Directed polymers in the presence of colununar disorder

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Directed polymers in the presence of colununar disorder

Joachim Krug, Timothy Halpin-Healy

To cite this version:

Joachim Krug, Timothy Halpin-Healy. Directed polymers in the presence of colununar disorder. Jour-

nal de Physique I, EDP Sciences, 1993, 3 (11), pp.2179-2198. �10.1051/jp1:1993240�. �jpa-00246862�

Classification

Physics

Abstracts

05.40- 74.60G 61.40K

Directed polymers in the _presence of columnar disorder

Joachim

Krug(~)

and

Timothy Halpin-Healy(~)

(~) ^{Institut fir}

Festk6rperforschung, Forschungszentrum Jfilich,

P-O- Box 1913, D-52425

Jfilich, Germany

(~)

Physics Departement,

^Barnard

College,

Columbia

University,

New

York,

NY10027-6598, U.S.A.

(Received

10 March 1993, reiised 28 June 1993, accepted 21

July1993)

Abstract. We consider directed

polymers

in

a random

landscape

that is

completely

_corre- lated in the time direction. This

problem

is

closely

related to

diffusion-reproduction

_processes and undirected Gaussian

polymers

in _a disordered environment. In contrast to the _case of _uncor- related

disorder,

_we find the behavior to be _very different at _zero temperature, where the

scaling

exponents

depend

_on the details of the random energy distribution, and at finite temperature, where the transverse

wandering

is

subballistic,

_~

r~

t/(logt)~

with _y

= 1+

2/d

for bounded distributions in d + I dimensions.

Numerically,

these strong

logarithmic

corrections

give

rise to

apparently

nontrivial effective exponents. Our

analytic

results _are based _on

appropriate Flory expressions

for the

(free)

_energy at T

= 0 and T _> 0. Some universal statistical

properties

of the

evolutionary hopping

of the

optimal path

_are also derived.

1. Introduction.

The

problem

of directed

polymers

in _a random

potential [1-3]

is _one of the few _cases in the statistical mechanics of disordered systems ^{where the}

properties

of the disorder-dominated

low-temperature phase

_can be

analyzed

in considerable detail

[4, 5].

^{In addition} to its value

as a

testing ground

for

concepts

and methods in the

theory

of ill-condensed matter, ^this

system

^has attracted much interest due to its ramifications into other classes of

problems,

most

notably higher-dimensional

manifolds in random media [6]

and,

somewhat

surprisingly,

the

nonequilibrium

fluctuations of

moving

interfaces

[7-9].

In the latter _case the directed

paths

encode the

growth

_sequence of the

interface,

and the disorder average becomes _an average

over

growth

histories

[9]. Physical

realizations of the directed

polymer

model itself include interfaces in two-dimensional

systems

^{with bond disorder}

ill

and flux lines in

dirty type

^II

superconductors [10].

Our

understanding

is rather

complete

for the _case of _a

single

transverse direction

(the

I + I- dimensional directed

polymer),

where the behavior is disorder-dominated at all

temperatures.

Of

primary

interest is the fluctuation in the transverse

coordinate,

_{x, as a} function of the time coordinate t

along

which the

polymer

is

directed, given

that its

starting point (t

₌

0)

is

pinned

at _x

= 0. One

expects

a

scaling

law

xc(t)

₊

iwi~/~

~

t~ (1)

where the

angular

brackets denote _a thermal _average and the overbar represents the subse-

quent

_average over the

quenched

disorder. The disorder-induced fluctuations _are

generally superdiffusive,

_so the

wandering

exponent

(

>

1/2.

A second

scaling

exponent ^{describes the}

fluctuations in the free _energy about its extensive disorder _average

fl

r~ t,

IV _T~l~/~

r~

t°. (2)

For _a random energy

landscape

that is uncorrelated both in _x and t the value

(

=

2/3

of the

wandering exponent

^{is known} to be exact in I ₊ I dimensions 11,

7]. Moreover,

_{an average}

Hookes law holds in the _sense that [5]

(x~) ix)~

+~

t, (3)

which enforces the

exponent identity

_11,

2,

8]

9 =

2(

-1

(4)

and thus 9

=

1/3. Remarkably,

these results

apply

both _at finite

temperature

^and at T

= 0

ill (where

the thermal average is

replaced by

a

single optimal path),

_a statement that _can be extended

beyond

the values of the

scaling

exponents to include

amplitudes

and

scaling

functions

ill].

A natural

question

_concerns the

dependence

of the

scaling

exponents on the statistical _prop- erties of the random _energy

landscape. Zhang [12]

^has

pointed

out that non-Gaussian tails in the distribution of the disorder _{may alter the}

scaling

of the

polymers,

_a behavior which is due to the dominance of

exceptional

_energy values _over

typical

_ones

_[13].

The influence of

long- ranged

^disorder correlations in the

spatial (x-)

direction _is also well understood

[6, 8, 14];

^the

identity (4)

remains valid in this _case, and the values of the exponents can be obtained

exactly.

In

contrast, (3)

and

(4)

break down in the _presence of

long-ranged temporal correlations,

and both

analytic

_[8] and numerical [15] ^treatments become _more difficult. In the

present

_paper

we consider the extreme _case of

temporally

correlated

disorder,

when the random

energies

be-

come

independent

oft- This type ^{of columnar} disorder has attracted much interest in recent studies of flux line

pinning

in

high-temperature superconductors _[16],

where it _can be induced

experimentally by

ion irradiation

[17].

^As will be mentioned in section

3,

this variant of the directed

polymer problem

has been addressed

previously

in different contexts.

Nevertheless,

we have encountered _a number of novel and

unexpected -features,

which _we describe in the

following.

We will

mostly

be

working

in I + I

dimensions,

however _many of _our results _are

easily

extended to _a

general

number d of transverse dimensions.

2. Model and main results.

We consider directed

paths starting

at the

origin

^of a square lattice and

proceeding upwards

into the half

plane

t > 0. The transverse

displacement

is restricted to at most _one lattice

spacing

_per time

step. Apart

from this

restriction,

_no additional _energy cost is associated with horizontal bonds.

Quenched

random

energies e(x)

_are

assigned independently

to the columns

of the lattice. At _a

given temperature T,

the restricted

partition

function

Z(x,t)

for all

paths running

^from the

origin

to the

point (x, t)

then evolves

recursively according

to

ill

Z(x,

t +

I)

₌

e~~~~~/~'[Z(x, t)

+

Z(x I, t)

+

Z(x

+

I, t)] (5)

with initial condition

Z(x, 0)

₌

_b~,o.

Note that due to the restriction _on transverse

steps Z(x, t)

vanishes outside of the

wedge

-t < _x < t. We will refer to this

setup

as the

wedge geometry.

The total

partition

function is obtained from

(5) by summing

_{over x,} and the free _energy is

given by

t

F(t)

= -TIn

~=-t ~j _{Z(x, t) (6)}

When

only

the free energy fluctuations

(2)

_are of

interest,

it may be _more efficient

computation- ally

to _use the substrate initial condition

Z(x, o)

₌ I with

periodic boundary

conditions _{on a}

transverse

strip

of width

L, corresponding

to the simultaneous

propagation

of L directed

paths.

The disorder average can then be

replaced by

_{an average over x}

ill].

The

price

to be

payed

is the introduction of finite size effects when tC becomes

comparable

to L. At _zero

temperature (5)

reduces to _a recursion for the restricted

ground

state energy

E(x, t)

=

limr-o

T In

Z(x, t) ill,

E(x,

t + I

=

min[E(x, t), E(x I, t), E(x

+

I, t)]

+

e(x). (7)

In this _case both the transverse

positional

and the energy fluctuations _can be obtained in the

substrate

geometry [18].

The focus of this work is _on bounded distributions for the random

energies e(x).

In

particular

we have considered _a

one-parameter family

of continuous distributions _on the unit interval

Pu(E)

₌

(v

₊

I)e" (8)

with _{v >}

-I,

_as well _as the

binary

distribution

p~(e)

₌

bd(ej

+

(i bid(e 1), (9j

0 < b _< I. In _a certain _sense

Pb

_can be

regarded

_as ^the v - -I limit of

P~,

since in that limit

P~ acquires

a finite

weight

at e = 0. We will also

present

some T

= 0 results for the

family

of

power-law

distributions

[12,

13]

P~(e)

₌

~(e(~~"+~~,

e <

-1, (10)

with _~ > 2.

Our central result is the breakdown

of

the

equivalence

between the T

= 0 and the T > 0

scaling properties

familiar from the _case of uncorrelated disorder. At T

= 0 the

scaling exponents

are

nonuniversal, showing

a sensitive

dependence

_on the details of the energy distribution

(Sect.

4).

In

particular,

_we show that

~ ~~~~

for the bounded distributions

(8), (

= 9

= o for the

binary

distribution

(9) (note

that

(

_- 0 for _{v -} -I in

(II)),

and

(=1, 9=1+1/~ (12)

for the _power law distribution

(10). Qualitatively,

the absence of

universality

at T

= 0 is due to

the fact that

optimal configurations

of the directed

polymer

_are localized at individual columns associated with

exceptionally

low

energies.

In

contrast,

in the

problem

with uncorrelated

disorder the

optimal path

encounters _an extensive

(r~ t)

number of

impurities,

^and a type ^of central limit theorem _ensures the convergence ^to a universal distribution.

At T _> 0

universality

is

largely restored,

however the

simple

_power law

scaling (I)

is

replaced by

an

asymptotic

behavior of the form

xc(t)

+~

~~(~~~ (13)

where _~i

= 3 for bounded distributions

(Sect. 6).

While

(13) implies formally

that

(

₌

I,

our

simulations show that the

large

value of _~i leads to

effective wandering exponents

^which remain

significantly

different from

unity

on

numerically

accessible scales.

In _a

single

realization of disorder the time evolution of the

thermally averaged position (x)

_or, at T

=

0,

the

position

of the

optimal path,

shows _an intermittent

pattern

with the total transverse

displacement occuring during

_a few

abrupt jumps.

In section 5 _we derive two universal relations for this kind of

evolutionary hopping. First,

the cumulative number of

jumps

_up to time t is

proportional

to In t.

Second,

the distribution of the range R covered

by

the first

jump decays

_as

^R~~

3.

Summary

of

previous

work.

The recursion

(5)

lends itself to _a number of

interpretations.

Nieuwenhuizen

[19]

^studied it

(with

the

binary

distribution

(9)

and _an additional

boundary condition)

_{as a} model for

wetting

ⁱⁿ a two-dimensional

system

with bulk disorder

fully

correlated in the direction of the substrate. If the time t is

thought

to label the _monomers

along

_a

polymer chain, (5)

describes the conformations of _an undirected Gaussian

polymer,

with _one end fixed at the

origin,

in

a one-dimensional random

potential _[20, 21].

^{In other}

applications

_{one may}

regard Z(x,t)

as the

population density

^of some

(chemical

_or

biological) species

which

migrates diffusively

and

reproduces (or

is

trapped)

at a

position-dependent

random rate

[21-23].

Here much interest has focused _on the

asymptotic

behavior of the moments of Z

[24].

^{In the}

biological

context the x-coordinate could

represent

either real _{space, or a} one-dimensional

phenotype

space, in which each

point corresponds

to _a set of

organismic properties [22].

^The

analogy

to

biological

evolution has

inspired

the _use of

(5)

_{as an}

optimization strategy

in

complex landscapes [25]. Finally,

in the _zero

temperature

^limit

(7)

the recursion becomes

equivalent

to the one-dimensional

polynuclear

model of

crystal growth

_[9] with

time-independent

random local

growth

rates.

These

(and

several

other)

versions of the

problem

have

generally

been treated within the continuum

approximation

to

(5),

which takes the form of _an

imaginary-time Schr6dinger

_equa-

tion

fi

fi2

h~~~~

~~ ~

3X~ ^{~ ~}

~~~~~

_~~~~

with _a Gaussian random

potential

V characterized

by V

₌ 0 and

v(x)v(xi)

₌

v2d~(x xl), (15)

where

ba(x)

is the d-function smeared _{over some}

microscopic

distance _a. The free _energy

F(x,t)

=

-lnZ(x,t)

then satisfies

~F(x, ^t)

=

D$F

_D

()F)

_V

₍₁₆₎

t _X X

~

which is the

Kardar-Parisi-Zhang equation

for _a

moving

interface

[7,

9]

subject

to time-

independent

random

growth

rates

[20, 21, 26].

The energy

landscapes

of the models considered here _can therefore be viewed _as

evolving, rough surfaces,

the

height

of which at _a

given

site x

and time t is the

(ground

state or

free)

_energy of all directed

paths

of

length

t which end _{at x} and start

anywhere

_on the substrate t

= 0

ill].

The

importance

of the disorder _can be estimated

[21] by comparing

the two terms on the

right

hand side of

(14)

_{on a} scale R » _a. Their ratio defines _a dimensionless

coupling

constant

g(R)

=

(V/D)R~~~/~

which increases with

increasing length

scale in transverse dimensions d <

4, showing

that the

large-scale

behavior is dominated

by

the disorder. In the weak

coupling

regime

_g <

I,

studied _e-g-

by

Cates and Ball

[20],

^the

polymer

_{averages over many}

impurity

sites and the effective disorder distribution becomes Gaussian

by

virtue of the central limit theorem.

However,

in the

strong coupling (large scale) regime

_g » I the

polymer

is

pinned by

individual

impurities,

and their energy

distribution,

which is _a

microscopic,

nonuniversal

feature of the

system,

becomes of crucial

importance.

In _most of the work based _on the continuum

equation (14)

the on-site energy distribution

has, explicitly

_or

implicitly,

been assumed _to be Gaussian. It _so

happens

that _a Gaussian disorder distribution

gives

rise to

weakly

subballistic

scaling

of the form

(13)

both at T

= 0 and at T _> 0. As _a consequence, the difference between the _zero and finite

temperature

_cases, as well _as the

possibility

of

nonuniversal

scaling

exponents at T

= o has been obscured in much of the earlier discussion.

The

problem

_was

recognized by Zhang _[23],

who

pointed

out that

(

₌

1/2

for _a uniform energy distribution _at T

= 0

(the

_{case v}

= 0 in

(II)),

and

observed,

in simulations of _a discrete model similar to ours, a crossover to _a behavior of the

type (13)

at finite

temperature. However,

as we shall _argue

below,

the finite

temperature scaling

cannot be attributed to the central limit theorem _or the fact that the

system effectively

_{averages over many}

impurities. Rather,

_our

analysis

based _on the connection to Anderson localization [22] reveals the Lifshitz

phenomenon occuring

at the baud

edges

of random

Schr6dinger operators

^{with bounded disorder}

[27]

to be

responsible

for the universal behavior at T > 0. We show that the correct behavior both at zero and _at finite

temperature

_can be extracted from

appropriate Flory-type expressions

^for

the

(free)

_energy; in this _{sense we propose a} unified

picture [28]

^{whicl~ should} cover

arbitrary

on-site distributions of disorder. In order to make _contact with the bulk of

previous work,

we

will also

apply

_our

approach

to the _case of _a Gaussian distribution.

4. Zero

temperature.

The

optimal path

has _a

simple

form

[25] Starting

from the

origin,

it travels to _a favorable

site at _some transverse distance R and remains there _up to time t. In order to maximize the

time

spent

at

R,

the transverse

displacement

_occurs at maximal

speed,

I-e- the

path

_moves

sideways

_one lattice

spacing

_per ^time step. The _energy of the _transverse

portion

of the

path

is

zR,

where z is the _average of the random energy distribution. The total _energy of the

path

can therefore be estimated _as

~(R, i)

₌ zR +

(i RjE~~~(R) (17)

where

Emin(R)

denotes the lowest energy

expected

to be encountered in the

spatial

interval

i-R, RI.

This

quantity

^is related to the extremal statistics

[29]

^{of the} energy distribution. For

a

given P(e),

the distribution of the smallest among N

independently

chosen random

energies

is

PN(e)

~

=

NP(e)[I ~co deP(e)]~~~, (18)

and,

since _a transverse

displacement

R

implies

that the

path explores

2R +1

sites,

Em;n(R)

₌

/ ~

_de

_{eP2R+1(e). (19)}

For the

family

of bounded distributions

P~(e)

defined in

(8)

this

yields, asymptotically

for

large R,

Emin(R)

cs

r(1/()(2R)~~/~"+~~ (20)

with

( given by (II ),

^while the power law distributions

(10) give

E~j~(R)

_m

-r(i 1/~)(2R)1/H (21)

Quite generally, _(Emin(

is _a

decreasing (increasing)

function of R for bounded

(unbounded)

energy distributions.

Consequently

the dominant _energy

cost,

^{which has} to be balanced

against

the _energy

gain tEmin

in

(17),

is

given by

the first term zR for bounded

distributions,

but

by

the third term

-REmin

for unbounded distributions

(note

that then

Emin

<

0).

In the bounded

case this

implies

that 4l

r~ R for the

optimal path, showing

that the energy fluctuations _are of the _same order _as the

positional

fluctuations and hence 9

=

(.

In

contrast,

^{for unbounded} distributions 4l

r~

Emin(R)R which,

for the power law distributions described

by (21),

leads to 9

=

((1+1/~).

To determine the actual value of the

wandering exponent,

we use the

appropriate expression

for

Emin

in

(17)

and minimize with

respect

to R. For the bounded distributions

P~

_we obtain

with

(

=

(v

₊

I) /(v

+

2)

_as claimed in

(11),

while the result for the _power law distributions

P~

reads

R _m ^~

t, (23)

2 _{+ ~}

so

(

₌ I and 9 ₌

1+1/~.

We note that 9 _>

1,

which

implies

that the

ground

state energy per unit

length diverges

as

t~/" (see Fig. 2).

For the

binary

distribution

(9)

the _zero

temperature problem

is

obviously

trivial: Once the

path

has found _a site with _{e =}

0,

which is

expected

to

require only

a small transverse

displacement,

there is _no incentive to venture further _away, hence

(

= 0.

In

figures

I and 2 _we show numerical results for the bounded distributions

P~,

and the power law distributions

P~, respectively.

While the

predictions

for the

exponents (

and 9 _are in excellent

agreement

with the

simulations,

the

prefactors

in

(22)

and

(23)

do not appear ^to

be exact. This is not

surprising,

since in _our derivation the disorder _average,

leading

to the

function

Emin(R)

in

(17),

_was

performed before minimizing

with

respect

to R.

The above

arguments

_are

easily generalized

to d _transverse dimensions. The number of

sites, N,

_among which the minimum energy is

sought,

then scales _as

R~,

which

implies

that

Emin(R)

_r~

R~~/("+~)

_{for the bounded} distributions

P~,

and

Emin(R)

r~

R~/"

_{for the power} law distributions

P~. Inserting

into

(17)

and

minimizing yields

the

exponents

(=9= ~)~~ (24)

for the bounded _case, and

(

"

1,

9

= 1 +

d/~ (25)

loo loo

I u= -I/Z I _u=I

O O

Z Z

/~'

[

~ ,l'

(IO

_,/~ ~_(IO

C II' (=1/3 C

~ l' ~

00 1' ₀₀

~ l' ~

( l' [

© /" _©

~ /" _~

c /~ c '

~ /~ ~ ,'

q ^~~ q ^I ,'

c c ,'

O O ,

Z Z ,'

G G (=Z/3'

~

(

o I

io loo I°°°

o

~~ ~~~ Ioo0 10~

t t

a) b)

Fig.I. a)

Positional fluctuation

[~opt(t)[ (upper curve)

and ground state energy fluctuation

[$-

~~]~/~ (lower curve)

for bounded disorder with _u

=

-1/2,

obtained in the substrate geometry with

a transverse system s12e L ₌ 5 x 10~. _{The dashed line indicates the}

prediction (

= 0

=

1/3; b)

Same

as

a)

for

u = I. These data _were obtained from 10~

independent

_runs in the

wedge

_geometry.

H"3 ^IO

H=3

,

~

,

c

,

~

/ /

)

^6=4/3

,/

,

/ ,

~ loo

,

w

~

/

b ,' ^{,, ,}

,/

j ^,

~ ^,

, II w

/

T ,' ,/ ^v

,/

~ ,' ^,'

/

~

,

/

( ^{lo ,'}

,,'

^,/

( ^{' "} ("I

I' slope1/3

j ^,,

, /

,,

/

,

/ /

,,'

^/

,

/ , , / / /

l 1° 1°° 1°°° lo loo loco

t

a) b)

Fig.2. a)

Positional fluctuation

[~opt(t)[ (lower curve)

and

ground

state energy fluctuation [E]~~

~~]~/~ _{(upper curve)}

_for

power law disorder with _p

= 3, obtained from 10~ disorder real12ations in the

wedge

geometry. ^{The dashed lines indicate the} predictions for

(

and 0;

b) Divergence

of the

ground

state energy per unit

length,

obtained from the _same simulations _as

a).

JOURNAL DE PH>s<0uE1 T ~, N'ii, NOVEMBER 1991 80

for the _power law distributions.

It is

amusing

to note that the

nonuniversality

with

respect

to the distribution may, in _some

instances,

_appear as a nonuniversal behavior with

respect

to the

precise

definition of the model.

Consider,

for

example,

_a ^modified model

[30]

^{in which the}

paths

_are not allowed _to remain at the _same site for two

subsequent

time steps. ^{Rather than}

looking

for _a

single

favorable

site,

the

optimal path

_now has to find the

pair

of

neighboring

sites of lowest _energy within the distance

R,

between which it _can then alternate. The distribution of the total _energy of the

pair

is the self-convolution of the

original

_energy distribution. For the

P~

this

implies

_a

renormalization _{v -} fi

= 2v

+1,

^{hence the}

wandering _exponent

for the modified model is increased _to

(

₌

(P

₊

I) /(P

+

2)

=

(2v

₊

2) /(2v

+

3).

To end this section _we

apply

_our

approach

to _a Gaussian energy distribution

The extremal statistics calculation

yields

_[21]

E~~~(R)

_~z

W(iu(R))1/2 ₍₂₇₎

Inserting

in

(17)

and

minimizing _[31] yields

2R In R _m t

(28)

which,

for

long times,

leads to _an

asymptotic

behavior of the form

(13)

with _~i

= I. Here _we

differ from Nattermann and Renz

[21],

^{who found} _~i =

3/4

for the _same

problem.

The

discrep-

ancy arises because

they, working

in _a continuum

model,

assumed _no limit to the transverse

speed

of the

optimal path,

I-e- to the

angle

between the transverse

portion

of the

path

and the t-axis

[25].

^{If the}

speed

is

optimized (using

_a conventional elastic energy cost

R~ It

for the

transverse

stretching

instead of the zR term in

(17))

_one finds that it

diverges

_as

(lnR)~/~,

which _accounts for the smaller value of _~i.

Possibly

this _case is realized in _a lattice model with

no

explicit

restriction _on the range of transverse

steps.

5.

Evolutionary hopping.

It is of _some interest to consider in detail how the transverse

displacement (I)

acccumulates for _a

single

realization of disorder. This

aspect

is of

particular importance

when

assessing

the

efficiency

of the transfer matrix

(7) (or (5))

_{as an}

optimization strategy

^used to locate

minima of the random

potential _[25].

In

figure

3 _we show the time

dependence

of the end

point

of the

optimal path xopt(t)

at T

=

0, along

with the evolution of the

corresponding

energy

e(xopt).

The behavior is

highly

intermittent: The

optimal

end

point

^{remains fixed for}

long

stretches of

time,

and the transverse

displacement

_occurs

during

_a few

abrupt jumps, corresponding

to _a switch between two

optimal paths

whose

energies

_cross. In the

language

of _current

evolutionary theory, long periods

of stasis _are

punctuated by

short

episods

of

rapid

change,

_a

pattern suggested by

the fossil record to be

typical

of

species

evolution

[32].

On _{a more}

quantitative level,

_we observe that the average number of

jumps

_up to time t increases _as In t

(Fig. 4).

This behavior _was

previously

found

by

Kauffman in _a

study

of

long-

range

adaptive

walks _on

rugged

fitness

landscapes _[33].

It follows from _a well-known order statistics

argument [29, 34]

^{if the full}

optimization problem

is

replaced by

_a

simpler

_process in which the

endpoint always occupies

the site with the smallest _energy available at time

t,

~(z~~~(t))

"

fifjl~iE(~)l'

29

loo

a) b)

50

d d

0 ~

o1

-50

1

t t

-

with =

2; b ) _Energies

of

the sites

visited by

the

path.

5

« 4

cu

~ H=3

~

~ 3

fl

u=I c

( Z

j

? fi

I

0

0 Z 4 6 8

in t

Fig.4.

Cumulative number of

evolutionary jumps

_up to time t for power law disorder with _{~ =} 3

(upper curve)

and bounded disorder with

u = I

(lower curve).

Each data set is the average over 10~

realizations.

A

jump

occurs at time t if _one of the two sites _x

= ~t which become

newly

available is _more favorable than all the

previously

available sites

-(t I)

< _x < t I.

By _symmetry,

the

probability

for this event is

2/(2t +1),

and therefore the

expected

number of

jumps

_up to time

t is

fi(t)

=

~j

^~

m In

t, (30)

~_~

2t +1

a universal result valid for _any continuous disorder distribution. The additional correlations

present

^{in the full}

optimization problem

show _up in the

prefactor

of the

logarithm,

which is less than

unity

and distribution

dependent (Fig. 4).

A

rough interpretation

of this observation is that the effective number of

independent paths

available to the

optimization

_{process grows}

as some sublinear _power of t.

The

simplified

_process discussed above

suggests

the

following approximate picture

for the evolution of the

energies

of the sites visited

by

_xopt. We _assume that

given

the energy

e(xopt)

=

en for the

endpoint

of the

optimal path,

the _next

jump

_occurs to _a site with _an energy ez~+i chosen at random from the disorder

distribution, subject only

to the constraint that _en+i < en.

We therefore

write,

for _an

arbitrary

disorder distribution

P(e),

f()

de

eP(e)

~~~~

J])

dE

P(E)

^~~~~

where the disorder _average is conditioned _on the value of _en. For the

P~

this

implies

that the _average site energy decreases

exponentially

with the number of

jumps,

G =

("

where

(

=

(v

+

I)/(v

+

2)

is the

wandering

exponent.

Equating

this with the power law

decay e(xopt)

_+~

Emjn(R)

r~

t~(~~~~

_we obtain

fi(t)

_m

j~j~j~

ml. ~~~~

While the

expression

for the

prefactor

in

(32)

is _not

quantitatively

correct, the

argument

does indicate how _a

prefactor

different from

unity

_may arise.

The

corresponding

calculation for the _power law distributions

P~ yields

_an

exponential

di- vergence G ₌

-(~/(~ l))", which, equated

to the time

dependence -e(xopt)

+~

t~/",

_also

results in fi

r~ In t

(cf. Fig. 4).

For the Gaussian distribution

(26), (31)

reduces to

@

"

)e~~i [erfc(-en)]~~ (33)

where

erfc(x)

is the

complementary

_error function.

Expanding

this function for

large arguments

we recover the known result

[22, 23] @

_en re

-1/(2ez~),

which

implies

that G re

-fi

for

large

_n.

According

to

(27),

the site

energies

decrease with time _as

-(Inn)~/~

,

so,

again,

we

obtain the

relationship

fi

r~ In t. These

arguments apply

in

arbitrary

transverse

dimensionality

d. Related considerations for the finite

temperature

_case will be

given

in section 6.

Exact results _are available for the statistics of the

first

transverse

jump

in the full

optimiza-

tion _process. Consider the

probability r(t)

that _no

jump

has occurred _up to time

t,

for the

family

of bounded disorder distributions

P~(e).

This

implies

that the

energies

of all

optimal paths

which _can be reached at time t _are

larger

than the energy

e0t

of the

path

that remains at the

origin,

I.e.

ix +

(t x)e(x)

> e0t

(34)

for _x

=

~l,...,

~t. Here _we have assumed that the energy of the transverse

portion

of the

path

can be

replaced,

for

large

_x and

t, by

its average ^ix.

Using

the

independence

of the

e(x)

and the

simple

form of the distribution

P~

this leads to

r(t)

m

~

deo jj(I

(eo

ix

/t)"+~)

~

~=i (35)

0

~

a) b)

o

~

° slope -3/2 0

-2

p _'s ~,

~ ~

~

E &

g ^I

O ~

-4

o oi

0 2 3 lo loo

logic t R

Fig.5. a)

Distribution of the time t of the first

evolutionary jump,

obtained from 10~ realizations of bounded disorder with

u = 0

(uniform distribution), using

_a maximal

path length

of 103. _{The full} line indicates the

predicted t~3/~ decay; b)

Distribution of the range R of the first

jump

for the _same

disorder distribution _as in

a).

The data _were

averaged

_over 4.2 _x 10~ realizations with

a maximal

path

length

of t ₌ 10~. The full line is the

rigorous

lower bound

(37).

The saturation at

large

R is due to finite size effects.

with x*

=

(eo/z)t. Evaluating

this

expression asymptotically

for

large

t results in

T(t)

m

r(()(i(v

₊

1)/21~ t~~ (36)

and the distribution of

jump

times

decays

_as

-dr/dt

r~

t~(~+()

_{Simulation results for the}

jump

time distribution _are shown in

figure

5a. We also note that for the

simplified

_process

(29)

the

argument leading

to

(30)

shows that

r(t)

r~

Ill,

which is consistent with

(36)

and the fact that

(

₌ I for this process.

Together

with the

scaling

relation R

r~

t( (36) suggests

that the

probability

distribution for the range R covered

by

the first

jump

should

decay

_as

^R~~ independent

of

(.

This follows from _a

simple

order statistics

argument

similar to that used in the derivation of

(30).

The end

point

of the

optimal path jumps

from the

origin,

with energy eo selected at random from the disorder

distribution,

to _a site at _a distance R with energy ei < e0. This site need _not be the closest site from the

origin

with _an energy less than _e0i

rather,

it is the site with the shortest

transition time

[22, 25]. Nevertheless,

if the

energies

of all the N

= 2R sites within range R from the

origin

have

energies larger

than _eo, the first

jump necessarily

has to _{cover a} distance

larger

than R. The

probability

for this event is

equal

to I

/(N

+

I),

I.e. the

probability

that _eo is the smallest among N ₊ I

independent

random variables

[29, 34].

This

implies

_a

rigorous,

universal bound _on the cumulative distribution of the range of the first transverse

jump,

P~um(R)

e

,~+i ~j P(R')

>

(37)

~~ ~

In

particular,

it follows that

P(R)

cannot

decay

_more

rapidly

than _as

R~~,

and the

expected

jump length

is infinite. Our numerical data shown _in

figure

5b indicate that the bound

provides

a

good approximation

to the actual distribution

(note

that in _a finite

system

of

length

t the cumulative distribution saturates at R

r~

t(,

since

jumps

of

larger

^size would not have had time to

occur).

In d transverse dimensions the number of sites within distance R scales _as N

r~

R~

and

consequently

the

decay

of

P(R)

is bounded

by R~(~+~)

6. Finite

temperature.

In

figure

6 _we show the finite

temperature equivalent

of the

position

of the

optimal path,

the

thermally averaged

transverse

displacement ix),

for _a

single

realization of disorder. The _punc-

tuated

pattern

is _seen to be very similar to the T

= o _case; the extent of thermal

smoothening

of the

jumps

_can be assessed from the

peaks

in the width

((x~) (x)~)~/~

of the thermal

wave

packet,

which

develop

because

Z(x, t) briefly

becomes bimodal _as the average

(xi

shifts between two

positions

_[25]

(Fig. fib).

Looking

at these

pictures,

_one is lead to

inquire

about the nature of the attractors that define the

preferential

locations of the

wavepacket.

In contrast to the T

= 0

situation, they

do not coincide with individual lattice

sites,

and cannot,

therefore,

be

directly

associated with local minima of the energy

landscape. Following Ebeling

et al.

[22]

^and

Zhang [23],

we

identify

the attractors with the localization centers of the

eigenstates

^of the

time-independent

version

of the

discrete, imaginary

time

Schr6dinger equation (5),

which

satisfy

iffy

(x)

_{+ iffy}

(x

+

I)

₊

iffy

(x I)

=

e~~~~/°~~~'ifij (x), (38)

where

Aj

^{denote the}

quasienergies.

For bounded disorder distributions with

e(x)

> 0 solutions of

(38)

exist for ln3 < Aj ^< cc, and the

long

time behavior of

Z(x,t)

is

governed by

the

low-lying

states close to A0 " -ln3.

Equation (38) belongs

to a

large

class of disordered

systems

for which the

low-lying

states _are

exponentially

localized

[35],

~~

~x~ ~

~-i~-~,

i/t,

~~~~

where xj ^{is the} localization center and

fj

is the localization

length. Expanding Z(x, t)

in the

~lj

^and

using

the initial condition

Z(x, 0)

=

b~,o

we obtain [22]

Z(x, t)

_~3

~j _{exPl-(lxj I/fj}

₊

_lx

_xj

_I/fj

₊

_{Ajt)1. (40)}

Clearly,

the localization centers

correspond

to local maxima of Z. To decide which state dominates the _sum at a

given

time

t,

the

argument

of the

exponentials

in

(40)

has to be maximized. This involves

knowing

the smallest value of

Aj likely

to be encountered in _a

region

of

given

size. We therefore introduce _a function

Amin(R),

which is the

analogue

of the lowest

energy on the scale

R, Emin(R),

used in the _zero

temperature

case. It _can be

computed

from the extremal statistics formula

(18),

if the bare disorder distribution

P(e)

is

replaced by

the

density of

states

p(A)

of the Anderson

problem (38).

Once

Amin(R)

and the localization

length f

_are

known,

the

typical

distance of the dominant localization center in

(40)

can be estimated

by minimizinj

the effective

Flory expression

~(R, i)

= R

It

+

n~~~(R). (41)

At this level of

generality,

_we conclude that the introduction of _a finite

temperature

has two consequences:

First,

the bare disorder distribution P is renormalized to the

density

of states p,

Second,

the _zero

temperature Flory expression (17)

is

replaced by (41),

0

a)

lo

AI

-20

-30

-40

lo loo loco

bj

_c~

~~ _Z$

i ~ ~*

~ ~

E S

[ +

~ Z$

~ ~

lo loo iooo to loo iooo

t t

Fig.6. a)

Motion of the

thermally averaged position

for _a

single

realization of

binary

disorder with b

= 0.3, at temperature T ₌ 0.5;

b)

Width

[(~~) (~)~]~/~

of the thermal

wavepacket.

Note the

correspondence

of the peaks with the jumps ⁱⁿ

a); c)

Decrease of the free energy per unit

length

#(t)

₌ -T III

Z/t

for the

same disorder realization and temperature _as in _a

)

and b

).

Our analysis shows that

#(t +1) #(t)

_c3

TAj

where Aj ^{is the}

eigenvalue

of the localized state at which the

wavepacket

resides at time t. For

long

times

TAj

- TAO _" -TIn 3 _@ -0.549306...

Bounded disorder distributions _are

universally

characterized

by

_a Lifshitz tail in the inte-

grated density

of states

k(A)

₌

f/ _dip(i)

near the baud

edge

A0, ^{I.e. in d} dimensions

k( j)

_~s

e-c(>->o>~~/~ (42j

with _a

positive

constant C. This relation holds

rigorously

in the _sense that

[27]

lid _l~)~~~~~

~~~~

Using (42)

ⁱⁿ the extremal statistics estimate

(18)

and

(19),

_we obtain

Amin(R) lo

+~

(In R)~~/~ (44)

Since the kinetic _energy of _{a state} of

spatial

extent f is of the order

I/f~,

the states which dominate the

partition

function

(40)

_occupy

regions

of size

(lnR)~/~

This scale is small

compared

to the distance R from the

origin.

^Hence the

overlap

of such _a state with the

origin,

which

gives

rise to the first term in

(41), decays exponentially

in R with _a localization

length f

=

O(I)

which is

independent

of R

[36]. Using

this and

(44)

in

(41),

and

minimizing

with

respect

to R

[31]

we obtain R

r~

t/(Inn)~

with

~i = 1 + 2

Id. (45)

The _crossover from the T

= 0 behavior described in section 4 _occurs when the

typical

_energy differences between

competing optimal paths

become of the order of T. Since these _energy differences decrease _as

t~(~~°),

where 9 _< 1 is the _zero

temperature exponent,

^the _crossover time scale is

given by

t*

~_

T-i/(i-o) (46)

The

resulting picture

for the time evolution of

Z(x, t)

is confirmed

by

recent

rigorous

results due to Sznitman

[37]. Studying

continuous Brownian motion in the presence of Poissonian

obstacles,

he _proves that at

long

times the _process is concentrated in ^"

clearings"

of size

(In t)~/~

associated with low local

eigenvalues

of the

corresponding Schr6dinger

operator which _are of the order

(In t)~~/~. Moreover,

his work shows that

(

₌ I in the _sense that

limt-co ^t( /xc(t)

= 0

for _any

(

<

I,

and

provides rigorous

bounds _on the

logarithmic

correction exponent ~i, viz.

~i > 2

Id

for d > I and _{~i <} 6 in d

=

I,

consistent with _our estimate

(45).

Severe corrections to the Lifshitz behavior

(42)

arise for

continuous,

bounded disorder dis- tributions such _as the

P~(e)

defined in

(8) _[38].

To

leading order, (42)

is

replaced by

~( j)

_~

~-cj in(>->a)j(>->o)-~/~ (47)

Note that this does not affect the

rigorous

result

(43), only

the rate at which the limit is

approached.

The

corresponding

correction to the _transverse

wandering

is of the form

(1n(Inn))2/d

~/ ~48)

'~

(lnt)1+2/d'

Needless to say, such corrections _are

extremely

difficult to detect

numerically [38, 39].

We therefore

begin

_our discussion of numerical results at finite

temperature

^with the

binary

disorder

distribution

(9),

for which _{a pure} Lifshitz tail is

expected.

In

figure

7a _we show that the data for _xc

(t)

=

[(x)2] ~/2,

_as well _as the free _energy

fluctuations,

are

reasonably represented by

conventional power law

scaling

with _a value

(

_re 9 _m 0.58 of the

scaling exponents

^{that is} _very ^{far from the}

asymptotic prediction (

= 9

= 1.

However, plotting

xc

(t) It

_vs. ln t _we find _a

long

time behavior of the type

(13),

with _an exponent _~i

approaching

the

predicted

value _~i

= 3

(Fig. 7b);

_a least squares fit for t

= 500 loco

yields

_~i

= 2.74 for

z~(t),

and _~i

= 2.86 for the second moment

x2(t)

₊

_(x2)~~~

In

particular,

_our data rule out