Strong Coupling Probe for the Kardar-Parisi-Zhang Equation

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Strong Coupling Probe for the Kardar-Parisi-Zhang Equation

T. Newman, Harald Kallabis

To cite this version:

T. Newman, Harald Kallabis. Strong Coupling Probe for the Kardar-Parisi-Zhang Equation. Journal

de Physique I, EDP Sciences, 1996, 6 (3), pp.373-383. �10.1051/jp1:1996162�. �jpa-00247190�

Strong Coupling Probe for the Kardar-Parisi-Zhang Equation

T.J. Newman (~>*) and Harald

Kallabis (~)

(~) Institut für Theoretische

Physik,

Universitàt _zu

KôIn,

D-50937

KôIn, Germany

(~)

Hôchstleistungsrechenzentrum, Forschungszentrum Jülich,

52425

Jülich, Germany

(Received

13 November 1995, received in final form 20 November 1995,

accepted

27 November

1995)

PACS.03.65.-w

Quantum

mechanics

PACS.05.40.+j

Fluctuation

phenomena,

random _processes, and Browman motion PACS.68.35.Rh Phase transitions and cntical

phenomena

Abstract. We present an exact solution of trie determmistic

Kardar-Parisi-Zhang (KPZ)

equation under the influence of _a local

driving

force

f.

For substrate dimension d < 2

we recover

the well-known result that for

arbitrarily

small

f

> 0, the interface

develops

_{a non-zero}

velocity

~( f).

Novel behaviour is found in the

strong-coupling

_regime for d _> 2, in which f must exceed _a cntical force fc in order to drive the interface with constant

velocity.

~ie find

v( f)

+~ f fc)"~~~

for f ~ fc. In

particular,

the exponent

a(d)

₌

2/(d- 2)

for 2 < d _< 4, ^but saturates at

a(d)

= 1

for d > 4,

indicating

that for this

simple problem,

there exists _a finite _upper critical dimension du ₌ 4. For d > 2 the surface distortion caused

by

the

apphed

force scales

logarithmically

,vith

distance within _a critical radius Rc

+~

if fc)~"l~~,

where

v(d)

=

a(d) /2.

Connections between these results, ^{and the critical} properties of trie

weak/strong-couphng

transition

m the _noisy KPZ

equation _are pursued.

1. Introduction

This article is concerned with _a

problem

which arises _in trie

study

of trie

Kardar-Parisi-Zhang (KPZ) equation iii

of interface

growth _{[2, 3].} Namely,

_we consider trie deterministic version of trie KPZ

equation

under the influence of _a

local,

constant force.

Although

this

problem

is

m

pnnciple

rather

simple,

_a detailed

investigation

has not been carried out to _our

knowledge.

The

results, although interesting

in their _own

right,

_assume

greater significance

when related to the

weak/strong-coupling

transition in the

noisy

^KPZ

equation.

Before entenng ^{into further} details of this

problem,

_we shall

give

_a brief _overview of the

key properties

of the KPZ

equation.

In, by

_now, standard notation this

equation

for the evolution of the interface

height h(x,t) (above

_a d-dimensional

substrate)

takes the form

ôth

=

uv~h

₊

(À/2)(Vh)~

+ _Y/

il

where

J/(x, t)

is _a white

noise, generally

taken _to be Gaussian distributed. The

Cole-Hopf

trans- formation

h(x, t)

=

(2u/À) In(w(x, t)) yields

_a linear

equation

for _w albeit with

multiplicative

(*)

Author for

correspondence le-mail: tn©thp.uni-koeln.de),

Address from lst

January

1996:

Department

of

Physics~ University

of

Manchester,

Manchester,

M13 9PL, U-K-

©

Les

Éditions

_de

Physique

1996

noise:

ôtw

=

uv~w

₊

(À/2u)wJ/. (2)

This

equation

_may be

re-expressed by

_means of the

Feynman-Kac formula,

such that _{w rep-}

resents the restricted

partition

function of _a directed

polymer

in the presence of

quenched

random defects.

Naturally,

trie above

equation

may also be

interpreted

_{as an}

imaginary-time Schrôdinger equation

for _a quantum mechanical

(QM) partide

in _a

time-dependent

random

potential (-J/).

It is well-known

iii

that d

= 2

plays

the role of _a lower critical dimension. For d <

2,

the interface will be

asymptotically ii-e-

^for

large times) rough

for

arbitrarily

small À. For d _> 2 there exists _a transition between

asymptotically

smooth and

rough phases, depending

_on the value of À. To _access the

rough (strong-coupling) phase,

must exceed _a critical value Àc.

Amongst

the most

important questions

in this field _{are: 1)} what _are the

scaling

_exponents

associated with the

strong-coupling phase; ii)

does there exist _an upper critical dimension

du,

above which these exponents take _on mean-field

values; iii)

what _are the critical

properties

of the

weak/strong-coupling

transition? There _{are no answers} to the first

question,

bar

predictions

from extensive numerical work

(except

for the _case d

= 1 where exact

analytic

results _are available

[2]).

^As

regards

the second

question,

the

community

_seems

split

between beliefs in

du

= 4 and

du

= oc. The former belief is based

primarily

_on ^the

predictions

of _various self- consistent

'mode-coupling'

theories

(see

for

example [4]),

whilst the latter belief _arises from numerical evidence for

d-dependent strong-coupling exponents

_[3]

(for

all accessible d

1 7).

With

regard

to the final

question,

_a recent

two-loop

renormalization _group calculation _[5]

(formulated

_{as an}

(-expansion,

with _c

= d

2) predicts

that at the

transition,

the

roughness

exponent x = 0 +

O(c3) (to

be

interpreted

_as

logarithmic roughness),

whilst the correlation

length exponent

^{has the form P} ₌

(1le)

₊

O(c3).

For the

one-loop

versions of these

results,

_see reference [6].

We _now concentrate _{on a} much

simpler problem.

As mentioned

before,

_we retain the essential

non-linearity

of the KPZ

equation,

but

replace

the noise J/

by

_a local

applied

force:

J/(x, t)

=

foô~(x).

In directed

polymer language

this

corresponds

to

replacing

the bulk disorder

by

_a

single

constant energy columnar defect. In terms of the

QM analogue,

_we have

replaced

the random

potential by

a delta-function

potential

with

amplitude (-fo).

There exists _a well- known result

[7],

^which is most

easily

stated in

QM language:

for d < 2 the

partide

will have

a bound state for

arbitrarily

small

fo

>

0,

whereas for d _> 2 there

only

exists a bound _state for

fo

>

fo,c

> 0. In terms of the

interface,

these results

imply

that for d < 2 _an

arbitrarily

small force _is suilicient to mduce _{a non-zero}

velocity

to the

surface,

whereas for d > 2 the

driving

force must exceed _a critical value in order to achieve this. The identification of _a lower critical dimension

(along

with the

properties

of the

system

^for d <

2)

_may be obtained from _a

perturbative analysis

of

equation (2) (in

terms of À, or

equivalently fo).

In order to

explore

the

properties

^of the

'binding/unbinding'

transition for d _>

2,

a non-

perturbative analysis

is

required

such _an

analysis

constitutes the

body

of the present work.

The _main results which arise _are the

following (which

_we shall describe in terms of the interface

formulation).

Whereas for d _< 2 _an

arbitrarily

small

driving

force will induce _a

velocity u( fo)

to the

interface,

for d > 2 the

applied

force must exceed _a critical value _in order to achieve this. The

dependence

of the

velocity

_on the _excess force

ôfo

₊

fo fo,c

_is found to be

u m~

(ôfo)°~~~

where

o(d)

=

2/(d 2)

^for 2 < d <

4,

^but saturates at

a(d)

= for d > 4. For d _< 2 the

applied

force will create _a non-hnear distortion of the surface _{over a} critical radius

R~

m~

f[~~~~~~~,

^{but the}

amplitude

^{of the distortion} is of order

unity.

For d _> 2 the distortion scales

logarithmically

with distance and exists within _a critical radius

R~

m~

(ôfo)~~~~~,

where

u(d)

₌

a(d)/2. Therefore,

for d _> 2 the distortion is _a macroscopic rupture ^{of the surface}

with _a

logarithmic profile (by 'rupture'

_{we mean a gross} distortion of the

interface, leaving

the continuous

properties

of the surface in

tact).

We shall leave the

interpretation

of these results for the final sections of the paper.

The outline of the _paper is _as follows. In the _next section _we shall

give

_a

precise

^{definition of} the model to be

solved, along

with _some

simple steps

which lead to _an accessible

strong-coupling (1.e. non-perturbative) analysis

of the model. In Section

3,

we

proceed

with this

analysis

and derive the

velocity-force

characteristics. The

spatial profiles

of the surface distortion for various

dimensions _are evaluated in Section 4. Connections between the

properties

of this

simple model,

and the

weak/strong-coupling

transition in the

noisy

KPZ model _are

pursued

in Section 5. We

end the _paper with _our

conclusions,

and _a

description

of

possible

extensions to the

present

work.

2. Deflnition and Reformulation of the Model

Replacing

the stochastic _source of the

noisy

KPZ

equation

^with a local

applied

force leads _us to consider the

equation

ô~à

=

uv2à

₊

(~/2)(và)2

₊

foôdjx). j3)

As for the

noisy problem,

_{we may}

apply

the transformation

h(x, t)

=

(2u/À) In(w(x, t))

which

yields

the linear

equation

ôtw

=

uv~w

₊

_(À fo/2u)

_w

ô~(x). (4)

Since _{we are}

primarily

interested in the interface

application

of this

problem,

_we choose the initial condition

w(x, 0)

₌

1,

which

corresponds

to _an

initially

flat interface

(h

=

0). [The

natural initial condition for the directed

polymer application

_is

w(x, ₀₎

=

ô~(x).

We shall not consider this

problem here.]

We may

formally _integrate equation (4)

_using the Green function of the diffusion

equation, g(x,t)

=

(4~vt)~~/~ exp(-~~/4ut), _leading

_to

wjx, t)

= i +

j~ fo /2v) j _{dt' g(x,}

t

t/)w(o, t') (à)

Therefore, knowledge

of the function

w(0, t)

is suficient _to determine _w

throughout

all _space.

For convenience _we shall define

çi(t)

+

w(0, t). Setting

_x

= 0 in the above

equation

leads to a

Volterra

integral equation

for _çi

t

çi(t)

₌ +

f / _{dt' a(t t')çi(t') (6)}

where

a(t)

=

t~~/~

_and

we have defined _an effective force

f

_e

(Àfo/2v)(4~v)-~/~. [Such

a

Volterra

equation

_may also be derived for the _more

complicated

situation of _a

time-dependent

force

f(t).

The _case where

f(t)

is _a stochastic variable is of

particular

interest in the directed

polymer picture

_[8].]

Clearly

this

equation

is ill-defined for d >

2,

since the kernel _is

non-integrable

over the range

(0, t).

To

regularize

the

equation

we have many

possibilities.

The most convenient is

to smear the delta-function force _{over a} small region of size

i~

with _a Gaussian

envelope,

_{i e.}

we

replace foô~(x) by fo(~i~)-~/~exp(-~~li~).

_The

_length

_scale constitutes the smallest

physical

scale in the

problem. Assuming

that

w(x, t)

_varies

slowly

_over the _core

region (x(

<

allows _us to rederive the above Volterra

equation,

where _now

ait)

=

(t +i~)~~/~ (after rescahng

- =

il(4v)).

Laplace transforming equation (6)

and

using

the convolution theorem

yields

_an

explicit

solution for this

Laplace

transform

£[çi(t)]. Inverting

the transform gives the result

1 ~~

~st/i2

~~~~ 2m

/

s

il

-12-d

fD(s)]

^~~~

where _~/ is the usual

integration

contour

along

the line _s

= c +

iv

^where c is a real constant chosen to lie to the

nght

of any

singularities

in the

complex plane.

The function

D(s)

is

given by

m

~ ~ e-8Y

~~~ ~

Î

~

(l

+ Y)~~~

(which

_may be

expressed

in terms of the

incomplete

Gamma function

[9]).

We _{are now in} a

position

to obtain all

physical properties

of the model from the

analysis

of

equation (7).

~ie stress _once

again

that this

analysis

is

non-perturbative

in

fo (or equivalently À).

^{All the results for d < 2} may be obtained

by perturbation theory (although

_one must evaluate all

orders),

whereas _none of the results concerning the

binding/unbinding

transition may be obtained

by

such _a

technique

for d > 2.

In the next section _we shall concentrate _on

evaluating

the

asymptotic (large time)

form of

fi(t)

,vhich will enable _us to determine whether the interface has _{a non-zero}

velocity.

The

spatial profile

of the interface distortion

requires

the evaluation of the convolution of the diffusion

equation

Green function with the function

#(t) (as

illustrated in

Eq. (5))

this

analysis

is described in Section 4.

3.

Velocity-Force

Characteristics

The

singularities

in the

integrand

of

equation (7)

consist of _a branch cut

along

the

negative

^real

axis,

and

(depending

_on the value of

f)

_a

pole

in the

right-half plane (RHP).

The

large-time

behaviour of

çi(t)

,vill be dominated

by

the contribution from the

pole (if

it

exists).

0ne _may

easily

_convince oneself that the existence of this

pole

is

required

for the interface at the

forcing

site to have _{a non-zero}

velocity,

_i-e- for

H(t)

m~ ut

(where H(t)

_e

(À/2v)h(0, t)

= In

çi(t).) By studying

the small-s behaviour of

D(s)

_{one may} show the

following properties

of the

integrand

in

equation (7).

For d _< 2 there

always

exists _a

pole

in the RHP for

f

>

0,

which for

f ~

0 is situated at sp =

f~/(~~~).

_{For d >} 2 the

pole

will

only

exist in the RHP for

f

>

f~

₌

i~~~(d- _{2) /2.}

_For

_{f ~ fc,}

the

position

of the

pole

is

given by

_sp

=

A(d)( f f~)~/(~~~)

for 2 < d < 4

(~vhere A(d)

is _a

complicated

function of d)~ ^{whereas for d} > 4 _one has sp =

[(d 4)/(d 2)](f f~).

This

abrupt change

of behaviour for sp ^{at d} = 4 will have

interesting

consequences.

Evaluating

the contribution to

çi(t)

from the

pole

leads _us to the

following

results for the

velocity-force

characteristics:

i)

^{for d} < 2 _one has for

f ~

0 the relation

u( f)

m~

f~/(~~~~j ii)

for d _>

2,

the

velocity

_{is zero} unless

f

exceeds

f~

_one then has for

f ~ f~

the relation

u( f)

_m~

lé f)"l~)

~&>here

ôf

e

f f~

and

a(d)

=

Ù

^{~ ~ ~ ~}

(9)

~

>4>

which

signals

the existence of _{an upper} critical dimension

du

₌ 4.

Table I.

Asymptotic

behauio~tr

of H(tÎ

the

interface height

at the

forcing

site _{as a}

function of

dimension d and

apphed force f.

The

symboi ôf

is the _e~cess

force: ôf

=

f- f~.

^The

constants _ci and _c2 haue the

form

_ci

Id)

=

[Fil d/2)]~/(~~~)

and _c2

Id)

₌

[r(2 d/2)]~/l~~~).

d f~

Hit) ~if)

f

<

fc f

=

fc f

>

fc

0 < d _< 2 0 ~~j~~ ^In

f~/~~~~~t)

⁰ ~ t _ci

Id)

f~/~~~~~

2 0 In

(In(t/t~)

0 _~ t

) exp(- il f)

2 < d _< 4 ~~i~~

t~~~

_in

Ii ))

_~~i~~

in(t/t~)

_ut

ifiiôf/fc)~/id-~~

~ ~~ ~~

~~

)

~~

~

Î

i~Î

Î/)ij)

d > 4 ~~i~~t~~~ _in

Ii )) init/t~)

~ t

i

_~~i~~

iôf/fc)

In the absence of _a

pole

_m the

RHP,

the

asymptotics

of

çi(t)

_anse from the contribution from the branch _cut. The results for this _case

(along

with the

vif)

characteristics at the critical

dimensions 2 and

4)

_are

displayed

for

completeness

_in Table1.

4.

Spatial

Profiles

In this section _we wish _to evaluate the

spatial profile

of the interface for

f

>

fc.

We de- fine the

positive quantity AH(x, t)

₊

(À/2v)(h(0, t) h(x, t)). l/Iaking

use of the

Cole-Hopf

transformation, along

with

equation (5),

_we have

(after rescahng

_{x - x}

=

x/(4v))

AH = In

lj

+

j /

dt' g(x, ^t'

+

i~) çi(t t') (10)

o

For

f

>

fc

and ut >

1,

_we have from the

previous analysis çi(t)

m~ e~~ In this case, the above

expression

reduces to

AH =

In[ fF(x)] (11)

where

m

F(x)

=

dt' e~~~' g(x, t'

₊ i~

). (12)

o

This

expression

is valid for

(x(

<

u~/~t.

_{For this entire}

range we see that the

height

deviation Ah is

independent

of time. For

extremely large

distances from the

forcing

site

(x(

»

u~/~t)

~

the influence of the force is insuficient _to induce _a

velocity,

and the surface distortion is

essentially

zero.

We _now concentrate _on examining the form of the

steady-state profile _given by equations

ii

and

(12).

In fact there _are two

spatial regimes

for this

profile, separated by

the

length

scale

Rc

=

u~~/~

_{which will}

emerge as a criticai radi~ts. We shall denote the regime

R~

<

(x(

<

u~/~t

as region A~ and the _{regime <}

(x(

<

R~

_as

region

B. It is

important

to stress that for _{ut >}

and _u < 1

(which

is the _case when

f ~ f~),

_regions A and B _are both

large

and

well-separated.

Region

A is less

interesting

and

simpler

to

analyse,

_{so we} shall focus _on this first.

4.1. AH IN REGION A. In this

region, F(x)

_may be evaluated _{using a}

steepest-descents

treatment. A

straight-forward

calculation

yields

the result

~Hix)

=

ÎÎÎÎÎÎI Î Î

ÎÎÎÎÎÎÎÎÎIÎÎ [

o

(in(R~ Ii)), Î _i13)

The form of the corrections is of interest. We _see that for d _<

2,

the

leading

term dominates for

(x(

>

R~

_as

expected.

This indicates that the overall

height

deviation within the critical

radius is of order

unity.

For d _> 2 the

microscopic length

scale enters into the corrections and the condition for the dominance of the

leading

term is _now (x( >

R~ In(R~ Ii

We _are therefore

given

^the hint that for d _> 2 the overall deviation of the

height

within the critical radius may be

large (of

order

In(R~ Ii)).

To _see this

explicitly

_{we now} turn _our attention to the form of

AH for _{x <}

R~.

4.2. AH _IN REGION B.

Although

the

integral determining F(x)

_appears

_quite simple,

the

analysis

in

region

B is rather

complicated.

We therefore refer the reader to

Appendix

A where this

analysis

is described in _some detail. The end result for AH in

region

^B takes the form

AH

=

ÎÎÎÎ((~~Îi)~~~ÎÎ~Î~~~~~~~ ₍₁₄₎

(d 2) Ill(Îxlli)

+ °

Il)

d _> 2

where

c(d)

=

2r(d/2) /(2 d). [It

is instructive for d

= 2 to rewrite the

prefactor

of the

height

deviation

exdusively

_in terms of

Rc making

_use of the relation 2

f

=

il In(Rc Ii

_)]

There _are two main features of these results _we wish _{to stress.} First. there exists for all dimensions _a critical radius

Rc

within which the surface

profile undergoes

_a distortion from the linear form shown _in

equation (13).

The cntical radius scales with à

f

_e

f f~

_as

R~

m~

lé f)~~

where

v(d)

=

a(d)/2.

In

particular

_we note that for 2 _< d <

4, v(d)

=

1/(d- 2);

and for d _> 4.

v(d)

=

1/2.

The second

important point

^is that

by calculating

the overall

height

distortion within the critical

radius,

1e.

AH(R~),

_{we see} that

AH(R~)

+~

~°(~) _Ill(Rc/1)

⁰_d

[

^{< d}₂ ^{< 2}

⁽¹⁵⁾

These results confirm the ideas

propounded earlier,

1-e- for d < 2 the distortion of the surface within the critical radius is

negligible,

whereas for d > 2 the distortion is _a

macroscopic

rupture which scales

logarithmically

with distance.

5.

Applications

ta KPZ

Weak/Strong-Coupling

Transition

As _we mentioned in the

Introduction,

there _are two _main characteristics of the

weak/strong coupling

transition for the

noisy

KPZ

equation (which

arise from _a RG

study

in powers of

c = d

2)

for d _> 2. These _are the

vanishing

of the

roughness exponent,

which

implies

_a

logarithmic roughness;

and the existence of _a correlation

length (

_m~

(g g~)~°,

where _{g is} the

dimensionless

couphng constant,

and P

=

Ile

+

O(c)3.

There _{is a} remarkable

correspondence

between these

results,

and those found in the

previous

section for the

simple

model of _a deterministic KPZ interface under _a local

applied

force. To reiterate these

results,

_we found that for d _>

2,

^the

applied

^{force induces} a

logarithmic

distortion of the surface within _a cntical radius which scales _as

R~

_m~

f f~)~~

with _v

=

Ile

for 2 < d _<

4,

and _v

=

1/2

for d > 4.

The natural

question

arises: _are the critical

properties

of the

noisy

KPZ

equation directly

related _to the

simple problem

discussed above? If _so, then _we have _a much easier way ^to understand the

weak/strong coupling phase transition,

and also _we may

identify

_{an upper} crit- ical dimension of 4

(for

the unstable fixed

point characterising

the

transition).

A

quantitative

connection between the two sets of results is

beyond

our reach. However _we shall make several

points

^which

certainly

make the connection _more

plausible.

Since _{we are} interested in the

weak/strong coupling transition,

_we henceforth restrict our attention to d _> 2.

The

coupling

constant g in the

noisy

KPZ

equation

_is

proportional

to the noise

amplitude

D which _we shall take _to be _our control parameter. ^{For D well below} the critical

value,

the _non-

linearity

in the KPZ

equation

_is irrelevant.

Thus,

the surface behaves _{as an} Edwards-Wilkinson

(EW)

interface

[10],

^{which is smooth. As} we increase the value of

D,

^it becomes _more and _more

likely

for small

regions

in the interface to

experience strong

fluctuations due to rare events in

the noise

spectrum.

If the

smooth/rough

transition is

triggered by

_an inherent

instability

of the interface with

respect

to

strong fluctuations,

it is

plausible

_{to assume} that such _an

instability

is localized due _to the local nature of _rare events in the noise.

Having

established

this,

_{we may}

connect the two models

by arguing

that these

strong

local fluctuations resemble _an effective

force _over

long

time scales. There _are three essential

steps

in this

argument

^which _we discuss in tum.

First,

_we must argue that the fluctuations _{appear as a} constant force _{over some}

temporal

interval _T.

Clearly

_any fluctuations in the 'downward' direction

(1.e. anti-parallel

to the direction of

positive h)

will have _no effect since

they

_are

completely suppressed by

the _non-

linearity.

We therefore consider _a noise

path (or

noise

'history')

which tends _{on average} to be

positive

over the interval _T within _some

region

À of size

l~.

_This

path

_may be considered _{as a} force of

magnitude f

_so

long

_as it lies between the bounds

f

+ a. The fluctuations within these

bounds _are irrelevant. This fact _may be established

by considering

the delta-function model

equation (3)

but

allowing

for the force to be _a stochastic variable.

By

_power

counting

one may show that the fluctuations of the force _are irrelevant

(about

the constant force fixed

point)

for d _> 1. For the noise

path

to exceed

f

_{+ a} is

extremely unlikely

_as the

path

is

already

a rare

event _even to have entered between these bounds. The

path

is much _more

likely

to

drop

below

f

_a. This _is deemed _as

switching

off the 'force'. The force will be switched _on

again

when the noise

path

returns within the bounds. Thus the interval _{T over} which the force acts is _a

random variable.

Second,

_we must argue that this random

switching

off and _on of the effective force is unim- portant. ^We assume that the force has been able to create _some disturbance _in the interface

over the

region À,

before it is switched off. How will the surface evolve? To _answer this

question

one may

simply

solve the determmistic KPZ

equation

in the absence of _{any source,} but with _an initial condition

corresponding

to _a localized disturbance. The result is that for any disturbance

exceeding

_a critical

height

h~ _m~

v/À,

the centre of the disturbance

decays logarithmically slowly

it is

essentially

frozen. This effect is due to the

strong

^{influence of} the

non-lineanty driving

the disturbance

upwards against

^the

smoothing

effect of the diffusion

term. So

during

the

periods

of _zero

force,

the disturbance _in the region À is frozen and _on

switching

_on the

force,

the evolution continues _as if there _{were no}

interruption.

The third and final

point

^is to argue that the effective force remains localized in the

region

À. This _is in

general

not the _case. The _noise has

short-range

correlations _{in space} and time

and _rare events _are

equally likely

to _occur

anywhere

in the surface. A

simple

resolution would

be that _once the

rupture

has been formed in

À,

then all other _rare events in the

vicinity

_may be

rotationally averaged

to

give

_an effective force within À. A _more subtle

point

is the

following.

Once the

logarithmic

distortion centered in À has been

established,

it is

essentially

frozen

during

the

periods

of _zero force. Therefore _{any rare event occurring in} À will continue the

evolution of the distortion. On the other

hand,

if the critical

height

for

freezing

is not

small,

then most rare events will fail to seed such _a distortion of the

interface,

_since the disturbances

due to rare events which _are of _a

height

less than

h~

will

simply

diffuse _away. ive therefore

have _a

picture

of

extremely strong

fluctuations

being required

to seed _a distortion of

height

>

h~,

with less

strong

fluctuations

being required

to sustain the evolution of the

logarithmic

rupture once it has been seeded.

If this

rough ^physical picture

^is correct then the

correspondence

between the local force model considered

here,

and the

noisy

KPZ

equation

at the

weak/strong-coupling

transition is

established. The surface

morphology resulting

from such _a picture ^{is that of} a dilute system of

macroscopic distortions,

each with _a

logarithmic profile.

This is in contradistinction to the

more conventional view of weak uniform fluctuations

giving

rise to

logarithmic roughness.

Such

a difference in

morphology

should be discernible from numerical simulations of models believed to lie _in the KPZ

universality

Mass.

[A physically plausible

connection between these two models may be drawn also _in the directed

polymer representation.

The

strong-coupling phase

of the KPZ model

corresponds

to the

low-temperature phase

of the directed

polymer,

in which the

polymer

_is

stongly

localized

onto

low-energy paths,

which have characteristic transverse fluctuations.

Conversely,

the ,,;eak-

coupling phase

of KPZ

corresponds

to the

high-temperature phase

of the directed

polymer

_in

which the

polymer

_is liberated from the low energy

path

and

performs

thermal

wandenng.

Hence the transition between these two

phases

for the directed

polymer

_may be ,rie,ved _as the localization transition to the lowest energy

path (at

this

temperature).

In this _case the

polymer

is

essentially subjected

to _a 'columnar'

defect, although

the defect is not of constant energy.

and also has transverse fluctuations.

Using

similar

arguments

to those

presented

above in the

interface

language,

_{one can} motivate the idea that ₁₎ the energy fluctuations _on the column _are irrelevant within some

bounds,

and

ii)

the transverse fluctuations of the column _ai-e

thermally averaged

such that the effective column is

straight

_{on soine} transverse scale of order

t~.

_We

believe _{a more}

quantitative

^connection _may ^be tractable in this directed

polymer picture.]

6. Conclusions

We have

presented

_a

non-perturbative analysis

of the deterministic KPZ

equation

under the influence of _a local

driving

force

f,

which for d > 2 must exceed _a cntical value

f~

in order to induce _{a non-zero}

velocity

in the surface. The central result is that for d > 2 and for

f ~ fc,

the force _{creates a}

macroscopic

distortion of the surface. This distortion has _a

logarithmic profile

and exists within _a cntical radius of _size

R~

m~

f fc)~~

,vhere

v(d)

=

1/(d 2)

for

2 _< d _< 4 and

v(d)

=

1/2

for d _> 4. In the

penultimate

section _we offered _some

physical arguments

as to

why

_one

might

consider this

simple

model to underlie the

weak/strong-coupling

transition _in the noisy KPZ model. The _main characteristics of such _a scenario _are that ₁₎ the

logarithmic roughness

at the transition is due to _a dilute

system

^of

loganthmic distortions,

_as

opposed

to _a uniform _sea of fluctuations with

logarithmic

_variance, and

ii)

the critical radius

Rc

is _a

physical

realization of the correlation

length

found from the RG treatment. A _more

quantitative

connection between these two models is

presently beyond

_our reach,

although

_one future

possibility

is to estimate the

d-dependence

^{of the}

weak/strong-coupling phase boundary

based _on the exact results obtained in this paper for the local force

model,

_in

conjunction

,vith the statistical

arguments presented

above.

It would be useful _to

apply

the

non-perturbative

methods

presented

here to the _case of _a random local force. Such _an

analysis

would allow _one to

tighten

_some of the arguments given

in Section 5. This

problem

is also of substantial _interest in its own

right

_as it may be

mapped

to a

system

^of two directed

polymers

with random _contact interactions

[8,11,12]

which has _a wide range of

applications.

Acknowledgments

The authors wish _to thank M.A. Moore and L-H.

Tang

for

illuminating

discussions. T.J.

Newman

acknowledges

financial support ^{under SFB 341.}

Appendix

A

In this

appendix

_we

analyse

the function

F(x)

in region

B,

1-e- for 1<

(x(

<

Rc

=

v~~/~,

_in

order to derive the relations shown in

equation (14).

The function

F(x)

has trie

explicit

foi-m

(cf. Eq. (12))

c<

F(x)

=

/

_dt

_(t

+

i~)~~/~ exp(-ut) exp[-~~ lit

₊

i~)], (A.1)

o

which may be

re-expressed

_as

F(x)

=

~~~~

_e~~~

[Ii(~) +12(~)] (A.2)

where

i

I~(y)

=

/

_dt

_{t~~/~e~~~ e~~/~, ~~'~~}

(i/~)2 and

~

12(~)

₌

/

_dt

_{t~~/~ e~~~ e~~/~ (AA)}

i

where _we have defined _{a e}

u~~

=

(~/R~)~

< 1 for notational convenience.

The first

integral

_may be

simply

evaluated to

leading

order _{in a.} ~ie have

c<

Ii (x)

=

/

_du

_u~/~~~

_e~"

+

Dia)

₊

0(e~~~/~)~ (A.5)

1

The

leading

terms of the second

integral

_may be extracted _as follows.

Referring

to equa- tion

(AA),

_we

expand

the last factor of the

integrand

_{as a power} series to

give

I21x)

₌

f

i~)~~Jnia) ^(A.6)

n=o

~'

where

~

Jn(a)

=

a"~~+~/~ /

_dt

t~"~~/~ _e~~ (A.î)

a

Integrating by parts yields

the recursion relation

Jn(a)

=

(~~

_~

~~~

~

~

~~~Jn-i(a). ^(A.8)

For d _<

2, Jo (a)

has the

expansion

Jo(a)

=

Fil d/2) ^a~/~~~ il d/2)~~

₊

01a) (A.9)

where

r(z)

is the Gamma function

[9].

^{This form of}

Jo implies Ji (a)

= 2

Id

+

0(a~/~)

and for

n >

1, Jn

₌

(n

1+

d/2)~~

₊

0(a). Retuming

to

equation (A.6)

_we have

12(z)

=

r(1 d/2) ^a~/~~~

+ C ₊

0(a~/~) _(A.10)

where

~

i

C =

£ ^~~ in

1+

d/2)~~

=

(d/2 1)~~

+

/

du

u~/~~~ [e~" ii. (A.Il)

~ n.

"" o

Combining equations (A.5)

and

(A.10)

then

gives

^for d < 2

Ii +12

"

r(1 d/2) ^a~/~~~ ₍₁ d/2)~~r(d/2)

₊

0(a~/~) (A.12)

From Table I _we have for d < 2

, u =

f r(1- d/2)]~/(~~~),

which in addition to

equations (A.2)

and

(A.12) gives

fF(x)

= i

f(i d/2)~~r(d/2) lxl~~~

+ 0

((lxl/Rc)~) (A.13)

Combining

the above

equation

with

equation il1)

then

yields

the first relation in

equation (14).

A similar

analysis

sullices for

higher dimensions,

which _we shall

briefly

sketch. For d

= 2

one finds

Ii +12

"

In(a)

_{2~/ +}

Dia In(a)), (A.14)

where _~/

= 0.57721.. _is Euler's constant. From Table I _we have for d

= 2

,

u =

i~~ e~~/f,

which in addition to

equations (A.2)

and

(A.14) gives

ffjx)

t 1

2fin(jxj Ii)

₊

o( f). jA.ià)

Combining

the above

equation

with

equation (11)

then

yields

the second relation _in equa- tion

(14).

Finally

^{for d} > 2 _one finds

Ii +12

_"

r(d/2 ₁₎

₊

0(a~) _(A.16)

where

fl

=

d/2

-1 for 2 < d _<

4,

and

fl

= 1 for d _> 4. From Table I _we have for d _> 2,

f

m~

f~

₌

(d/2 1)i~~~,

which in addition to

equations (A.2)

and

(A.16) gives fF(x)

=

na/2) lillxl)~~~

₊ 0

(llxl/Rc)~~lillxl)~~~) IA-1?)

Combining

the above

equation

with

equation (11)

then

yields

the third relation _in

equation (14).

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M.,

Parisi G. and

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J. and

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