the

logarithm

of the number of sites, and _propor- tional to the order of ramification, the

magnetization

is not

trivial,

with local maxima. This is due _to the fact that is is

possible

to cut the

graph

into many

large

_parts,

by breaking only

a small number of links. We have introduced _a

magnetization-dependent

Edwards-Anderson order parameter, the variations of which _are correlated with the

magnetization

distribution. It would be

interesting

to have _an

analytical proof

of this fact. We have shown that the _cross-over

temperature ^T* of

percolating

clusters is related to the order of ramification of the _structure.

We find C

= 2.3 + 0.2 for bidimensional

percolation

clusters. The _cross-over temperature T*

can be estimated in two different _ways: 1)

by calculating directly

the

probability

distribution at different temperatures,

2) by calculating

the correlation between

equilibrium

states with _a

given magnetization.

Our result is in agreement ^{with the fact that the}

percolation

cluster _at threshold is not

quasi

_one dimensional

(which

_was known _a

long

time ago I).

However,

it is clear that _our method _to determine the order of ramification is _not accurate, since _one has to estimate _{a cross~over} temperature, ^{which is determined} up ^to a certain

incertainty.

It would be

interesting

to

generalize

this

study

to

magnetized spin glass phases.

This will be done in _a

near future.

Acknowledgments

acknowledge

J.C.

Anglbs

^{d'Auriac who} lent _me his Swendsen _program, and I thank B.

Dougot

for comments on the

manuscript.

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