On the replica approach to random directed polymers in two dimensions

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On the replica approach to random directed polymers in

two dimensions

Giorgio Parisi

To cite this version:

On the

_replica

_approach

to

random

directed

_polymers

in

two

dimensions

Giorgio

Parisi

Dipartimento

di Fisica, INFN sezione Tor

_Vergata,

II Università di Roma Tor

_Vergata,

Prolungamento

di Via del Fontanile di Carcaricola, Roma 00173,

Italy

(Reçu

le 8 janvier 1990,

accepté

sous

_{forme définitive}

le 17 avril

₁₉₉₀₎

Abstract. 2014 In this note _we

study

directed

polymers

in a two dimensional random medium with short range noise

using

the

_{replica approach.}

We find the

_predictions

of the

_{replica symmetric}

theory

and we compare them with exact results. We consider the

_possibility

of _spontaneous

symmetry

breaking

and we _suggest that

replica

symmetry is

weakly

broken also in this two

dimensional model. Classification

Physics

Abstracts 05.40

1. Introduction.

The

_study

of directed

_polymers

in a random medium is very

interesting [1].

In a nutshell the

problem

of directed random

_polymers

consists in

_finding

the

_probability

distribution of

where

_{d W[ w}

_{J xt}

_is

the usual Wiener measure for

_going

from 0 to x in time t

_{and ’0}

is the noise.

We notice that

_{equation ( 1.1 )}

can be also read as the

_probability

distribution of an

heteropolymer (without

self

_{excluding effects)}

on a

_substrate,

where the interaction _among the substrate and each

_component

of the

_{heteropolymer}

is random.

Typical quantities

we would like to

_compute

are

where the brackets denote the _average over the

noise q

and

_{P n (x, t)}

is the normalized

probability

for

_finding

a directed

_polymer

at

_{point x}

at

_{time t,}

which is

_{given by :}

We are also interested in

quantities

like the

_probability

distribution of the self

_overlap

_q, which is defined as :

Different models are obtained

_by

_choosing

different

_probability

distributions for the noise

[2,

3].

In two dimensions

_(one

space, one

_time)

a _very

_interesting

case is for the white noise

limit,

i.e. the noise is

Gaussian,

with zero _average and with variance

The

_{replica approach}

is based on the

_following

observation : the

_expectation

value of the

products

of

G’s,

e.g.

satisfies a

_Schroedinger

_type

_equation

at

_imaginary

time

_(i.e.

_aglat

= -

HG)

with an

_n-body

Hamiltonian

_{given by}

The most

_likely

behaviour of

_{Gn (x1, t)}

can be related to the

_properties

of

G (n)

near n=0.

This model is

_{particularly interesting}

as far as the Hamiltonian

_(1.7)

is soluble. The

_ground

state wave function and energy are

_{given by}

where we have added to the Hamiltonian a constant

_proportional

to nA 5

₍₀₎

in order to

remove the

_divergent

terms which appear when i = k in

equation (1.7).

These observations where used

_by

Kardar in his

_{original study}

of the model

[4].

In this note

we discuss more

_carefully

the

_properties

of the model. After this

introduction,

we will

_analyze

the results that we obtain

_neglecting

the

_possibility

of

_breaking

of the

_replica

_symmetry.

In the third section we _{compare these}

_predictions

with the known results

_coming

from the

_mapping

onto the random stirred

_{Burger equation.}

In the fourth section we will show that

_replica

symmetry

is

_(in

some

_sense)

broken and

_finally,

in the last

section,

we

_present

our conclusions.

There are also two

_appendices

in which some technical

problems

are discussed. In the first

appendix

we

_study

some

_properties

of random

walks,

while in the second

_appendix

we

_present

some exact

_computations

of the Green function for the Hamiltonian

_Hn

for n =

0.

2. The

_{replica symmetric approach.}

Indeed we have that

The absence of a term

_proportional

to

n 2 in

the energy lead Kardar

[4]

to the conclusions that the fluctuations in

_F’TI

_(t)

are of order t" with w =

1/3. Using scaling

arguments

this result

implies

that

with

These results agree with the

analysis

done

_using

the

_mapping

onto the random stirred

Burger

equation

[5] (see

next

_{section) :}

one _{proves that the}

stationary probability

distribution for

constant

is

_proportional

to

i.e. to the Wiener measure where now x

_plays

the role of the time. It is crucial that

cp (x)

is defined

_apart

from an overall _{constant ;}

_indeed,

if we use the measure

_(2.6),

we

_get

cp 2(x) = 00,

but

_{(cp (x) - cp (y) )2) - 00.}

The consequences of this form of the

ground

state wave function have never been

investigated ; they

will be the

_subject

of the rest of this section.

Let us assume that for

_large

time and fixed x

This

_assumption

is

obviously

valid for

_integer

n, but it is less clear if it is correct also in the limit n - 0.

It is evident that

where

_{G (x)}

is a short hand notation for

_{G, (x,}

_t).

Using previous equations

we

finally

obtain for n = 0 :

The

_apparent

decrease of the wave function at

_infinity

has transformed itself in an

exponentially

increase !

where we assume that

Only

if equation (2.11)

is satisfied

_{(or n}

= 0 in

Eq. (2.8))

the result is

independent

from the absolute normalization of

G,

otherwise we should

_always

write

_proportional

to instead of

equal.

By

now the educated reader should have

_recognized

the

_equivalence

of

_{equation (2.6)}

and

equation (2.8).

The form of the

_ground

state wave function

_equation

_(1.8)

_implies

the

stationarity

of the

_probability

distribution

(2.6).

In

_particular

the

_exponential

_decay

of the

ground

state wave function

_implies

that

At this

_stage

one _may be inclined to _argue that the

_{exponential decay}

of the wave function is

generic

for bound states

_{independently}

from the

dimensions,

and therefore the value

y = 1 should be

superuniversal,

i.e. valid in

any

dimensions,

of course in the

_region

where a bound state is

_present

in the limit n --> 0. This

_argument

is

_{strongly suggestive,}

but it is not

clear to me if hidden

_loopholes

are

_present.

If we want to

_compute

the functions

_{P (x, t ) something}

must be said about the Green functions. We know that for

_large

times the Green functions must be

_proportional

to the

ground

state wave

_functions,

while for short time

_they

should be

_{given by}

the free one.

The

_{simplest hypothesis}

is that for

_large

times

where

_{D (x,}

_t)

is the free

_propagator

_{given by}

Equation (2.13)

is the

_{simplest interpolation}

between the behaviour at t =

0,

where the free

term

_dominates,

and the

_asymptotic

behaviour for

_large

times. An other

_simple

approxi-mation,

i.e. the

_ground

state function

_{multiplied by}

the effect of diffusion of the center of

mass, has the serious

disadvantage

of not

_reproducing

the correct behaviour of the Green

functions at small time. Now we have

If we use

_{equation (2.13)}

in

_{equation (2.15)}

the result for P is reduced to

_{quadratures ;}

unfortunately

1 have been unable to find a nice

_{representation}

for the results of the

_integral

and to extract the result in the limit n - 0. A _way out is to use the

_{following representation}

for

the wave function

Putting everything

together

we

finally

find

It is not

_strange

that

_equation

_(2.17)

can also be derived

_starting

from the stirred

Burgers

equation (see

next

_section).

The evaluation of this formula is non trivial. Some results may be

obtained

by

dimensional

_{analysis using}

the fact

_that

_{1 cp (x) - q (y) 1 =}

0

₍

_{x - y}

_ll2).

The

_integrand

will be concentrated for

_{large t}

near the maximum which is located for

i.e. for xM oc

t2/3

in

_agreement

with

equation

(1.5).

It would be

_interesting

to evaluate the width of the

_region

where the difference of the

argument

of the

_exponential

with its maximum remains of order 1. In

_principle

we can use the formula

and

_compute

It has been shown

[6, 7]

that

for

_large

t. A

_{probabilistic}

argument

which leads to

_(2.21)

is

_presented

in the

_appendix.

More

_precisely,

if we define

the

_probability

distribution of

_L117

has a tail

proportional

to

in the

_region

d «

(4/3.

The linear increase of d in

_{equation (2.21)}

is due to rare events in the tail of the

_probability

distribution,

where d = 0

(t4/3) ;

these events small

probability

proportional

to t-

1/3.

3. The random stirred

_Burger

_equations.

It is evident that

_{Gn (x,}

_t)

is also a solution of the stochastic differential

_{equation :}

where we have omitted the obvious

dependence

of G on _q.

We can now use the

_{mapping (2.5)}

for

_writing

an

_equation

of evolution for

_{cp (x, t ).}

We

readily

get

which is the random stirred

_Burger

_equation,

which has been well studied in the

_past

_[5].

Equation

(3.2)

induces a functional differential

equation

for the

probability

distribution of

ç as function of the time.

Using

usual

techniques

for stochastic differential

_equations

one finds the formal

_{equation :}

If the term

_(aç

_laX)2

were absent from

equations (3.2), (3.3),

an

equilibrium,

i.e. a time

independent,

solution of

(3.3)

would be

given

by

equation (2.6).

This result is not

_surprising

as far as

_{equation (3.2)}

reduces in this case the well studied

_{Langevin equation.}

The

_surprise

comes when we reinstall the term

_(aç

/ôx)2;

if we use the form

_(2.6)

for

p [ Q, t )

and we

_compute

_{ôP [ Q,}

_{t )/at,}

we find the extra

_{pieces :}

and

_they

both vanishes

_{by integration by}

_part.

If the

_long

time behaviour of P

_{[ cp, t )}

is

_unique,

as it

_happens

in a finite size

_box,

it is

_given

by equation (2.6).

However we have

_already

remarked that the

_problem

we need to solve is to find the solution of

_{equation (3.1)}

with the

_{following boundary}

conditions at time 0

The exact

_computation

of the

_probability

distribution of G is not _easy

_(we

shall see some of the difficulties in the

_{appendix II).}

In absence of exact results we could

_try

to _guess an

approximate

solution. The

_{simplest possibility}

consists in

_writing

where D is

_{given by (2.14).}

In this case we find

that cp

satisfies the

_following

differential

_equation

Q (x,

0)

=

0).

A

_{perturbative expansion}

in this _{extra term seems}

possible ;

_a careful

analysis

may be necessary to understand if the Ansaltz

_(3.6)

is correct at

_large

times. However no

major

inconsistencies are

_apparent

in

_{equation (3.6)}

and we can assume

tentatively

to be

essentially

correct.

We have seen that the direct

_analysis

of the

_problem,

_by

_mapping

it on the random stirred

Burger

equation,

seems to

_reproduce

the results of the

_replica

_method,

with unbroken

_replica

symmetry,

however we shall see that

_replica

_symmetry

is in some sense

weakly

broken.

4.

_{VVeakly breaking}

of the

_replica

_symmetry.

The

_analysis

of the

_previous

section seems to confirm the correctness of the unbroken

_replica

symmetry

approach,

however we shall see in this section that the

_replica

_symmetry

is

weakly

broken.

Let us consider the wave function for the n

_{particle problem}

in which

n /m

groups of m

particles

are bound

_together

with the wave function

(1.8) (where

here m

plays

the role

of n)

for each group of

particles.

In the infinite volume

limit,

this wave function becomes an

eigenvalue

of the Hamiltonian

_(1.7),

if we

neglect

the

_region

of

_phase

_space where

_particles

of the two _groups are near each other. In other words we consider the case where the n

_particles

form

_{n /m}

bound states of m

_particles.

The energy for such wave function can be

readily computed

and one finds that the energy per

particle

is

given (neglecting proportionality factors) by

If we assume that a sensible value of m must

_satisfy

the

_inequalities

the function

_{En (m )}

has a minimum for m = n.

In the

_strange

situation where n is less than

_1,

_{inequalities (4.2)}

are

_usually

substituted in the

_{replica approach by}

When n is

_equal

to 0 the maximum of

_{En (m )}

is located at m = 0 and

according

to the

standard folklore in the

_{region n}

1 we must _{maximize the energy}

_(not

minimize

!)

in order

to find the

_ground

state.

The

_ground

state seems at m = 0 and the

replica

_symmetry

is

apparently

_not

broken ;

if the minimum of

_{Èn (m )}

is located at

_{m # 0,}

as

happens

on the

Caley

tree or in the

_large

dimensions limit

_[8-11]

the

_replica

_symmetry

would be

_definitely

broken.

However the difference in energy vanishes

quadratically

with m and therefore the solution with

_replica

broken

_symmetry

is

_{nearly degenerate}

with the

_symmetric

one. This fact has serious consequences on the

_overlap

distribution.

If the

_overlap

distribution

_{Pr ( q )}

is a

single

delta

_function,

the

replica

symmetry is

usually

considered not to be

broken ;

however it was remarked in reference

_{[12] that,}

if we want to

study

the

_breaking

of the

_replica

_symmetry

in a

_{thermodynamic}

_sense, we must consider the behaviour of the function

_Z(e)

for

_positive

and

_negative

s. If the function

has a

_{discontinuity}

in the derivative at t =

0,

i.e. if

the

_replica

_symmetry

is broken.

Here the term

_proportional

to E in

_{equation (4.4) corresponds}

_{in the}

_replica

_formalism to

adding

a new term in the

_potential

between

_{replicas belonging}

to _{groups of}

_{n /2}

elements

each ;

the Hamiltonian becomes

where

_Hn

is

_{given by equation (1.7).}

The new term is an attractive

_potential

for

_positive

e and it is

_repulsive

for

_négative

e. We

can now _compare,

_using

E as

perturbation,

the _energy of the

_{fully symmetric}

solution

(ES )

and the energy we have

_starting

from the situation in which we have two bound states of

n/2

particles

each

_(EB),

i.d. m =

n/2.

We

_readily

find

For small values of n, at fixed

negative

e,

EB

is smaller than

ES

and the second solution

should be

_preferred.

In the limit where n is

_going

to zero

first,

we obtain that

and

_replica

_symmetry

is broken. The

_breaking

is

_extremely

weak because it

_disappears

for finite values of n when E goes to zero first.

The reader may be

puzzled by

the fact that the smallest of the two

_energies

(EB

and

_EM)

is

_taken,

while 1 have stated before that for n 1 we should

_maximize,

not

minimize the energy. In the same _way the choice of

_{taking m}

n was

_supposed

not to be allowed in the

_{region n}

1. In the

_replica

formalism the difference between minimization and

maximization is rather subtle :

_{usually [11]}

_]

the correct

_approach

consists in

_verifying

the

stability

of the

solution,

a task whose

_meaning

is not

_perfectly

clear to me in this context and that 1 have not

_yet

started to

_analyze.

However the choices 1 have made seem to me to be the

most reasonable ones.

The final

_predictions

are related to the behaviour of the solutions of the

_Schroedinger

equation :

If E is

_positive

G should be concentrated in the

_region

_where the

_{difference xl -}

_x2 is not

_large,

In the same _way, if the fluctuations in thé total free _energy

are

_proportional

to C

t2/3

for e >

_0,

_they

_{sould be reduced} to C

_{(t/4 )2/1}

for e 0. This

change

in the behaviour of the fluctuations is related to _{the presence of}

n2/4

in the formula for

EB.

The result seems rather reasonable. In the

_{replica symmetric phase configurations}

with two

well

_{separated nearly equal}

_peaks

in

_{G (x)}

are

possible

with a

_probability

which vanishes

slowly (as

1 It1/3);

these

_{configurations}

become dominant as soon as e is

negative. They

dominate the

_large

time behaviour of

_{G n (x1, x2,}

t )

and the simultaneous presence of two

peaks

reduces the fluctuation.

The behaviour at E

_strictly

zero is not

_easily

found.

_Simple

minded

arguments

may

suggest

that in this case the function

_Pr(q)

is

asymptotically

a

_single

delta function

_{( d (q - q M) ),}

while extra contributions vanishes

_{(as 8(q)lt1/3),}

however one should be _very careful in

drawing

conclusions in this delicate

_region.

A numerical check of the

_prediction

of this

_weakly

broken

_replica

_symmetry

would be

welcome,

especially

if we consider that some of the

_steps

done in the derivation are not

_crystal

clear.

5. Conclusions and outlook.

We have seen that in the one-dimensional case

replica

_symmetry

is

nearly

broken,

because the

replica symmetric

solution is

degenerate

with the solution with broken

replica

symmetry.

The behaviour is

quite

similar to the one found for the directed

_polymer

on a

Caley

tree

_{[8, 11 ],}

with the difference that m becomes zero on the

_Caley

tree

_only

at zero

_temperature,

while m

is

_always

zero in this two dimensional case.

The most

_interesting

result is the

_decoding

of the information contained in the

_replica

approach

in the wave

function,

in

_particular

the relation between the form of the wave

function,

its

_exponential

decrease at

_large

distance and the

_asymptotic

increase

_(with

the distance

_x-y)

of the

_expectation

value of the ratio

_{Gn(X, O)/Gn (y, 0).}

We have

_already

remarked that this result may be the

starting point

for

understanding

the

_{superuniversality}

of some of the critical

_exponents.

The extension of the

_techniques

used in this paper to the

higher

dimensional case will be

_hopefully

done in the near future.

Acknowledgements.

It is a

_pleasure

for me to thank M. Mézard and

_{Zhang Yicheng}

for

_interesting

discussions and

suggestions ;

in

_particular

1 am

_grateful

to M. Mézard for

_pointing

to me references

_[6,

_7].

1

am also

_grateful

to B. Derrida for an

_illuminating

discussion.

Appendix

1.

In this

_appendix

we

_present

an heuristic derivation of

_{equations (2.22), (2.23),}

which

supplement

the more

_{rigorous study}

of references

_[6,

_7].

It is clear that sizable contributions to the

integral (2.21)

will come from those

has two distant local maxima which do no differ too much as far as the value of

F(x)

is concerned.

In order to

_simplify

the

argument

it is useful to consider at first the

_{following simpler}

function

where the function ç is defined in the interval 0-X.

1 will argue that the

probability

of

having

cp

(xM

+

_{y) = Q (xM) - 8}

is

_given

for

_{large y}

and

small 8 by

The crucial

_point

_{is the presence}

of d 2

in the

_prefactor

of the

_{exponential (the}

_power

y3/2

follows from the normalization

_{condition) ;}

it can be

_justified

as follows.

The

_{starting point}

to derive

_{equation (AI.3)}

consists in

_estimating

the

_probability

Pr(

ç , x,

ê)

for

having

a

_path

_ço

_(x),

which satisfies the initial conditions _lp

_{(0) = -}

e and

ço (x) =

ç, such that :

Using

the method of

_images

one

_readily

finds

In the

_large

x limit Pr

_simplifies

to

The total

_probability

for

_{having equation (AI.4)}

satisfied

_{(independently}

of the value of

’P ( x»

is

_{given (in}

the

_large

x

limit) by

Pr(x, e)

factorizes in the

_product

of two terms : a term is

_proportional

to E, which is the

probability

of not

_crossing

the

_boundary

_(’P

=

0)

for small _x, the other _term is

proportional

to

x1/2

and it is the

_probability

for not

_crossing

the

_boundary

in the

_{region x > e 2}

If we have a

_{configuration}

which has a maximum of _ç near x = 0 with

’PM =

0,

the

probability

of

_having

’P

(xo)

=

8,

with 5 = 0

( 1 )

and xo

large,

will be

proportional

to the

normalized

_probability

for the

_trajectory

to

_satisfy

the condition

_{(AI.4) (i.e.}

_{d /xo1/2 Go( d, xo)}

multiplied by

the

_probability

that ç remains

negative

in the

region x just

greater

than Xo

(i.e. 5). Putting everything together

and

adjusting

the factor x in order to normalize

correctly

the total

_probability

_we find

_{equation (AI.3).}

Equation (AI.3) implies

that the

_expectation

value of

is

_proportional

to

1 /y 3/2.

In the

_original

case the _arguments would run

_quite

_similarly

and the results would be

multiplied by

smooth functions of

_{(X _ Y)3/t2 .}

The

_probability

law

_(2.23)

should be modified

only

in the end of the tail where

(x -

y)

oc

t 2/3 .

Therefore

t2/3

_play

a role of a cutoff in the same

way as

_{W ; equation}

_(AI.9)

is thus

_{replaced by}

The reader who

_correctly

doubts the soundness of this derivation may be comforted

by

the

fact that 1 have tried to check

_numerically

if the

_equations

derived in the

_appendix

are reasonable. To this end 1 have extracted

105

random

_{path (cp)}

of

104 steps

and these simulations are in very

good

agreement

with

_{equations (AI.9, AI.10).}

The behaviour of the most

_likely

value of

_i,

if it is defined as _exp

₍

_{(log A> ),}

is more

complex

and 1 have not been able to reach definite

conclusions ;

it seems to increase like

t03B2(t),

where

_f3

_{( t )}

is a

_decreasing

function of the time in the range 0.5-1.0. The

extrapolation

of

the numerical data at infinite time is

_ambiguous

and

_{preasymptotic}

terms must be

_{présent ;}

if we include corrections to

_scaling

no conclusion can be

_{reached ;}

the data are also consistent with the

behaviour : d

= _{constant -}

t - ",

with a = 0.15.

We could also introduce

_Am,

i.e. the value of 2l at which the

_probability

distribution

P ( 4 )

has a

_{maximum ;}

_{strickly speaking 4M,}

not d,

is the most

_likely

value. It turns out that the

probability

distribution of à is

_quite

flat,

also in

_logarithmic

scale, and i

is

_definitely

larger

than

_4M.

_Although

it is

_likely

that

_dM

and

J

_{asymptotically}

coincide,

they

may behave

in different _way for not too

_large

time : indeed in the simulations

_àm

behaves as

ty,

with y

weakly dependent

on the time

(’Y = 0.2-0.3 ).

In

_principle

we could

_analytically

_compute the

probability

distribution of

(0394) ,

in the

simpler

case where cp is defined in the interval

0-X,

by evaluating

the

integrals

For any

given n

it is

_possible

to find a closed form for these

_integrals.

However the number of terms in the

_computation

increases with n and 1 have not been able to find

_enough

_compact

formulae to allow the

_analytic

continuation in n up to n

_equal

to 0. It may be

possible

that

using

an

_approach

similar to that of reference

_[14],

an exact

_computation

_{may be done.}

Appendix

II.

In this

_appendix

we will

_compute

the Green function

G (n) (x1,

_{x2, ..., x n ;}

_{t ).}

We start

_{by looking}

to a

_{simpler problem,}

i.e. the

_computation

of the Green function

G (x ; t )

which is the solution of the

_equation

The function

_{G (x ; t)}

can be found for

_{positive x}

to be

_equal

to

where the

_integral

over p is done in the

complex plane

from - oo + iA to + oo + iA where A

is

_{large enough}

in order that the

_pole

of the

_integrand

remains below the

_{integration path,}

(i.e.

A >

A).

For

negative

x, G can be found

_{by using}

the relation

_{G (x ; t )}

= G

(- x ;

t ).

G is

_obviously

a solution of

equation (AII.1)

at x =

0,

because it is a linear combination of

D

_{(x, p ; t ) ;}

it satisfies the

_boundary

condition at t =

0,

because

for x #

0 the

integral

_{may be}

shifted up to

_ioo.

We check

_by

an

explicit computation

that

_{equation (AII.1)}

is satisfied also at

x=0.

In the limit where the time goes to

_infinity

the

_integral

may be deformed on the real axis

by

peaking

the contribution of the

_pole,

which coincide with the bound state contribution.

Similar formulae holds for the G

_(n)(X1

_{X2, ..., x n ;}

_{t ).}

For

_example

in the case n = 3 we can write in the

_region

x, x2 x3 :

where the

_integral

over

_{the p’s}

is done

_along

an

_{appropriate path}

in the

_{complex plane}

similar

to the

_previous

case.

Although

it is

_possible

with some work to write down the

_{generalization}

of

_(AII.4)

to

generic n,

the formulae are

quite complex (they

involve

multiple

integrations

over the

p’s)

and 1 am not able to

_compute

the

_analytic

continuation up to n = 0.

References

[1]

For an introduction to random directed

polymers

see KARDAR M. and ZHANG Y. C.,

Phys.

Rev.

Lett. 58

₍₁₉₈₇₎

_{2087 ;}

MCKANE A. J. and MOORE M. A.,

Phys.

Rev. Lett. 60

₍₁₉₈₈₎

527, and references therein.

[2]

MEDINA E., HWA T., KARDAR M. and ZHANG Y.-C.,

Phys.

Rev. 39A

₍₁₉₈₉₎

3053.

[3]

PARISI _G., A soluble model for random directed

_polymers

in the infinite range limit, Rendiconti

dell’Accademia Nazionale dei Lincei, to be

_{published (1990).}

[4]

KARDAR _M., Nucl.

Phys.

B 290

₍₁₉₈₇₎

582.

[5]

See for

example

FOSTER _D., NELSON D. R. and STEPHEN M. _J.,

_Phys.

Rev. A 16

₍₁₉₇₇₎

732 and references therein.

[6]

VILLAIN _J., SEMERIA B., LANÇON F. and BILLARD L., J.

_Phys.

C 16

₍₁₉₈₃₎

2588.

[7]

SCHULZ U., VILLAIN J., BREZIN E. and ORLAND H., J. Stat.

_Phys.

51

₍₁₉₈₈₎

1.

[8]

DERRIDA B. and SPOHN H., J. Stat.

_Phys.

51

₍₁₉₈₈₎

817.

[9]

DERRIDA B. and GARDNER E., J.

_Phys.

C 19

₍₁₉₈₆₎

5787.

[10]

COOK J. and DERRIDA B.,

Europhys.

Lett. 10

₍₁₉₈₉₎

_{195 ;} J. Stat.

Phys.

57

₍₁₉₈₉₎

89 and submitted

to J.

_Phys.

A.

[11]

PARISI G., in

_preparation.

[12]

PARISI G. and VIRASORO M., J.

_Phys.

France, to be

_published.

[13]

MEZARD _M., PARISI G. and VIRASORO M.,

Spin

Glass

Theory

and

_{beyond (World Scientific,}

Singapore)

1988.