Stability of the Mézard-Parisi Solution for Random Manifolds

(1)

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Stability of the Mézard-Parisi Solution for Random Manifolds

D. Carlucci, Cirano de Dominicis, T. Temesvari

To cite this version:

D. Carlucci, Cirano de Dominicis, T. Temesvari. Stability of the Mézard-Parisi Solution for Random Manifolds. Journal de Physique I, EDP Sciences, 1996, 6 (8), pp.1031-1041. �10.1051/jp1:1996114�.

�jpa-00247229�

(2)

Stability of the Mézard-Parisi Solution for Random Manifolds

D.M. Carlucci

(~),

C. De Dominicis

(~,*)

^and ^T. ^Temesvari

(~)

(~) ^Scuola ^Normale

Superiore

di Pisa, Piazza dei Cavalieri, Pisa 56126,

Italy

(~) ^Service ^de

Physique Théorique,

CE

Saclay,

91191 Gif

sur Yvette, France (~) ^Institute ^for Theoretical

Physics,

Eôtvôs University, 1088

Budapest, Hungary

(Received

4

January

1996, received in final form 16

April

1996,

accepted

18

April 1996)

PACS.05.20.-y Statistical mechanics

PACS.05.50.+q

Lattice

theory

and models of chaotic systems PACS.02.50.-r Probability

theory,

stochastic processes, and statistics

Abstract. Trie

eigenvalues

of trie Hessian associated with random manifolds _are constructed for trie

general

_case of R steps ^of

replica

symmetry

breaking.

For trie Parisi limit R

~ oJ

(continuum rephca

symmetry

breaking)

which is relevant for trie manifold dimension D _< 2, they are shown to be _non

negative.

Résumé. Les valeurs propres de la hessienne, associée _avec _une variété aléatoire, sont _cons-

truites dans le _cas

général

de R

étapes

de brisure de la

symétrie

des

répliques.

Dans la limite

de Parisi, R

~ oJ

(brisure

^continue ^de ^la

symétrie

des

répliques)

qui est

pertinente

_pour la dimension de la variété D < 2, _on montre

qu'elles

sont _non

négatives.

1. Introduction

The behaviour of

fluctuating

manifolds

pinned by quenched

random

impurities

encompasses a

rich

variety

of

physical situations,

^from ^the ^directed

polymers

_m a random

potential (with

_a

manifold dimension D

=

1)

to interfacial situations

ID

= d

-1,

d

being

the total dimension of the

space).

Trie directed

polymer problem,

for

example,

is itself related to surface

growth,

turbulence of the

Burgers equation

and

spin glass.

The literature

concerning

vanous

aspects

^is wide and

florishing,

_a

good

^deal ^of ^it

being quoted

in _some recent _reviews

Il,2].

The

approaches

taken have been

quite

diverse and it is not the purpose of this paper ^to describe them. This paper is _a first

step

m an

approach

to random manifolds that _uses field theoretic

techniques

m the

replica

formalism. Field theoretic

techniques

have been used with _success

(in

their functional renormalization _group

variety) by

Balents and Fisher [3] ^to describe the

large

N

(N

= d D is the transverse

dimension)

for the 4 _e dimensional manifold. In the context of

directed

polymers

Hwa and Fisher [4],

using

_more conventional field

theory,

have

emphasized

the role of

large

but _rare fluctuations.

Alternately

_a unified

description

of the

general

^random

manifold has been

developed by

^Mézard ^and ^Parisi

[si (MP), where,

^via ^the use of the

replica technique, they

have laid down the first step

(mean

field and self-consistent Hartree-Fock

level)

of _a

systematic

field

theory 1IN expansion (~).

^This ^is ^the

approach

_we want to pursue here and this paper may be considered _as _a first

sequel

to MP.

(*)

Author for

correspondence je-mail: cirano@amoco.saclay.cea.fr)

(~)

The intricacies of trie first

loop

correction _were

briefly

considered in further work [6]

©

Les

Éditions

_de

_Physique

₁₉₉₆

(3)

In this _paper _we

study

the

eigenvalues

of the Hessian associated with the

1IN expansion

as laid down

by

MP. We do it in _a

general

form

by keeping

discrete trie R

steps

of

rephca symmetry breaking,

and _we show that the MP continuous solution

(R

₌ _oo, _as is

appropriate

for D _<

2)

does _not

generate negative eigenvalues.

The

general

discrete

expression

derived here allows _to control

carefully

the

limiting

_process

to the MP continuous solution. It will be also of _use in further discussions of the solutions for the _case D _> 2

(and

in

particular

D _~

4)

_an unclear _case to date.

In Section 2 _we summarize the MP

approach

down _to the

quadratic

fluctuation _terms

defining

trie Hessian

(see

also

I?i ).

Section 3 is devoted to trie introduction of _a discrete Fourier-like transform _on trie

replica overlaps.

In Sectioù 4 _we

briefly

discuss the

equation

of state at mean field level. In Section 5 _we discuss and exhibit the

replicon eigenvalues.

Sections 6-8 _are devoted _to the

tougher

_case of the

longitudinal-anomalous eigenvalues.

2.

Summarizing

MP

Approach

We thus consider the MP random manifold model defined

by

~7i(°')

"

~Î / _d~~ If

~~)j)~

~

+

2v(~> ~'(~))

₊

l~'~(~)j ₍₁₎

~=i where d

= N ₊ D is the dimension of the space, N the dimension of _w, _i.e. of the _transverse space, and D the dimension of the _vector _~, _the

longitudinal

directions. _The

quenched

random

potential

V has _a Gaussian distribution of _zero _mean and correlation

V(~, w)V(~', w')

=

-à(~l(~ _~')N _f

^~~'

/~~~ ⁽²⁾

~~~~

~~~~ 2(1~

_+f) ^~~ ^~ ^~~~ ^~ ^~~~

where the function is

regularized

at small arguments

by

the cut-off Ù.

Averaging

_over the random

quenched

variable V via

replicas replaces il) by

-7irep(Wa)

=

/ _d~~

IL ^L ^~)[)~

⁺

^J1Wli~)

⁺

^PL ^NI ^'~'~~~~ ^~'~~~~~~ ⁱ

a

~i

^~

a,b

(4)

In

studying (4)

MP have _used first _a variational

approach (identical

_to _the Hartree-Fock

approximation),

which becomes _exact _in the

large

N limit. Then

they

introduced

auxihary

fields with the

advantage

that it

generates

correction _to

Hartree-Fock, namely

(1) The _zero

loop

term

embodying

the effect of

quadratic fluctuation,

1.e. the

logarithm

of the determinant of the Hessian matrix

(ii) 1IN

corrections in _a

systematic

_way

(~).

In this work _we _are interested in (1)

studying

the Hessian and its

eigenvalues

around the

mean

field (Hartree-Fock

_or

vanational)

solutions. We leave the

study

of

(ii)

^for _a

separate publication

_[8].

(~)Note

that in reference [8] trie _case

1-RSB,

relevant for short

ranged correlations,

_is

extensively

studied both for its

stability

and its

1/N

corrections

(given

m closed

form).

(4)

Following

MP _we introduce trie

n(n +1)/2 component auxihary

fields

Tabi~)

_"

j~'ai~)~'b(~) ^là)

and its associated

Lagrange multipliers sab(~).

We need to

compute

trie

generating

function

@

=

/ ^jj _d[wa] / ^jj d[rab]d[sab]

x exp

~ ~j / _dmab(~) _[Nrab(~) wa(~)wb lx)]

a=1 a<b

~

ah

~ ~~~

/

_~~

( ~

~ÎÎ~~

~

~'~~~~l

fl2

d~

£

_N

_f _(raa(~)

₊

_rbb(~) 2rab(~)) (6)

~

ah

now

quadratic

in

w's,

hence after

integrating

over wa

@

#

/ _fl djTab]djsab]

eXp

jNçjT( Sjj

a<b

çjr. s)

₌

^~ £

_d~

[sab(~)rab(~)

+

fl f(raa(~)

+

rbb(~) 2rab(x))]

₊

S[s] (7)

' 2

exp

Sis]

₌

/ _fl _d[wa]

exp

~ ~j /

_d~

_[wa(~) _((-V~

₊

~1)ôab sab

lx)) _wb(~))

2

a ah

Assuming

_an uniform saddle

point (for large N)

with

Tab(~)

" pab +

ôrab(~)

3ab(~)

_" _OEab +

ô3ab(~) (8)

one can

expand

around it. MP thus obtains

ii)

The _mean field contribution

Ç(°1(r; s)

(ii)

^The

stationarity

condition

(vanishing

of linear

terms)

that defines _p and _a

(identical

to the variational _answer

(~)).

.

Stationarity

with

respect

to

sab(x):

Pab "

j _~jGab(P) ₍₉₎

(~) Only ⁱⁿ ^trie

large

N limit. If N is

finite,

_m ^trie ^vanational answer

f(y)

_is

replaced by /(y)

=

ù

_fo°°

doEa'~/~~~e~" J(2ay/N)

with

f

-

J

_as N _- _ce.

(5)

.

Stationarity

with

respect

to

rab(~):

°ab "

2flf' (paa

+ _pbb

2pab) (ii)

°aa "

~

OEaà

(12)

b(~a)

(iii)

The Hessian matrix of the

quadratic

form in

ôsab (after integration

of the

quadratic

terms m

ôrab

which in

particular

_says that

ôsaa

=

£~

~~ sab, 1-e-

W

~ exP

lNç°lP; ail / _fl _dlôsaài

exp

fl fl ôà(p)M>~~(p)ôscà(-p) (13)

a<b

a<j

p

~~~~~~~~~

2

j"jpaa Î~Î

2pab) ~ ~~~~~~~

~

~~~

~

lGac(P _~)

₊

_Gbd(P _~) Gad(l'~ _~) Gbc IF ~)l (14)

3. Discrete Fourier Transform

We shall work in _a formalism where the level R of

replica symmetry breaking

is

kept

discrete

(R

= 0

replica _symmetric,

R

= one

step

of

breaking,

R

= oo Parisi

breaking,

1-e- continuous

overlaps).

In the

end,

if _necessary,

we shall let R _~ _oo.

For _an observable _aap, with aap = ar for _an

overlap

on

fl

= r, define the discrete Fourier transform _as

R+1

âk

"

~j

_Pr

(ar ar-1) (là)

r=k

and its _inverse

ar =

f (âk âk+i) _l~~~

~=o

Pk

Here the

pr's

_are the size of the _successive Pansi

boxes,

_po _e _n, _pR+i ₊ 1. The above transform has the

properties

ar ar-1 "

(âr âr+1) (17)

~ _a°7b7P

r

" ~°P

(18)

7

âklk

" ék

(19)

In

particular

(a _~)k

"

(~°)

k

All this _is but _a discrete _version of the

operations

introduced in MP for the continuum

(R

_~ _oo

_).

Note that in the above all

quantities

with indices outside the _range

(0,1,.

,

R

+1)

_are to be taken

identically

null. If the limit R _~ _oo _is understood _as

âr

_~

â(u)

ar ~

a(u) ₍₂₁₎

(6)

then the _connexion with MP

is,

m their

notations,

à(u)

₌ à

la) lai(u)

lai(U)

=

/~ dta(t)

+

Ua(U) (22)

4. Trie

Equation

of State With the above definitions _we have

Pr =

j _L _GTIP) ₍₂₃₎

p

Ôk(P)

"

~ ~

(24)

and the

equation

of state

~~ =

2p Il

^~

fl _jGR+i(q) Gr(q)]lr

⁼

^0,1,.

_, ^R

⁽²⁵⁾

fl

~ R+1

"

~j

pt

(at at-1) (26)

t=0

The last

equation,

^identical to

(12),

also _writes

R+1

à~

=

fl _p~(a~ a~_~)

= 0

(27)

r=o

To solve for the

equation

of state one uses the transform that allows to write

i~r(~) ~r-1(~)i

"

) [Ôr(~) Ôr+1(~)j ^(~~)

and

Yr ⁺

j _fl _{IGR+i (q)} _Gr(q)1

"

j _fl ) _lâk _âk+i _)Ôk _(q)Ôk+i _(q)

₊

_ÔR+i ^q)1

~ ~ ~~~~~ ~

j29)

The

equation

^of state then writes, with

f(vr)

" 16 +

vr]~'

°r "

2fl/~iYr) (~°)

1-e-

~fl k

~~~

~~~~

^~~~

~ ~

~ ^~~~ ^k

~~~

~~~~~~

_{~~~ ~}

~~~~

_~~~

(31)

(7)

Here _we have

Ôk(q)

"

~ ~

(32)

q ~ ak

~°k ~

Î°Î(~)

p~ ~ ~

(in

the Parisi

gauge) easily restituting

the MP solution _as _given for the _case of noise with

long

range correlations.

For later _use, let _us take _a _one

step

difference _m the

equation

of _state

°r °r-1 "

) (Ôr àr+1)

"

2fl1/~(Yr) J'iYr-1)1 (33)

with

Pr-i Yr =

j _L _lGr(q) _Gr-i(q)1

=

j _L

[Ôr(q) Ôr+i

(q)] (34)

q q

1-e-

"

~

~~)

flÔr(~)Ôr+ll~) _(3à)

r ~

For small yr-i Yr

(infinitesimal

in the

continuum)

_one

_gets

1 =

-4£Ôr(q)Ôr+1(q)Î"(Yr) _(~~)

q

a formula used below.

5. Hessian

Eigenvalues:

The

Replicon

Sector

The Hessian

(14)

written _m terms of

overlaps

becomes

M'(P)

_"

MÎ(Î (P) (37)

with

r=anb

,

s=cnd

u =

max(a

n c

,

a n

d) (38)

u=max(bnc

,

bnd)

In ultrametric _space,

only

three of the four

overlaps

_are distinct. In the

replicon subspace

r = s, and _we have two

independent overlaps

_u,u > _r +1. In the

longitudinal

anomalous sector, if _r

#

_s, _one _can

keep max(u, u)

to

parametrize

M.

For the

replicon

_sector, and _on

general ground [9-11],

the

eigenvalues

_are

given by

the double discrete Fourier transform

~~~~

^~ _~~

Î

~~

Î

~"

~~~ ~i~ ~~~~-i

+

Miii,~ ii _j~~~

fork,1>r+1.

(8)

Using

the Hessian

expression

Mjjj

_#

~~j~Î~~~~ £ _jG~(q)

₊

_Gu(q) _2Gr(~)l _lG~(P _~)

₊ _GV

_(P _~) 2Gr(P _~)l

~ ~

j40) Hence,

for the double transform

Àplr; k,1)

₌

-) _L

[Ôklq)Ôilp ^q)

⁺

Ôilq)Ôklp _q)j 141)

and

using (36),

valid in

particular

in the

continuum,

ÀPI~Î k, i)

"

fl (2Ôr(~)Ôr+1(~) _(Ôk ^(~)Ôl ^(P ^~)

⁺

^{Ôl (~)Ôk} ^(P ~)j

i,k à r~+1 _j42)

[rhe

propagators Ôr(q)

_are

decreasing

functions of _(q( and of _r

(since -âr

_is an

mcreasing function).

Hence the most

dangerous eigenvalue

is

ÀPI~,

~ +

l,

r +

Î)

" 2

L (àri~)Ôr+li~) Ôr+li~)Ôr+liP ~)) ^1~3)

or m the continuum limit

>Pi~.

x

~)

- 2

~l~D / _dq~ ^q~

+

~

à~~~

~p q~~

/

~

à~~~ lq~

₊

^~ _à~~~ ¹⁴⁴⁾

1?or D <

4,

_one _can set the ultraviolet cut-off _to

infinity

and rewrite the

dangerous rephcon eigenvalues.

m the continuum

limit,

_as

>~j~;

~,

~)

m

fl _[à~jq) à~jp ^)j~ j4à)

where the

marginal stability

is

obviously displayed (~)

Ii. Hessian

Eigenvalues:

The

Longitudinal

Anomalous Sector

'fhe

analysis

of

longitudinal

anomalous sector is

notoriously

_more diflicult. It has been _con-

siderably simplified by

the introduction

[10-12]

^of

kernels,

1-e-

by going

to _a

block-diagonal iepresentation

^of the ultrametric matrices. Take in

(37)

_a

component

of the

longitudinal

<~nomalous sector

M[1],

r

#

_s,

which, given

that

only

three

overlaps

_can be

distinct,

_can be

written

(5)

^~

M)"~

t ₌

max(u; u)

'To

M)"~

is associated _a kernel

(in

fact _a Fourier

transform) K["~,

^where ^k =

0,1, 2,...,

R

+1,

k = 0

corresponding

to the

longitudinal

sector. Likewise for the _inverse

M~~

_of _the _mass

(~) ^{It is} ^worth mentionmg at this

point

that trie hmit D

- 4 is delicate to take. Indeed

solving

for trie

t~quation

of state

(30-32),

_one finds that trie

breakpoint

_xi

[[a](x)

= 0 for _~ _>

xii

_crosses its

boundary

value _xi

" 1, at _some intermediate dimension

DOIT),

2 < Do < 4. This delicate point ^is ^left out for

further consideration. See also [13].

(~)

Except for trie components

M[j[

that also mcorporate _a

rephcon

contribution.

(9)

operator

matrix _we bave _a kernel

F["~

The

relationship

between them is the

Dyson's equation [10,11]

Éj,S

₌

~r,S f

~r>t /lk(t)

/ht;s

_~~~

~ 2

~_~

~

Ak(t)

^~

Éj'~

=

-Àkir)Fj'~Àkis) ₁₄₇₎

Here the

Ak(r)

_are

special replicon eigenvalues

defined

by

lÀ(r;k,r+1)

^k>r+1

Ak(r)

=

j48)

À(r;r+1,r+1)

k<r+1

with _as in

(39)

and

Î(Pt Pt+i)

t < k 1

/lk(t)

=

)(pt 2pt+1)

t ₌ k 1

(49)

(Pt Pt+i)

t > k i

The kernel

K["~

^itself is _a transformed of the _mass

operator

matrix

M)"~

It writes _as _a

generalized

discrete Fourier transform

lr

^s ^R+1

Ki~

=

L

₊₂

L

₊₄

L

p~

imi~ Mil[) iso)

t=k t=r+1 t=s+1

where _we have

displayed

the _case k < _r < _s

(for

different

orderings

the summation

£~~~

is

only kept

in the allowed

summands). Knowing K["~,

one then

computes F["~

via

(46, 47)

and

(M~l)["~

via _an _inverse transform. That is if _one is able to solve the

integral equation (46)

which is

only possible

with

especially simple

kernels

K["~.

^This turned out to be the _case when

studying

the bare

propagator

of the standard _spm

glass _[10]

where

~

K["~

=

Ak(min(r; s)) (si

4

The _same functional form also _appears for the

spin glass

model introduced

by

Nieuwenhuizen

[14].

For the system studied

here,

let _us construct the kernel

K["~

^with ^the ^matrix

M)"~

_as

given by

the Hessian

(14) manifestly symmetric

_{in r,} _s. Let _us choose

r < s and _we then find

R+i

jKi'~ ^Iv)

=

L L

pt

iiGtlq)Gt (p _q) Gt-i (q)Gt-i (p q)i

q t=max(s+i,k)

Gs lq) iGtlP q) Gt-i (P q)i iGt (q) Gt-i lq)i G~ iv q)j (52)

which also _rewrite _as

j~r;s

lBk(S)

^r ^< ³ ^< ^~ ⁱ

~ ^~

Bs+i(S)

_s _/ k 1

(53)

where

Bkls)

=

~ (Glq)S _q))

~

Gs (q)Ôk (P q) Ô~ (q)G~(p _q) (54)

(10)

The

important point

is that the kernel

only depends

_on

max(r, s)

jKj>s

^m

B~(max(r; s)) (àà)

tbus

rendering

soluble the

integral equation (46).

7. Solution of trie

Dyson Equation:

The

Propagator

Kernel

Ii the _case of

(K["~

=

Ak(min(r; s))

the solution of the

Dyson equation (46)

has been

given (in

the continuum

limit)

_m

_[10].

We wish to extend it here to the _case

(K["~

=

Bk(max(r; s))

Ii this section _we work for _a _given value of

k,

1.e. _we remain in _a sub-block of the

longitudinal

anomalous sector

(k

₌ 0 for the

longitudinal).

The

propagator

^kernel ^of

equations (46, 47)

is obtained _as

i~ Ak(r)/lkAk(s)

^~~~~

where

/lk

is _a Wronskian and

il

_the

_regular

and

irregular

solution of _a discrete Sturm-Liouville

equation

and

Ak

the

special replicon eigenvalues

of

(48). Namely

we have for the Wronskian

/~~

₌

~~Î~ ~Î~~~~~~~

_~~~

D~Bk(s) (57)

with

1~~

j~s~

_e

j~

~~

j~ (à8)

The functions

#*

_are

given by

lit _js)

m i +

f jB~(t) B~(s)j

^/~k~~~

ij(t) (à9)

2

tmo

Àkit)

jij(s)

m

( jB~(t) B~js)j (kjjj ^{ii (t)} ^B~(s)c~ ⁽⁶⁰⁾

with

Bk(s)

_as in

(53).

Note

that, contrary

to the solution obtained for

(si ),

where

Ck

"

1,

here, #j

and

#)

_remam

coupled

via

Ck

~~

=

( ~k(t)

~+

~~~

Ak(t)

^k

In _terms of

#)

the Wronskian _can also be written _as

/~k

"

4Ck

1 +

~j _~k(S) _)~))) Î(S)1.

~~

~~~~

k

(11)

8.

Stability

Discussion in trie LA Sector

Given the

explicit

expressions

(53)

of

Bk(s)

in terms of

Ôt (or Gt)

itself _a

positive

definite

function, decreasing

with

increasing

t

DtGt

"

Ptli ~# ^GjGÎ ~/~ (62)

one can

easily

show that

(1)

B(r)

< °

(63)

(ù) DrB(r)

_e

B(r

₊

₁₎ B(r)

> 0

(64)

Under these conditions _we _can _now

give

_a iower bound for the

longitudinal

anomalous

eigen-

values. We _can _wnte _out the determmant of the

eigenvalues

restricted to the sub-block k:

det

(M))) ÀI)

⁼

jj [Ak(r)

_À]

l

+

( Bk(t)

~

)~~~~ #)(t)j

(65)

r=0 t=0 ~

which is

directly

related to the Wronskian

(apart

from the

Ck factor).

Note

that,

from

equation (59),

_one also obtains

Ds#t _là)

=

-2DsBk(3) É ()~~

~

#t(t). ₍₆₆₎

With the

boundary

value

#)(0)

= 2 and

D~B(s)

>

0,

_one infers

that,

in _case of _<

Ak(t)

for all

t,

4t(S)

_> o

(67)

D~#)(s)

_> 0

(68)

Consider

now in

equation (65)

_a small value of such that <

Ak(t)

for ail t For that value the determinant of

(65)

cannot

uanish,

it is

always positive

on account of

(63, 67).

Hence the spectrum ^of the

longitudinal

anomalous sector is

certainly

bounded

by

the bottom values of the

rephcon

_spectrum

Àp=o(r;

r

+1,

r +

1).

This establishes

stability,

_a

marginal stability

_as

m the continuum limit where

Àp=o(~;

_~,

~)

= 0.

9. Conclusions

We thus have _set _up for random manifolds the formalism _to discuss the

stability

of _mean field solutions. For the well established continuous

(R

=

oo) solution, adequate

for _D _<

2,

_we are

able to prove its

marginal stability.

The formalism used here allows _a discussion of _any

type

(discrete,

continuous _or

mixed)

of _mean field solutions.

It is worth

noticing that,

when

discussing

the

stability,

the

only place

where

properties

of the noise correlation function _may

play

_a direct role

(outside

of the

fully

continuous R

= oo

solutions)

is _m

(33-36).

There the

equation

of state _is used to find the alternative

analytical

form for

f",

that cast in evidence the

non-negative

character of the

rephcon eigenvalues.

(12)

Acknowledgments

One of _us

(CD) gladly acknowledges

useful discussions with M. Mézard.

References

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J.,

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5.

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