Tempering Dynamics and Relaxation Times in the 3D Ising Model

HAL Id: jpa-00247134

https://hal.archives-ouvertes.fr/jpa-00247134

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Tempering Dynamics and Relaxation Times in the 3D Ising Model

Luis Fernández, Enzo Marinari, Juan Ruiz-Lorenzo

To cite this version:

Luis Fernández, Enzo Marinari, Juan Ruiz-Lorenzo. Tempering Dynamics and Relaxation Times in the 3D Ising Model. Journal de Physique I, EDP Sciences, 1995, 5 (10), pp.1247-1254.

�10.1051/jp1:1995195�. �jpa-00247134�

J.

Phys.

IHance 5

(1995)

1247-1254 OCTOBERI995, PAGE1247

Classification

Physics

Abstracts

75.10N

Tempering Dynamics and Relaxation Times in trie 3D Ising ^Model

Luis A. Fernàndez

(~),

Enzo Marinari

(~)

and Juan J. Ruiz-Lorenzo

(~)

(~ Departamento de Fisica Teônca, Universidad

Complutense

de

Madrid,

Ciudad Universitana, Madrid 28040

Spain

(~)

Dipartimento

di Fisica and

Infn,

Università di

Cagliari,

Via

Ospedale

72, o9100

Cagliari Italy

(~) Dipartimento di Fisica and

Infn,

Università di Roma La

Sapienza,

P. A. More 2, oo185 Roma Italy

(Received

26

January

1995, ^{revised 24}

April

1995,

accepted

30 June

1995)

Abstract. We discuss the

tempering

Monte Carlo

method,

and its cntical

slowing

down

behavior _m the 3D

Ising

model. We show that at Te the

tempering

does not

change

the critical

slowing

down exponent _z. We also discuss the

exponential slowmg

clown for the

flip-flop

from plus to minus state in the cold

phase,

and _we show that tempenng reduces it to _a power law

slowing

down. We discuss the relation of the

flip-flop

rate to the surface tension for local

dynamical

schemes.

1» Introduction

The

tempering updating

method

[Ii

has been introduced

recently

in order to

speed

_up sim- ulations of statistical

systems.

^Methods very similar in nature _were

already

in _use in the framework of molecular

dynamics

simulation [2]. Multicanonical methods _[3] also have

deep

similarities with the

tempering approach.

The main idea of

tempenng (see

also

[4-10])

is to let the

temperature

become _a

dynamical variable,

and to let it vary in the _course of the

dynamics. Tempering

^{is in} some sense a

sophisticated

kind of

annealing,

where the

system

^is

always

at thermal

equilibrium. Already

in its first

proposai

and

implementation

of reference

[Ii

the

tempering

method turned out to be very effective when

employed

to

study

the low T

phase

of the 3D Random Field

Ising

^Model.

There _{are some}

important advantages

in the

tempering approach (and

in this note _we will

try

to establish _{a new}

positive evidence).

The first _one is

maybe

that

tempering

^is very sim-

ple,

and that the

spin configurations

_one

_generates by

the

tempering dynamics

_are distributed

according

to the Boltzmann distribution. There is _no need for _any

weighting

or

reanalysis (im- proved

estimation schemes _can

obviously

be

employed).

We also want to stress that

tempering

parallehzation

is trivial. The

algorithm

is

local,

but for the _use of _{a rare}

global communication,

in the form of _an add-reduce function.

©

Les Editions de

Physique

1995

1248 JOURNAL DE

PHYSIQUE

I N°10

Apart

from the technical

advantages

we have

just

recalled there _are

important physical

reasons for which methods like

tempering

_are

becoming important.

Indeed _an

improved

Monte

Carlo

approach

_seems

crucial,

for

example,

for _an effective numerical

study

of finite dimensional

spin glasses [4,7-9, iii.

The

problem

of the behavior of three dimensional

system

^{is still}

quite

open, and

improved

numerical methods look here crucial. The main issue is the transition among different local minima of the free

energies.

In the _course of the

dynamics

the

system gets trapped

in local

minima,

and the

warming

_up allowed

by

^the

tempering

^favors the

exploration

of different local minima.

Here _we

study

the

normal,

_non

disordered,

3D

Ising

model. Here _in the cold

phase

there

are two states, ^and we will look at the transition _rate among these two states as

prototype

for _a

general

transition among local minima.

Ising

mortel is

interesting

since it is the

simplest possible mortel,

and since in this _case it is very clear what

going

from _a

given

state to another

one means

(one

_uses the

magnetization

to monitor the state to state

transition).

We will show

that

tempering

^allows one to reduce the

exponential slowing

clown local methods like heat bath

updating

encounter to a power law

slowing

down.

In Section 2 we define the

quantities

that _we will discuss in the

following.

In Section 3 _we discuss the autocorrelation time at the critical

point

^T =

T~.

We show

that,

_{as we} would have

expected,

in this _case

tempermg

cannot

change

the critical

exponent

z. In Section 4 _we discuss two main issues. First _we show that

by counting flip-flops

of _a local

dynamics

_{we can}

get

^hints about the surface tension. Second _we show that if _we let the

tempering

wander

(in fl space) only

in the cola

phase

the

exponential slowing

down _survives. In Section 5 _we show that if _we let the

tempering

_pass the border of T~ and

change

T among the two

phases

the

slowing

down becomes power law behaved. This is the main result of this note, and _we

shortly

describe its main

implications

_is Section 6.

2» A Few Definitions

In the

following

we will

give

_a few definitions _concerning

typical

time scales of the random

dynamics underlying

_our evaluation of the

Ising

model

partition

function. So

doing

_we will be

mainly following

Alan Sokal is very

comprehensive

book-like lecture _notes

[12].

Let _us start

by stressing

that correlation times _can be defined for the different observables

(A,

let _us

say)

of the

theory,

and

genencally they

will not need to be

equal.

We define the

e~ponential

correlation time

by

the

large

time

decay

of the connected correlation function

pAit)

e

(A(0)A(t)jc

cf

Cexpj- ^(~ j

,

(1)

TA ^~

for t _- cc, where

Ait)

is the value that the observable

quantity

A takes at the time t, ^and

we take the connected

part (indicated by

the

subscript c) by subtracting(~) (A)~.

As _we have

explained

_{we are}

exphcitly remarking

the A

dependence

of this time

scale,

and

by

the suflix

(exp)

the fact that it has been defined from the

exponential decay

of the correlation function.

This

asymptotic

^{behavior of the} correlation function is

corrected,

at finite

time, by sub-leading

contributions. We _can

have,

for

example, logarithmic

corrections _in the

exponent,

and _{a sum} of _a series of time

dependent

contributions to the correlation function.

Integrating p(t)

_over all times t we can define the

integrated

correlation time

(~ We _are assummg we have reached thermal

equihbrium,

and

our time _{serres is} invanant under time

translations.

N°la TEMPERING DYNAMICS IN THE 3D ISING MODEL 1249

oe

~~~~~

2P1°) ~Î~~~~~

^{' ~~~}

where _we have assumed _a discrete time

dynamics (the

extension _{to a} continuum time

dynamics

is

straightforward),

and the factor 2 makes the

integrated

time

equal

to the

exponential

time

in the _case of _{a pure}

exponential decay (at

ail

times)

of the correlation function.

The

integrated

correlation time has _a

special relevance,

since it is

strictly

connected to the statistical _error which affects the numerical data.

Indeed, again following [12],

one can relate

directly T~~°

to the true statistical _{error over} A. Let _us define a(~~~~ as the naive fluctuations of A. For _a

dynamics

of T

steps (starting

^after

thermalization,

1-e-

already

at thermal

equilibrium)

if

(A)

is the

expectation

value of A _we define

~~~~~

z~(At (A))~

~A ~

T(T i)

^'

(3)

that would be the _{true error over} A if the

configurations

collected

during

the

dynamics

_were

uncorrelated. On the

contrary

we call

a(U~

the true statistical _{error over} A.

a(U~

_can be estimated for

example by binning

the measured

quantities At

in bins

large enough

to make the

a estimated

independent

from the bin size. One finds that

~(ml)

_~ ^~A

j~)

A

r~~~e

^~

A

One _can

roughly

estimate

that,

_as far _as the value of the observable A is

concerned,

the number of

independent configurations produced by

the

algorithm

is

(

~~ ^m~j.

A

At _a critical

point,

where _a correlation

length (

is

diverging

when T _-

T~,

the correlation times do

typically diverge.

One

gets

that the

integrated

correlation time

T~~~~ ^~ Î~~~~~

là)

The usual

argument

^tells that _a local

dynamics

has _z >

2,

smce the

system

^is

undergoing

_a random walk in

configuration

_space.

Transmitting

information from site _{~ to} site y ^at distance à with _{a pure} random walk takes _a time of order

ô~.

_Since

we need to carry information at _a distance of order

(,

2 turns out to be _a lower bound for _z

(additional slowing

down

phenomena

can make _z

larger).

In order to

study

the

tunneling

relaxational

dynamics

_we ^define _a

tunneling

time _TT

by counting

the number of steps needed to the

system

to go from the

(let

_us

say) plus ground

state to the minus

ground

state and back. On _a limite lattice this definition _can be

plagued

from

ambiguities,

since it is not

completely

clear when the system has

completed

_a transition.

That defines _a

fl

window in which for _a

given

lattice size

things

_go

smoothly.

We _are in the broken

phase. fl

cannot be _too close _to fl~ ^{since in} this _case the transition

signature

^becomes

murky.

But

fl

_cannot be too

high

_{or we never}

_get

transitions

(which

_we call

jfip-jfop),

at least

when

using

the normal local Monte Carlo

dynamics.

In this paper we have tried _to work with

fl

values which

satisfy

this constramt. We dia not _seem to have

problems

in

defining

in _a

non-ambiguous

_way the transition from _a

ground

state to the other _one.

So,

_our

operational

definition of

tunnehng

time _was the time taken from the

magnetization

of the

system

to

change

sign

twice

(from plus

to minus and

back).

Since _{we were}

sitting

at _a value of T low

enough,

and the absolute value of the

typical

instantaneous

magnetization

_was

large enough,

this definition dia _turn out to be

non-ambiguous.

1250 JOURNAL DE

PHYSIQUE

I N°la

When

using

the

tempenng

^all correlation times _are defined

by

first

selecting

the

configura-

tions which _were characterized

by

the relevant

fl

value. That _means that the real

computer

time taken for _a

given tempered

measurement is

Np

times

larger

than the _{one we}

quote

^here.

The

scaling

of

Np

has

always

to be accounted for when

analyzing

the

scaling properties

^of the correlation times. The factor

Np

_one loses when

using tempering

_can be

regained by using

_any

histogramming algorithm,

the relevant

point being

that information _{at a}

given fl

value _is also relevant at _a

nearbying fl

value.

3» The Autocorrelation Time ai

T=T~

In this Section _we will discuss the behavior of the correlation time at T

=

T~.

We will

compare the usual local heat bath

dynamics (HB)

with the

Swendsen-Wang (SW) algorithm (which severely

reduces critical

slowing down).

We will

apply tempering

to both

approaches (obtaining

HBT and

SWT,

with _a

quite

^obvious

notation).

We will show

numerically

that in

both _cases

tempering

does not

change

the cntical

exponent

z.

It is easy ^to

develop

_an intuitive

argument(~) _suggesting

that

tempering

Carnot eliminate the critical

divergence

of the correlation time which is proper of the

underlying algorithm (for example

Heat Bath

dynamics).

It is clear indeed that if _we want the critical

slowing

down to be defeated _we have to let

fl

variate in the range

[flm(L), PM (L)]

such that the correlation

length

at the two extremes _is fixed in tattice units. But in order _to do that in the infinite volume limit _we will need _a

divergent

^number of

fin

values around fl~

(since

the

ôfl

allowed to

keep

fixed acceptance of the

fl change

will be

asymptotically infinitesimal).

In

slightly

more

quantitative

terms _one can start from the fact that the

optimal ôfl (which guarantees,

let us say, ^an

acceptance

^{factor of the order of}

50%)

is

[1, 4,

9]

~~~

~ ^{i 1}

(H2) jHj2 "1 ¹⁶⁾

1-e- is connected to the fluctuations of the internal energy of the system, ^and to the

specific

heat of the

system. Using

the fact that at the critical point ^the

specific

heat scales _as

cv

~

L"IV _ii)

we

get

that the

optimal

value of

ôfl

scales _as

L~i(~+

Il

(or using

the

scaling

relation _as

L~l _),

and goes ^to zero in the infinite volume limit. Notice that at fl~

+ôfl

the correlation

length

scales

as L and

consequently

with _a fixed number of

fl

values the _z

exponent

^should not

change.

If _we

try

to

keep

the

system

at _a fixed correlation

length

for _some

fl

the number of

fl

values needed

diverges.

The

argument

that _we have

developed

before will

apply

also in this _case

(fixed fl

limits and variable number of

fl values).

In order to check this scenario _we have simulated the 3D

Ising

model

by

_using the Heat Bath

and the Swendsen

Wang algorithm.

To bath of them _we

eventually

added _a

tempenng

part

(and

_{as we}

already

said _we denote the two modified

algorithms by

HBT and SWT

respectively).

In all _{cases we} have allowed 5

fl

values to the

tempering procedure.

The central

fl

value has

been

kept

fixed to fl~ m

0.22165,

and the

ôfl

has been

changed

when

changing

the lattice size.

We have used

ôfl

= 0.005158 for L

=

16, ôfl

= 0.001642 for L

= 32 and

ôfl

= 0.000527 for L = 64. So

flm(L)

= fl~

26fl(L),

and

flM(L)

= fl~ +

26fl(L).

This

scaling

of the

ôfl

values with L is not

optimized

but is

surely

reasonable. We do _trot

expect

that _{a more} careful

scaling

coula

change

the cntical behavior.

(~)We

thank

Sergio

Caracciolo for

a discussion of this point.

N°la TEMPERING DYNAMICS IN THE 3D ISING MODEL 1251

Table I. The number

of futt

lattice sweeps

(in

unit

of ^10~ ),

and _ouf best estimates

for

the _au- tocorrelation times. Ail _ruas haue been started ut

equilibrium, alter

_a number

of

thermalization sweeps.

L _{TE TM2}

16 4.7

32

with

Tempering

16 13.9 5.

Table II. Ouf results

for

the critical

dynamical _ezponents for

SW and SWT. We aise

report

the data obtained

by Woljf

with the Swendsen

Wang algorithm (SW(Woljf)).

Method i

zE zjmj zM2

SW(Wolff) 0.50(3) 0.50(3)

SW

0.54(2) 0.49(2) 0.51(2)

SW & T

0.46(2) 0.43(2) 0.44(2)

To estimate the autocorrelation times and its _error bars _we have used _a self-consistent _trun- cation window

algorithm

with width 6 Tjni,A as described in the references

[12,13].

We

give

the number of full lattice sweeps and our estimates for the

integrated

correlation time for SW and SWT in Table I. We have estimated the

integrated

correlation time for different observable

quantities (the

_energy, the absolute value of the

magnetization

and the

magnetization squared).

We have

analyzed

the data of Table I

using

the form

(5).

Dur best estimates for the _z values

are

given

in Table II. For

completeness

_we add the Wolff results obtained for the 3D

Ising

model with the

Swendsen-Wang algorithm [14,15].

Dur results for SW _are

compatible

with the Wolff results

jour

statistical

sample

_is 4 times

larger

than the former

one).

The values _we obtain for SWT _are smaller than the SW _ones, but

by

_{a very} small amount,

already quite compatible

if _we

only

consider the statistical _error. If

we also consider the

large (unknown) systematic

_error

(we

_are

asking

_an

asymptotic,

correction free value from three not so

huge

lattice

sizes)

_{we can}

safely

state that the _two sets of results

are

fully compatible.

The HB and HBT data _are also

completely compatible

with what _we

expected.

In the local

dynamics

_{z m} 2 in ail _cases, and the

tempering only changes

the non-universal constant factor.

So, tempenng

^does not

change

the

divergence

rate of the correlation times at trie critical

point.

This _was to be

expected,

since the

tempering

^is not

really changing

the critical

dynamics

of the system, ^{but is}

favoring

the

jumps

from _{one sector} of the broken

phase

to another. Next

we will

study

the correlation time in the broken

phase,

and will _see that here trie

tempering

can be _a crucial

help.

1252 JOURNAL DE

PHYSIQUE

I N°la

4. Below

T~i

the Relaxation Time when

Tempering

in the

Wrong

Phase

In this Section _we will obtain two results. One will be based _on trie HB method. We will show that

by measuring directly

the

tunneling

time

by

heat bath _we

get

an estimate of _a

quantity

that thanks to the

computation

of reference

[16]

is indeed connected to the surface tension

(even

if trie

quantitative agreement

^is trot

perfect).

The second result will be

negative (as expected),

and useful to prepare the

positive

result of the

following

section. We will show that

when

running

the

tempering

_{on a} set of

fl

values which remain in the cola

phase

and do not

enter the _warm

phase _{(1.e. flm}

>

fl~)

the

tempered

Heat Bath

algorithm undergoes

the usual

exponential slowing

down

(with

a coefficient _{very close to that}

one would

expect

to

get

^{at the}

highest

values of T induded in the values selected for the

tempenng scheme).

In the low

temperature phase

the

equilibrium

distribution between coexistent

phases

is

r~

exp(-cL~~~),

since the coexistent

phases

_are

separated by

free _energy barriers of order

L~~~ (surface

free

energy).

In this _case the conventional local

algorithms je-g-

Heat-Bath _or

Metropolis) undergo

an

extremely

_severe

slowing clown(~).

The autocorrelation

time, tunneling

or

ergodic

time in this context, ^behaves _as

T r-

eXp(cL~~~) (8)

So while at T~ trie

system undergoes

_a critical

"power" slowing clown,

for T _< T~ _an

"expo-

nential"

slowing

down

exists,

induced

by

the slow motion from _one pure ^{state to} the other.

The

previous

^formula

(8),

for the D.dimensional

Ising mortel,

_cari be deduced

through

^the

calculus of the _{mass gap, e,} trie inverse of the

tunneling

time

using

semiclassical methods and

considering only

the contribution of _one mstanton

[16].

^Une

gets

ⁱⁿ this way an estimate for the constant c of

(8).

The results is:

T r-

eXp(2a(T)L~~~ (9)

One _cari

identify a(T)

with the surface free _energy. The two m the

argument

of the

exponential

comes from the _use of

periodic boundary

conditions in the semiclassical calculus

(when using

periodic boundary

conditions _one has _{to create} two interfaces

).

This formula

(9)

is also relevant when

discussing

the

Ising

mortel _{m a}

cylindrical _geometry iii].

Dur results obtained

by using

the Heat Bath

dynamics

show that indeed the result

(9)

_cari

be substantiated

numerically.

For

example

at

fl

= 0.223 and lattice sizes L

=

10,16,

20 and 24

we fit _our results for T~~~l

by

ÎOgT # ai + 0E2L~

, ~~~~

and _we find _a X~

Id.o.f.

close _{to one,} and _{ai =} 4.52 +

0.04,

_{a2 "} 2a

= 0.0041+ 0.0002. A very

good fit,

and _a value of the surface energy very close to the best known value of 0.0025 from the simulations of reference

[18,19].

^{We also}

try

a power law

slowing

down

fit,

for

ÎOgT # ai + _£l2 ÎOg^L

, ~~

and _we

get

an unreasonable

X~/d.o.f

₌ 8.

Things

do not work _so well for lower values of T.

We _are still

seeing

the correct

exponential slowing clown,

but the number _we estimate for the surface tension does trot coincide with the number estimated in the literature. We believe the

(~)

In this context _we will not discuss about duster

algorithms,

_smce

they

_use the _a

priori knowledge

of the 22 symmetry

relating

^the two broken states to allow

a

(trivial) flip

from the plus to the _minus

state. in the context of the disordered systems, where there exist many

equihbrium

states _non related

by symmetry

operations,

this feature _is not

interesting.

N°la TEMPERING DYNAMICS IN THE 3D ISING MODEL 1253

main _reason for this mismatch is the poor estimation of trie

tunneling

time due to the little

number of

flip-flops

between the two minima and

consequently

the

large

error in the inverse of this number of

flip-flops,

1-e- the

tunneling

time. Also _we should notice that _{we are on a}

symmetric

cubic

lattice,

while _an

elongated

lattice would be needed for _a fair and accurate estimate.

We have _run trie

tempering

scheme in lattice sizes L

=

10, 16, 20,

22 and 24

by allowing

values of

fl

which do non lead the

system

in the _warm

phase.

The

system

is confined in the cold

phase,

and _we do not

expect

to make the

tunneling

^from _one

phase

to the other easier. This is indeed what

happens.

We simulate the

tempering algorithm

with five

temperatures

with

flm~x

₌ 0.225 fixed and

flm;n depends

_on the lattice

size,

for instance

flm;n(L

₌

24)

₌

0.223,

and _we

only analyze

the data

corresponding

to the lower

temperature (fl

=

0.225)

the best fit is

by

far the

one to _an

exponential slowing

clown

(10),

with

parameters

_ai

"

3.44(5),

_a2

"

0.0097(2)

with

x~ Id.o.f.

= 0.67. In the next section _we will discuss how does the effective

implementation

of the

tempering

^method

perform.

5.

Tempering

_among Cold and Hot Phase

As _a last step we have clone what had to be done. We have set up a

tempered

simulation where the

system

is allowed _to

flip

in and out the cold

phase.

The _{same way} of

reasoning

_we

have used before to show that

tempering

cannot eliminate the _power law critical

slowing

clown at T~ tells _{us now} that _on the

contrary

we cari

expect tempering

to transform the

exponential flip-flop slowing

down in _a power law behavior. Indeed here the number of

fl steps

^needed to

correct warm and cold

phase only

increases with _{a power} law

(this

is _a somehow unavoidable

feature of _a method which

improves

the

sampling scheme).

The

point

is that after _we have

flipped

to the hot

phase

_we loose _memory of the state we are

coming from,

^{and when}

going

back _to the cola

phase

_we will fall _{m a} random state.

So,

since the _way in which the allowed

fl

windows around fl~ shrink is dictated

by

_{a power} law _we expect to

only

find _a residual _power law

slowing

clown.

The results for the _pure heat bath

algorithm

are those _we have discussed in the

previous

section. We

get

there _an

exponential slowing

down

basically governed by

the surface tension.

In the

tempered

heat bath scheme _we simulate lattices L

=

10,16,

²⁰ and 24 and _we have allowed five

temperatures

with

flm~x

= 0.223 fixed and

flm;n, again, depending

_on the lattice

size,

_e-g-

flm;n(L

₌

24)

₌ 0.2212. We first select the

points

at the

higher fl

value

(in

order to be able _{to compare to} the heat bath

results). Again

_we

_try

the lits to the functional forms

(10),

and

(11).

In this _case the fit to _an

exponential slowing

clown is trot _a

good

fit. The

x~ Id.o.f.

is close to

15,

more than 50 times

larger

than trie best fit to _a power

dependence.

Trie power

we find in _our best fit

(a2)

is 1.76

+.06,

1e.

TCfL~'~~ (12)

The values _on which the fit is based _are

T(L)

=

58, 129, 204,

274

respectively

for L

=

10, 16,

20,

24. The _{error we} have

quoted

before is

purely

statistical. The

systematic

_error on this

number _is far

larger (we

have

only

4 not _so

large

L

values).

We consider _our evidence to be substantial for

establishing

_{a power} law

behavior,

but not for

determining

^{the critical}

exponent

with _a

good precision.

Let _us make clearer that in the

large

volume limit the number of

fl

values needed to tunnel from the low T to the

high

T

phase

will _increase, but

only

_{as a power} law. So _we will trot

get

back _an

exponential slowmg down,

but

only

_{a power} law like _one. On the other side the exact

1254 JOURNAL DE

PHYSIQUE

I N°10

pattern

of the

rescaling

of such limits and

ôfl

values will not

change

the conclusions of this section.

6» Conclusions

Dur numerical

experiments

have shown that the

tempering

method does transform _{an expo-} nential

slowing

clown into _a power law effect. Even if _we have discussed the

tunneling

in the cola

phase

of the 3D

Ising mortel,

where these states _are related

by

_a

22

_{symmetry, our con-}

clusions do not

depend

_on such _a

symmetry.

^{The fact that}

tempering

works

remarkably

well for

speeding

_up 3D

spin glass

simulations in the cola

phase ii,

_8] is connected to the behavior

we have discussed here.

Acknowledgments

We thank

Sergio

Caracciolo for _an

interesting discussion,

and for

communicating

to _us the

unpublished

results of

[20].

^{We also}

acknowledge

relevant communications with Ferdinando Gliozzi and

Margarita

Garcia. JJRL is

supported by

_a MEC

grant (Spain).

LAF and JJRL

acknowledge CICyT (AEN93-776)

for

partial

financial

support.

References

[1] ^{Marinan E. and Parisi} G., Europhys. Lett. 19

(1992)

451.

[2] ^Torrie G. M. and Valleau J. P.,

Camp. Phys.

23

(1977)

187.

[3]

Berg

B. and Neuhaus

T., Phys.

Lent. B 267

(1991)

249;

Phys.

Reu. Lent. 68

(1992)

9.

[4] Kerler W. and

Rehberg P., Simulated-Tempering Approach

to

Spin-Glass Simulations,

cond- mat

/9402049 (February 1994).

[Si ^Kerler W. and Weber A.,

Phys.

Reu. B 47

(1993)

11563.

[6] ^Marinan

E.,

Parisi G. and Ritort F., J.

Phys.

France 27

(1994)

2687.

[7] ^Marinan

E.,

Parisi G. and Ritort

F.,

to be

published.

[8] ^Marinan

E.,

Parisi

G.,

Ritort F. and Ruiz-Lorenzo

J.,

to be

published.

[9] ^Bartocciom B., Il Modello di Vetro di Spin di

Heisenberg

in 4 Dimensioni, Tesi di Laurea, Univer- sità di Roma La

Sapienza (July 1994);

Numencal Simulations _on the 4d

Heisenberg

_spm

glass.,

cond-mat/9410053.

[loi

Vican E.,

Phys.

Lent. B 309

(1993)139.

[Il] Berg

B. and Celik

T., Phys.

Reu. Lent. 69

(1992)

2292; int. J. Med.

Phys.

G3

(1992)

1251.

[12] ^{Sokal A.}

D.,

Monte Carlo Methods _m Statistical Mechanics: Foundations and New

Algonthms,

Cours de Troisième

Cycle

de la

Physique

_en Suisse

Romande, (Laussane,

June

1989).

[13] ^{Madras M. and Sokal} A.D., J. Star.

Phys.

50

(1988)

109.

[14] ^{Swendsen R. H. and}

Wang

J.

S., Phys.

Re~. Lent 58

(1987)

86.

[15] ^Wollf

U., Phys.

Lent. B 228

(1989)

379.

[16] ^{Niel J. C. and} Zinn-Justin

J.,

Nucl.

Phys.

B 280

(1987)

355.

[17] Zinn-Justin J.,

Quantum

Field

Theory

and Critical Phenomena,

(Oxford University

Press, Oxford

U-K-, 1989).

[18]

Gupta

S.,

Phys.

Lent. B 325

(1994)

418.

[19] ^{Caselle M. et} ai.,

Rough

Interfaces

Beyond

the Gaussian

ApproXimation,

cond-mat

/

9407002.

[20] ^Caracciolo S., Pehssetto A. and Sokal

A., unpublished.