A Puzzle on Fluctuations of Weights in Spin Glasses

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A Puzzle on Fluctuations of Weights in Spin Glasses

Th. Nieuwenhuizen

To cite this version:

Th. Nieuwenhuizen. A Puzzle on Fluctuations of Weights in Spin Glasses. Journal de Physique I,

EDP Sciences, 1996, 6 (1), pp.109-117. �10.1051/jp1:1996132�. �jpa-00247171�

A Puzzle

_on

Fluctuations of Weights in Spin Glasses

Th. l/I.

Nieuwenhuizen(*)

Van der Waals-Zeeman Institute, Universiteit _van

Amsterdam,

Valckenierstraat

614/5,

1018 XE

Amsterdam,

The Netherlands

(Received

20

April

1995, revised 27

July

1995,

accepted

6 october

1995)

Abstract. In certain

mean field mortels for _spin

glasses

there _{occurs a one} step

replica

symmetry

breaking

_pattern. As _an

example

of

general 1/N-corrections

in such systems, ^the fluctuations in the internai energy are calculated. For this

quantity

the outcome is known from the

specific

heat. It is shown that the correct result _can be derived

by

_assuming that the both the

height

and the location of the

breakpoint

fluctuate. This eRect

enlarges

the

commonly

considered _space of allowed fluctuations of the order parameter in

loop

calculations for short range

spin glasses

of this type. ^The

phenomenon

does not _occur in _spin

glasses

with infinite order

replica

symmetry

breaking.

PACS. 75.10Nr

Spin-glass

and other random mortels.

PACS. 75.40Cx- Static

properties (order

_parameter, static

susceptibility,

heat

capacities,

critical exponents,

etc.).

PACS. 75.50Lk

Spin glasses

and other random magnets.

1. Introduction

The theoretical

understanding

of the spin

glass phase

relies _on the Parisi

rephca _symmetry breaking

scheme [1]. ^{Parisi considered} a variational

problem

with k

breaking

of

replica

_sym-

metry.

The

abjects

that show up are the values of

possible overlaps

_q~

(0

< <

k)

and their cumulative

weights

_x~

il

< <

k).

The free energy was maximized in these variables. In the

Sherrington-Kirkpatrick

model the

physical

result arises in the hmit k

- oo.

Eliminating

trie i's _a continuous order

parameter

^function

q(x)

arises.

Loop

corrections to this _mean field

theory

have been studied in

great

detail in _{a serres} of papers

by

de

Dominicis,

^{Kondor and}

Temesùari

_[2]. In

particular,

the fluctuation matrix has been inverted in fuit

glory

_[3].

Many

_aspects of this and related

rephca analyses

_are ill defined and often Ill understood.

Nevertheless

replica

results for the SK-model have been rederived

by TAP-analysis

and

cavity

methods [4]. ^{For the random} energy model

("simplest spin glass"

the

replica

results _{are in} fuit

agreement

with the direct evaluation of the

partition

_{sum [5].}

It is _our belief that _a

replica analysis

_can

produce

the

physically

correct _{answer m a}

quick

way, but its

interpretation

should

always

be understood from _{a more}

profound analysis.

If the

answer is

unsatisfactory,

then _one should

simply

abandon it _{or improve} the denvation

(the

celeber _case

being

Pansi's improvement of the

Sherrington-Kirkpatrick saddlepoint.)

Such _a situation _occurs in

particular

in

systems

^{which exhibit} one

step

^of

rephca _symmetry breaking

(* ^e-mail:

[email protected]

©

Les

Éditions

_de

Physique

1996

110 JOURNAL DE

PHYSIQUE

I Nol

(IRSB).

Here there _occur two transition

temperatures, Tg

^{for the statics and}

Tc

_>

Tg

^{for the}

dynamics.

Ilirkpatrick

and

~iolynes

_[6] considered the Potts

glass,

which exhibits _a IRSB when the number of

components

^{exceeds four.} In their

analysis

of the metastable states

(TAP states) they

observed that there _{occurs a} critical temperature

Tc

>

Tg

^where

unexpected

behavior

sets in. On _a static level the

system

^condenses _{for T~} < T _<

Tc

in _a multitude of states with extensive

complexity,

ail of which have _non-zero

self-overlap

but

vanishing

mutual

overlap.

The free _{energy is} therefore still

equal

to the

paramagnetic

free _energy. As _a result. this transition _is not noticed in _a direct

rephca analysis

of the

partition

_sum when _{one uses} Parisi's

prescriptions.

Nevertheless. it _can be described

by replica's

if _one modifies the rule for

fixing

the

breakpoint.

The value of the

selfoverlap

_can be obtained

by solving

the

equation ôF/ôqi

=

o,

with the

additional rule that the limit _{xi ~} l is taken afterwards. In this

temperature interval, namely,

it

gives

the maximal free energy in the

physical

_range 0 < xi < 1.

Loosely speaking,

that limit equates the i~alue of the

selfoverlap

to the mutual

overlaps

in _a

limiting

small cluster of

states. This

procedure

also

automatically implies

that the free _energy of this

phase

_is

equal

to the

paramagnetic

_one. For T _<

Tg

^the extremum can no

longer

be located at xi = 1 and the

additional restriction has to be relaxed.

A related

mystery

of the

replica

^calculus occurs in the

dynamical approach

to the systems with IRSB. The prototype ^is the

spherical p-spin

interaction _spin

glass,

_see

equation il)

beloi~:.

The

long-time

limit of

dyiiamics

has _a

vanishing

fluctuation

eigeI1value,

aI1d is therefore called

"marginal" iii

_{[8] [9].} The

loI1g-time dyI1amical

values for _qi aI1d _{xi caI1} be derived from the

replica

free _eI1ergy

by

_assumiI1g

margiI1ality

withiI1 the

replica.

calculatioI1 [7] [8]

[10).

^The mystery ^remaiI1s

why

_a

replica

calculatioI1 _can

explain

the

loI1g-time dyI1amics.

After

ail,

the Gibbs

weight

is

thought

to be related to

exponentially large

times where the

system

has time

eI1ough

to visit ail states.

lIargiI1ality

is

thought

to be related to

algebraically large

times. It therefore _seems

paradoxical

that Gibbs

weight

at

margiI1ality

does describe the results of

long

time

dynamics.

It has beeI1

suggested

that _an esseI1tial rote is

played

here

by

the

complexity.

It is _so

large

that the

replica

free _{eI1ergy is} the lowest of the available states, an old

thermodyI1amic

law,

but _Ilew in

glassy systems il1].

In this _{note i~~e} ivish to draw attention to aI1other

aspect

of the location of the break points of IRSB solutions. This time _we coI1sider trie _more standard

statioI1ary

case, where the Parisi

description applies, Ilamely

the

optimal

value of the break points follows

by maximiziI1g

the free _eI1ergy. ~VheI1

calculatiI1g 1IN

contributions to _a

physical quaI1tity,

_we ^fiI1d that _care

should be takeI1of fluctuations in the location of the

plateaus.

In Ilext section _we

preseI1t

in detail the calculatioI1 of the fluctuations of the internai _eI1ergy

in a

p-spin

interaction _mean field _spm

glass

mortel with IRSB.

SubsequeI1tly

this

aI1alysis

_is

geI1eraIized

to models with OORSB. The paper closes i~~ith _a discussion.

2.

Analysis

of _a Model with IRSB

For _a system ^{with N}

spiI1s

in _zero field _we consider the HamiltoI1iaI1

~~

~ ~0t2"'tp~b~t2 ~~

~~~

q<~2<"'<~p

with

indepeI1dent

Gaussian random

coupliI1gs,that

have average zero aI1d variaI1ce

J~p!/2NP~~.

The

spiI1s

are

spherical

aI1d

subject

to the condition

£~S)

= N. A

rephca

calculation _was

performed by

Crisanti and Sommers

(CS) _[12].

The

replicated

^free _energy reads

Fn

=

~~ ^~ £ _q[~ ^~tra _log

_q

₍₂₎

ad

~

At _zero field the order

parameter

^{fuI1ction has the}

one-step

^form

q(x)

=

q10(x -xi),

and the free _energy reads

=

-~(

₊

_~( fia( ) _{iogli fi qi)}

+

))~ ^{iogli qi) 13)}

where

fi

= 1 _Xi The internai _eI1ergy reads

U "

~~~ Il Îiql) (4)

FollowiI1g

Crisanti aI1d Sommers _m their

TAP-approach _[14],

_we deI1ote

by

_z the value 0 < z <

2/p(p 1)

^{where the}

complexity

1=

)

^~ ~

log

^~~ +

~

~

z~ £ _là)

P

~

p z

is

non-extensive,

_viz.

I/N

= 0. The

equatioI1s ôF/ôqi

=

ôF/ôxi

= 0 theI1 _caI1 be cast in the form

Pli q)ql/~~~

~

" Z( Xl "

~~

(~

^~~

16)

PZ q1

From these results _{oI1e can} calculate the

specific

heat C

=

dU/dT usiI1g

ÎÎ Î~ _{Î~- Î~}

~~~

One obtains

C =

~~ ~~~ (l ^{fi qi 2q(}

^~~~~~~

~/

^~~

₍₈₎

Pqi P +

Let _{us Ilow} coI1sider the fluctuations _in the internai eI1ergy,

~U~

*

l~i~l l~il~

"

j _fl / Ds~ilsal~ilsdl

_eXP

~fl fl~ils~) ₁₉₎

,Î=i Îi

where the last relation iI1volves

replicas,

aI1d the hmit _{n ~} 0 is

implied.

The

leading

terms

in N _are of order

N~n,

so

they

do Ilot survive for11 _~ 0.

II1deed,

U is

expected

to have fluctuations of relative order

1/@.

The first

tJ(N)

contribution to

AU~

_comes wheI1all sites in the

pre-expoI1entials overlap pairwise.

It

yields

j2fil j2

fil

AU~I~~

=

< Lqip

=

iii ^liai) no)

ad

Before

calculating

the other terms _one has to average over the

couplings.

It

implies

the shift

j _~ j ~

~~~P~ ~ s~ s~ (ii)

~i ~p ~i' ~p ~p~p-1 ~i ~P

~

112 JOURNAL DE

PHYSIQUE

I Nol

One

gets

the contribution

~~2

j2) fl~

J~N2 pipi

j

j

"°~~

~i

2<...<~p

~[

^l<.

^-<~j

where _the replicated reads

e~~~n

=

exp£ _~fl~J~q[~

₊

~papqnp £pnpsfsfl(13)

ad

~ ~ ~

~

and

dpdq/27ri

is _a short haI1d for the

integratioI1measure fl [dpapdqap/27ri] fl[dpaa/27ri]

x

n<p a

il

+

O(n)),

while further _p and _q are

symmetric [12].

^{It is} iI1deed seen that the

leadiI1g

terms m N _are of order

N~n,

_so

they

do Ilot contribute when ₁₁

= 0.

We _{can now} express the

prefactor

_as

q's

aI1d _carry out the

spiI1 iI1tegrals,

to obtaiI1

~~~

~~~

~~ ~~~ ~~ _~ô

~~~~~~~~~ ~~~

Î~

~~ ~~~~~

^~

~~~~~~~

^~~~~~

l

(14) ExpaI1ding

arouI1d the

saddlepoiI1t

_oI1e sets qnp =

qnp +ôqnp,

_pop

=

fInp +ôpap,

with

ôqnn

= 0.

where q ^aI1d fl ₌

q~~ momeI1tarily

denote the saddle

point

values. Due _caI1 theI1

iI1tegrate

out the

ôp's,

_so that

oI1ly

the fluctuations in the

q's

remaiI1

~~~~~~ 4nÎ~ÎÎ~~/~ ~~Îô~~~~~~

~~~

ô~~~~~~~~~~~~~~

_~~~~

where

Mnp~~ô

=

ô~flfn/ôqapôq~ô

is the fluctuation matrix. Its _matrix elemeI1ts aI1d

eigeI1val-

ues are discussed

by

CrisaI1ti aI1d Sommers

[12].

Now _{oI1e caI1} go ^to

eigeI1modes

of M

[12].

^The traI1sversa1 _{ones can} be

iI1tegrated

out, which

brings

the _square root of _a aI1other determinaI1t _as

prefactor.

At fixed break

point

xi

only

the

longitudinal eigenvector ôqap

_{r- qap} ^will contribute due to the contraction _in the

preexponential. However,

also

q's

with _a small shift in the

breakpoint

will be _seen to

contribute

[13].

^{We therefore consider} fluctuations descnbed

by

order parameter functions

qnP ~

qlx)

=

Il

₊

iJ)q101x

_xi

à) l16)

where _qi and _{xi are} the

saddlepoint values,

and where _~

gives

fluctuations in the

height

of the

plateau

and à in the location of the

breakpoint.

The

shape (16)

of

q(x)

leads to n-times the free energy

(3)

with _{qi ~}

qi(1+ ~)

and xi -+ xi + à. The value at à

= ~ = 0 is of order _n and has been omitted

already

from

equation (15).

The linear terms vanish due to the

saddlepoint condition,

and _were also omit- ted

already.

The

quadratic

terms therefore

yield immediately

L ôqnpmnp.~ôôq~ô

~ ~

l~~qlflfqq

+

21JôqiflFq~

+

ô2flF~~) Ii?i

np~ô

We _{now assume} that for11 _~ 0 the summation _over diflerent

breakpoints

_can be

replaced by

an

integral

over their location _{xi +} à _viz.

f

_WI

(xi

₊

à)d(xi

₊

à),

where _wi is _an unknown but

smooth

weight.

Since à is of order

1/@,

_it then follows that its

weight

_WI

(xi

+

à)

_m

w(zi

is uniform _m à. The total

weight

_w consists of _wi and of _square roots of determmants that arise after

performing

the

ôp integrals

and the transversal

ôq

modes. Next _{we assume} that _w is such that the

remaining

Gaussian

integral

_over à and _~/ is normalized to

unity.

One then lias

AU~I~)

=

~~~~ (18)

~ ~

J d~dôji ii ql

₊

ôql pli ~qll~

_exp

Î _l~~

_QI

_flF~~

₊

_2~ôqiflFq~

₊

ô~flf~~

~

f d~dô

_exp

)" (~2q)flfqq

+

2~/ôqiflfq~

+

ô2flF~)

Let _us recall that the appearance of the

integral

in trie denominator _is due to _our assumed value of _{w, now seen to} be of order Nn. As the Gaussian fluctuations

bring

a factor

1/Nn, they

indeed

yield

terms that survive in the limit _{n -+} 0. There _are several fluctuation contributions.

We should

only

take into _account the _cross terms

[ôq( p(i~qi]~.

This

yields

AU~

i~~

=

~~~~ ~~~

^~

lp2ilRqq 2piiqiRq~

₊

qlR~x) i19)

where R is the inverse fluctuation

matrix,

viz.

R[~

=

flfq,

with

= q, x;

j

= q, x.

Inserting

the field

equations ôF/ôqi

=

ôF/ôxi

= 0 _one has

~'~

^~~~~~

~)Î-Î~~Î/~ÎÎÎÎ~

~~~~~

~~~~

~~~

2

Il

IÎÎÎ~~

fi

_qi)~ ~~~~

flfxx

-

illl~~~i~llll~jll~l~ ¹²²⁾

The _inverse matrix reads

~~~ 12 p ⁺

pÎÎÎlp

^~~

_ÎÎÎI ^ÎÎÎÎ

p +

1~~~

_~~~~

~~Q "

ii

~~~~~~ ~~~~~ ~ ~ ~~ ~~~~

Sqx

= pxi

Il

_qi

(25)

Sx~

=

~(~ llp i)qi

_{p +} 2 +

fi

_qi

IF

i pqi )1

126)

qi

Now _one

only

has to insert these results _in

(19).

One obtains

AU~

^l~)

=

~~~~~~~

_~~

~~)

_~ ^{~ ~ ~~~}

⁽²⁷⁾

Pqi P

Note that several

unpleasant

denominators have

canceled, ailirming

that _our

procedure

_is

correct.

Putting things together

_we find

~~2 ~u2

(ij _~

~u2

^j2)

J~~

(i ^j~

qP

2qP Îl

_{(P~~ P ~ ~~}

(28)

2 ^~ _{Pqi P +} 2

Comparing

with

equation (8)

one sees that indeed

AU~

=

T~dU/dT,

_as

required by

_a standard

thermodynamic

relation. If variations in _xi had not been taken into account,

AU~

_{would have} been

proportional

to

1/Fqq,

which is

certainly incorrect,

since the

comphcated

factor _in the

numerator of

Fqq

^{will not be canceled.}

114 JOURNAL DE

PHYSIQUE

I N°1

3. Infinite Order

Replica Symmetry Breaking

Recently

the author

proposed

_a

spherical

model with infinite order

replica symmetry breaking.

It is the model

given by equation (1),

but summed _{over p}

[15].

^In

particular,

random _pair

(p

=

2)

and random

quartet

interactions

(p

₌

4)

_occur, while random

triplets

should be absent. Then the model has _near the _spin

glass

transition the _same critical behavior _as the

Sherrington-Kirkpatrick

model. Its benefit is that it _can be solved

explicitly

in the whole low T

phase;

this arises due to the

spherical

nature of the spins and the _mean field character of the model. For _our

present

_purpose this allows _an

explicit

check at any

temperature

in a model with infinite RSB.

Assuming

that the variance of the _p-spin

couplings equals J)(p 1)1/NP~~,

the free _energy for _a

k-step

RSB

pattern

can be derived

using

results of _an

appendix

of CS

~

=

-(fl~fli)

+

(fl~4 ) _{iogli qk) iog ~f~ 129)}

z=i ^{z z}

where

m

j2

flq)

=

£

^~

_{qP 130)}

~~~ P

and

k k

4 "

£(X~+1

_X~)

/jq~),

A~ # i qk +

~j

_X31~3

_{Q3-1) 131)}

1=0 3"~+l

As usual _we defined _zo

= 0 and xk+i = 1. The variables

(q~, x~)

^{will be} denoted

by

_a 2k +1

component

vector ~~,

ii

=

0, ,2k).

From U

=

-(flN/2)( fil) -16)

_one obtains

by

diflerentiation with respect ^{to T the}

specific

heat

p2

p~

p4

p~ ^2k

C =

-(f(l)

_{16) +}

~j _{lb~R~lbj (32)}

~ ~

~,J=o

where

R[~

₌

ô~flf/ô~~ô~j

and lb~ =

ôlb/ô~~.

We again consider the fluctuations _in the internai energy. The

analog

of

equation (10) immediately explains

the first _{two terms} of

T~C.

The

analog

of

equation (19)

is

AU~l~l

=

~/ _£

lb~Rqlbj (33)

,Î~o

l/Iultiplied by fl~

^this indeed

equals

the last term in

equation (32).

Dur

approach

thus

reproduces

the correct result for the fluctuations of the internai energy at any fixed k

and,

in

particular,

_in the

physical

hmit k

-+ oo. In this continuum limit the meaning of the

breakpoints

_x~ becomes ill-defined. It _was

conjectured by

Parisi

[16]

^{that the} discussed eflect then

disappears.

In other

words,

the final result

might

be due to fluctuations

in the

plateaus

_j

only.

In order to

investigate

this

question

i~~e go ^to the Parisi free energy functional for trie SK mortel [1]

k

~~ ~~~~~

_~~~

~Î~~ Î~~

_~

Î ^~~

~~~~

where the

paramagnetic background

has been

subtracted,

_r

=

(fl~J~ 1)/2,

and

Ti

_%

12x~+1

x~iql

_{+ q~}

~ixj+i

xj)qj

₊

(izj+i ^{xj)qi 135)}

j-,+ _j-

For this model

equation (32)

still

applies

with

f(q)

=

J~q~,

while the term

involving fil)

=

J~

comes from the

paramagnetic background.

The nice

thing

of the Parisi model is that the solution for _{q~, x~} is knoi~~n

exactly

at any k.

It holds that

~~

ÎÎÎÎ~~

_~'

_~WÎÎÎ1)

_~~~~

where qk satisfies

~~~~

2

(2k

₊

)21

~°~~ ~ ~ ~ ~~~~

This is _an

immediate,

exact

generalization

of trie Parisi solution of trie model

(34)

to the

region

where ryi

/w~

is _not very small.

From Parisi's results it follows that _now

~

~~~ ^~~~~~ _~~~~~ ~~~ ^~~ ÎÎÎÎ~ÎÎ~~Î

~~~~

Differentiating

with

respect

to T _one finds that

Ab~~

^l~) should be

equal

to

p2 j4 filq~

^W

§iqk(3 _j2kÎi)2

~~~

~~~

(39)

W W vi

~k(3 ~ù~

Since this

expression

_{converges as}

1/(2k +1)~,

_a

_good approximation

_is

already

obtained for k = 1. The fuit result for trie

specific

heat converges also _as

1/(2k +1)~ [l].

We have also calculated

AU~I~) by taking

into account

only

trie fluctuations _in the _q~.

Numerically

we have inverted the matrix

ô~flf/ôq~ôqj

_up to

large

sizes

(k

=

150)

and find at

w = vi =

1,

_{r =} o-1 that

~ ôlb

jô~flfj~~

^ôlb

~~~~ ôlb

jô~flfj

~~ ôlb 0.446 0.446

~~ _{ôqz ôqôq}

~~

ôqj (

ô~~

ô~ô~

~~

ô~j

^~

(2k

+

1)3

^{~'~~~~~~~~ ~}

(2k

₊

₁₎₃

~ 3" ~ 3"

(4Q)

where the _sum in the middle involves fluctuations _in both q and _{z, viz.}

(~~)

=

(~0>~l>~2>'

ÎXI>X2,'

'1'

From this

example

it is _seen that Parisi's

conjecture

_is

probably

true in

general:

the effect of fluctuations of the

breakpoints disappears

for continuous order

parameter

^functions.

4. Conclusion

We have

put

the basis for

one-loop

calculations in _{a mean} field _spin

glass

model with IRSB.

We calculated the _energy fluctuations because their

strength

is known

already; they

_{are pro-}

portional

to the

specific

heat. In order _to

reproduce

the _{correct ansiver, we} had _to take _into

account fluctuations in both the

height

and the location of the

breakpoint

of the IRSB solu-

tion.

(It

should be admitted that the normalizatioii of the fluctuation

integral

deserves _a sohd derivation Dur assumed form _is the natural one, and leads to the correct

results.) [13].

116 JOURNAL DE

PHYSIQUE

I N°1

It is the _neiv

point

^{of this} paper that in

integrating

over ail qnp one must not

only integrate

over the fluctuations around the

saddlepoint

at

given parameterization (given breakpoint

_xi

),

but also _over ail

possible parameterizations (locations

of the

breakpoint).

In

hindsight,

this _is not so unnatural. In _mean field both the location of the

breakpoint

and the

plateau

value _are

determined

by

_a saddle

point. equation. Generally

saddle points _appear ^when one has to

perform sharply peaked integrals.

The _very notion of _a saddle

point

can

only

_occur due to smallness of fluctuations around it. Saddle

points

without fluctuations do not exist. Fluctuations must

always

be

considered; oI1ly

in _{some cases}

they

_are

Ileghgible.

The _new

question

i~~hich arises is: what

happens beyoI1d

_mean field in finite range

systems?

Physically,

the i~~eight ^{of the} IRSB solution _is the

probability

for _occurrence of _non-zero

overlap.

At before

hand,

there is _{no reason}

why

it should be _constant. One would thus expect ^{that it} fluctuates. This is

exactly

i~*hat _we showed in _an

explicit

_mean field calculation.

Beyond

_mean field _one expects ^{that it} fluctuates in space. In the reI1ormalization group

approach

of Cwilich

~nd

I(irkpatrick il?]

such fluctuations _are not taken into account.

An

important question

is whether

they

_are

compatible

with

ultrametricity.

The _answer is affirmative.

Indeed, gii~en

^{Parisi's division of the}

replica

number _n

=

(n/xi)(xi /x2) (xk Il),

one can

impose

at each lattice sites the allowed -ultrametric states" _xi,

., xk. In _mean field

only

_one

'~state",

_say T, ^will be

occupied;

this

happens umformly

in space. For

describing

fluctuations of the

breakpoint

at site _r, the state with

breakpoint x(r)

= Î +

ôx(r)

i~~ill be

occupied

at the _cost of the

occupation

of the state with f. This will

modify

the

longitudinal

sector of the fluctuations. As this sector is _massive at the transition and

below,

it is

expected

to

bring only quantitative effects,

and not to

change

the renormahzation floiv

[17].

In mortels with infiI1ite order

replica symmetry breaking

the order parameter fuI1ctioI1 has in

zero field _a contiI1uous

part

up ^to x = xi and is constant

beyoI1d.

Here the effect of fluctuations

in the location of the

breakpoiI1ts

_{was seeI1 to}

disappear.

For short _range

systems

^this

imphes

that the

breakpoiI1ts

do Ilot fluctuate in space. The fluctuations of the

probability

of the self-

overlap,

_I.e. l _{xi, are}

probably already

takeI1 iI1to accouI1t

by

the fluctuations _in the

height

of the

plateau.

Acknowledgments

The author beI1efited fi.om

participation

_in the coI1ference " Low

temperature dyI1amics

and

phase-space

structure of

complex systems",

held at

Nordita, KopeI1hageI1,

Nlarch 1995. He Mso

ackI1owledges

discussion with J-M- Luck and

hospitahty

at the CEA

Saclay,

where part

of the work _i~~as doue. He thaI1ks

Giorgio

Parisi for _urgiI1g him to

publish

this material. This

ivork _was made

possible by

the

Royal

Dutch

Academy

of Arts aI1d Sciences

(KNAW).

References

[1] Parisi

G.,

J.

Phys.

A13

(1980) Llls;

ibid. l101.

[2] de DomiI1icis C. aI1d

1(oI1dor1., Phys.

Reu. B 27

(1983) 606;

de DomiI1icis

C.,

KoI1dor1.

aI1d Temesvari

T.,

Int. J. Mort.

Phys.

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(1993)

986.

[3j ^{de DomiI1icis}

C.,

Kondor I. and Temesvari

T.,

J.

Phys.

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[4] Mézard

M.,

Parisi G. aI1d Virasoro

M.A.,

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A.,

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L.F. and Kurchan

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Th.M., Phys.

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Th.M., Complexity

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Physik

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[13] ^As a certain hnear combination

of'replicon'

_or

'ergodon'

fluctuations also represents a shift in the

breakpoint,

this _may look like

overcounting

the fluctuations.

However, equation (là) only

describes the

small,

Gaussian fluctuations

ôp

_r-

ôq

r-

tJ(N~l/~).

After

completion

^of

the

manuscript,

^it was reahzed that

equation (14)

should also have fluctuations

ôp

r-

ôq

_r-

O(N°),

that _are

effectively

taken _{into account}

by

_our shift in _xi We thank M. Ferrero and _an unkno~vn referee for discussion _on this item.

[14]

^{Crisanti A. and Sommers}

H-J-,

J.

Phys.

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Th.M., Phys.

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G., private

communication

(March, 1995).

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Kirkpatrick T.R.,

J.

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Cwilich

G.,

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