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arXiv:cond-mat/0212526v2 [cond-mat.dis-nn] 19 Aug 2003

EPJ manuscript No.

(will be inserted by the editor)

Universal properties of shortest paths in isotropically correlated

random potentials

Roland Schorr and Heiko Rieger

Theoretische Physik, Universit¨ at des Saarlandes, 66041 Saarbr¨ ucken, Germany

Received: date / Revised version: date

Abstract. We consider the optimal paths in a d-dimensional lattice, where the bonds have isotropically

correlated random weights. These paths can be interpreted as the ground state configuration of a simplified

polymer model in a random potential. We study how the universal scaling exponents, the roughness and

the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results

using Dijkstra’s algorithm to determine the optimal path in directed as well as undirected lattices indicate

that the correlations become relevant if they decay with distance slower than 1/r in d = 2 and 3. We show

that the exponent relation 2ν −ω = 1 holds at least in d = 2 even in case of correlations. Both in two and

three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations.

PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.50.+q Lattice

theory and statistics (Ising, Potts, etc.) – 64.60 Ak Renormalization-group, fractal, and percolation studies

of phase transitions – 68.35.Rh Phase transitions and critical phenomena

1 Introduction

Optimal paths have been a subject of intensive studies

during the recent years. Besides being one of the simplest

problems involving disorder, this interest can be traced

back to the relevance of this problem to various fields,

such as polymer models [1,2,3], surface growth [4], ran-

dom bond ferromagnets [5,6,7], spin glasses [8], and the

traveling salesman problem [9].

The model under considerationis easily sketched: given

an arbitrary weighted graph, each edge has a particular

cost. The optimal or shortest path connecting two sites is

the one of minimal weight, which is the sum of all costs

along that path. We do not restrict to a particular geome-

try yet as well as we do not specify the costs more precisely

so far. In the simplest case, we choose them from a set of

random numbers that are uniformly distributed. In this

context, the directed polymer model has drawn the most

significant attention during the past years [1,2,10,11,12],

where one assumes in d = 2 a simple square lattice being

cut along its diagonal and oriented as a triangle with the

diagonal as its base. One allows only paths in the direc-

tion to the base, i.e., the path cannot turn backwards. The

costs of the edges that belong to the shortest path are in-

terpreted as potential energies for a polymer configuration

that passes through these edges (or bonds).

We may now ask whether the properties (scaling ex-

ponents) of the shortest path are either influenced by the

distribution of the random numbers or the geometry of

the lattice. The former is still discussed [1,13,14,15,16].

As far as the latter is concerned, it seems to be clear that

the universal properties are not changed if the random-

ness is uncorrelated. For this case Schwartz et al. [17] in-

vestigated directed and undirected paths in d = 2,3 using

Dijkstra’s algorithm to find the shortest path and Marsili

and Zhang [18] used a transfer matrix method approach

considering directed and undirected paths up to d = 6.

Both state that overhangs exist but nevertheless they sug-

gest that both problems belong to the same universality

class, even in high dimensions where overhanging config-

urations are entropically favored. It is not a priori clear

that this observation remains true for correlated disorder.

In fact, as we will point out below, the average number of

overhangs increases for strongly correlated disorder indi-

cating that they might become relevant for strong enough

correlations.

In the present study, we study the universal proper-

ties of shortest paths and focus on the effect of isotropi-

cally correlated random weights on the scaling exponents.

To this end, we consider directed and undirected lattice-

graphs in two and three dimensions with bond weights ηj,

where the d-dimensional index vector j = i1,...,id ∈ Zd

denotes the position of a particular bond in the lattice.

The total energy or cost of a path P from one end of the

lattice (e.g. from one special site or node in the top layer)

to the opposite end (e.g. to an arbitrary site or node in

the bottom layer) is simply the sum of these bond weights

E =

?

j∈P

ηj. (1)

The weights ηj are correlated positive random variables,

which we define below. We choose the index vector j of the

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2 Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials

bonds in such a way that it is identical with the position of

the center of the bonds in an Euclidean lattice. E.g. for the

square lattice (2d) in which we define the upper left corner

as the origin, all indices take on odd values. We refer to

isotropic correlations if the connected correlation function

of the costs ηj decays with a power law with Euclidean

distance of two bonds, viz.

G(j,j′) = ?ηjηj′? − ?ηj??ηj′? ∼ |j − j′|2ρ−1, (2)

where ρ < 1/2. Here ?···? denotes the disorder average,

i.e. an average of the probability distribution of ηj.

For ρ = 0 model (1) with (2) represents an effective

single vortex line model for interacting vortex lines in a

random vector potential [19]. We would like to empha-

size that isotropic correlations have not been investigated

so far: For historical reasons one discriminates between

d − 1 transverse or spatial directions and 1 longitudinal

or time direction. This can be traced back to the relation

between directed polymers and the KPZ-equation. If the

randomness is only correlated in d−1 dimensions and un-

correlated in the remaining direction one refers to spatially

correlated randomness, if the correlations are only present

in time direction one refers to temporal correlations. Let

j = (x,t) = ((i1,...,id−1),t). Then we can describe both

cases by G ∼ |x − x′|2ρ−1· δ(t − t′) (spatial correlations)

and G ∼ |t − t′|2θ−1· δ(x − x′) (temporal correlations).

The first systematic numerical work done on spatially

correlated noise is due to Amar, Lam and Family [20].

Their results are in agreement with the dynamical [21]

and functional RG [22,23] predictions. Subsequent careful

work by Peng et al. [24] and Pang et al. [25] was done. As

far as temporal correlations are concerned, we refer to the

simulation of a ballistic deposition model by Lam, Sander

and Wolf [26].

In this paper we study isotropically correlated disorder,

it is organized as follows: In section II we introduce the

models, the numerical method and the quantities that we

are interested in. In section III we present our results and

in section IV we summarize our findings.

One remark on the notation: We will use the words

graph and lattice as well as costs and energy, node and site

and edge and bond synonymously throughout the paper.

2 Model

The undirected graph can be described as follows (see

Fig.1b)): we choose a simple lattice structure and define

one longitudinal and d − 1 transversal directions. We will

refer to them by means of t and x respectively. We assume

the lattice to be periodic in space (x) with period H and

L to be the longitudinal size. We choose the origin of the

coordinate system as being the starting point (the source)

of the path. We assign a particular amount of energy to

each bond, whereby these energies are isotropically cor-

related. Generating these random numbers we follow the

method introduced by Pang et al. [25]. We infer period-

icity and symmetry of the correlator gρin any direction,

where

gρ(∆j) := G(j,j + ∆j) = |∆j|2ρ−1

(3)

with gρ(...,(∆j)i+ pi,...) := gρ(...,(∆j)i,...) and

gρ(...,(∆j)i,...) = gρ(...,pi/2 − |(∆j)i− pi/2|,...) in case

of a period pi in direction i. In contrast to pure spatial

and temporal correlations where the correlator is taken

to be the product of two separate ones, one for the time

direction and one for the remaining spatial coordinates,

here it is due to a generic vector ∆j. Adapting the corre-

lator to the lattice, we require a period H in the transver-

sal and 2L in the longitudinal direction. By means of the

factor 2 we guarantee that gρ(t) has the required form

(3) in the range 1 ≤ t ≤ L. Its Fourier transform yields

Sρ(k) such that the correlator in the k-space is given

by ?ηkηk′? ∼ δk+k′,0Sρ(k). Choosing ηk≡

1/2) exp(2πiφk), that relation can be fulfilled, where rk

and φk are random variables uniformly distributed be-

tween 0 and 1. A transformation back to real space pro-

vides the random numbers correlated according to the

power law rule (see also Appendix A).

?Sρ(k)(rk−

a)b)

t

x

L

H

L

Fig. 1. a) The directed graph can be regarded as a square

lattice that was cut along its diagonal and oriented as a triangle

with the diagonal is its base. The paths are directed downwards

to the base. b) In the undirected case the path is allowed to

turn back on a lattice with periodic boundary conditions in

the spatial direction(s). In both cases one fixes the source •,

whereas the target ? is the most favorable node of the base.

From the set of Ld−1optimal paths that connect the

source (x,t) = (0,0) with nodes of the bottom layer with

coordinates (x,L), we select the shortest one. Technically

this is achieved not by repeating the same calculation Ld−1

times (i.e. once for each end-point) but by introducing

an extra target node connected to the bottom layer by

zero-weight bonds. The algorithm that computes the op-

timal paths in polynomial time is Dijkstra’s algorithm that

works in any graph with non-negative weights.

Our study is focused on the universal characteristics

of the optimal path, i.e. on the scaling exponents ν, the

roughness exponent, and ω, the energy fluctuation expo-

nent. ν describes the fluctuations of the path with regard

to a line parallel to the t-axes that is shifted to the ori-

gin by an amount matching to the mean position ?x? of

the polymer. We refer to these fluctuations by D. Due to

the direct mapping between the model of shortest paths

and growing interfaces [4], it is immediately seen that the

energy of the polymer also fluctuates, where we consider

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Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials3

several realizations of disorder: we have

D ∼ tν

∆E ∼ tω

(4)

There is no need to modify these relations when we con-

sider directed paths. For this purpose we introduce directed

bonds, e.g., we restrict to the positive axes. In order to de-

termine the roughness of the polymer, we refer to a line

parallel to the bisecting line (x,t) ∼ (1d−1,1) that crosses

the origin. For each size L we can determine the distance

of the target node to that line by considering the projec-

tion of its position vector onto the bisector. That is, a

directed path connects the source (x,t) = (0,0) and the

sites on the line between (0,L) and (L,0), what has to be

extended to the notion of a plane in d = 3.

As far as undirected paths are concerned one has to

make clear how x(t) can be defined if overhangs appear. In

that case x(t) is not a single valued quantity anymore. We

have checked the relevance of several choices but finally

we took x(t) = max{xi(t)} with t constant for all xi. The

numerical results that we present are independent of this

choice.

As there are O(2n) undirected paths across the lat-

tice, where n is the number of nodes, an efficient method

is needed that terminates within a reasonable time, such

as Dijkstra’s algorithm [17,27], that is able to generate

the shortest path in polynomial time. In this algorithm

the path is successively constructed, i.e., one obtains not

only a single path but a cluster of them with energy la-

bels smaller than a certain limit, so that a growth front

is established. This cluster contains the so called perma-

nently labeled sites. The algorithm proceeds by extending

the front by this site that is the nearest neighbor to it

with respect to the energy. If the growth front reaches the

base of the lattice we are enabled to reconstruct the short-

est path. The growth process of that front can directly

be mapped onto a growing Eden cluster [4] providing the

same scaling behavior as directed polymers.

If we assume uncorrelated costs the roughness of the

growth front scales like the energy fluctuations of the poly-

mer. Since the properties of the surface are described by

the KPZ-equation and, consequently, the height-height

correlation increases according to a power law with ex-

ponent ω = 1/3, we expect in d = 2 the scaling exponents

of the optimal path of size:

ωOP= βKPZ= 1/3νOP= 1/zKPZ= 2/3

Exact values are only accessible in d = 2. The value of

the exponent ν can be extracted from ω if we take into

account the exponent relation

2ν − ω = 1

which holds for uncorrelated noise. This relation follows

from the Galilean invariance of the KPZ-equation and the

transfer to shortest paths afterwards. In d = 3 there exist

no analytical predictions. Nevertheless, the estimates of

numerous numerical studies [2,1,17] yield

ω ≈ 0.19ν ≈ 0.62

As long as the Galilean invariance holds, the scaling rela-

tion remains unchanged. This invariance is not altered in

the presence of spatial (transversal) correlations but it is

broken in the presence of temporal (longitudinal) correla-

tions.

A Flory type argument [11] leads to the following es-

timate νF of the roughness exponent as a function of ρ. A

continuum Hamiltonian of the energy (1) has to include

an elastic part, since this is generated in a coarse graining

procedure.

H =

?

dt

?λ

2(∇tx)2+ η(x,t)

?

.(5)

Rescaling t with a factor b and x with a factor bνthe

elastic term scales with b2ν−2and the disorder term with

bρ−1/2(because of the power law decay of the disorder

correlations), resulting in

νF=1

2ρ +3

4. (6)

In case of uncorrelated random numbers the disorder term

scales with b−ν(d−1)/2−1/2, resulting in

νuncorr.=

3

d + 3. (7)

Hence we expect

ν =

?

νuncorr.

νF

forρ ≤

3(1−d)

2(3+d)

3(1−d)

2(3+d)

for ρ >

(8)

This simplified scaling picture should yield at least a lower

bound for the roughness exponent.

3 Numerical results

In addition to the relations in (4), we define the two ex-

ponents γ and δ by l ∼ Lγand B ∼ Lδ, where B is the

number of backward bonds with respect to time, and l is

the total length of the path. We also determine the fractal

dimension dcof the shortest path cluster M ∼ Ldc, where

M is the mass of its surface. The shortest path cluster

consists of all nodes with labels smaller than a maximal

one given by the shortest path weight from the source to

the base. As far as Dijkstra’s algorithm is concerned, its

surface is constituted by all the sites that are part of the

growth front with at least one nearest neighbor that is not

yet permanently labeled.

The scaling exponents are extracted from a set of data

that reproduces the simulation of several lattice sizes (Fig.2).

The statistical error is usually smaller than the symbol

size. We adapt the transversal expansion H to the size

L in such a way that we eliminate further effects on our

data by increasing H, even if ρ = ρmax. Finally, we con-

sider lattices of size H ≥ 4L if L ≤ 2048 and H = 4096

if L = 2048 in d = 2, as well as H = 128 if L ≤ 32 and

H = 256 if L ≥ 64 in d = 3. At any time, we forbid the

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4 Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials

1

10

100

10 100 1000

D ( L )

L

ρ=0.40

ρ=0.25

ρ=-0.50

0.1

1

10

100

10 100 1000

∆E ( L )

L

ρ=0.40

ρ=0.25

ρ=-0.50

Fig. 2. Scaling of the height-height correlation D(L) and the

energy fluctuations ∆E(L) in d = 2 for NDOPs. The straight

lines are least square fits of the data for L > 100 to a power

law D(L) ∝ Lνand ∆E(L) ∝ Lωwith ν = 0.66,0.77,0.85 and

ω = 0.32,0.50,0.67 for ρ = −0.50,0.25,0.40. The statistical

error of the data for D(L) and ∆E(L), which are averaged

over at least 10000 disorder realizations, is smaller than the

symbol size.

shortest path cluster spanning the lattice in order to avoid

saturation effects. This requirement cannot be satisfied for

all L = 128 in case of d = 3 and ρ close to ρmax.

Our results are averages over more than 10000 disor-

der realizations per size L. The generation of correlated

random numbers is the most time consuming part. For

each sample we have to perform a Fourier transformation

of N = 2dLHd−1numbers twice. On the average approxi-

mately 90% of the CPU time is needed for this execution.

Some more information about the generation is given in

Appendix A.

As a first check we studied the directed polymer prob-

lem in d = 2 with only spatially correlated bond weights

as it was done by Pang et al. [25]. It can be seen from

Fig.3 that we are in very good agreement with their re-

sults. In case of strongest correlations (ρ = 0.40) we obtain

ν = 0.71 ± 0.01 and ω = 0.43 ± 0.01. It is quite evident

that correlations are only relevant in the regime ρ > 0.

0.25

0.30

0.35

0.40

0.45

0.50

0.55

-0.50 -0.250.00

ρ

0.25 0.50

ω

DOP

Fig. 3. Our result of the directed path in d = 2 for spatial cor-

relations. A direct comparison with the results of [25] (Fig.3)

shows that they are in good agreement. The full line is theoret-

ical prediction from a one-loop KPZ dynamic RG calculation

[21] and a DPRM functional RG [23] calculation.

Two dimension (d = 2)

In d = 2 we find that the roughness exponent ν does not

depend on the directedness of the lattice. Overhangs do

not play an important role and the undirected path can be

regarded as a directed one (Fig.4). Independent from the

strength of the correlations all exponents ν are smaller

than the critical value 1. As far as undirected shortest

paths are concerned, exceeding that critical value leads to

fractal objects that cannot become directed, even on large

scales. The errorbars depicted in Fig.4 are not the result

of the least square fit but estimates of the minimal and

maximal slopes being in nearly perfect agreement with our

data.

In contrast, we obtain a less significant data collapse

with respect to the energy fluctuation exponent ω, if ρ > 0.

This may be affected by a statistics that has room of im-

provement but it indicates a tendency that is especially

noticeable in d = 3: the stronger the correlation the more

significant variations in ω occur. The reason for this rela-

tion is not clear to us. In both cases the exponent relation

can be satisfied (Fig.5) where we are in better agreement

with 2ν − ω = 1 for undirected paths. We learn from

Fig.3 that isotropic correlations in the randomness are rel-

evant for ρ > 0. The scaling of the energy fluctuations is

much more sensitive to passing from white noise to weak

correlations (ρ = −0.5) in comparison to the roughness.

Whereas ν keeps constant (ν = 0.66 ± 0.01), ω reduces

from ω = 0.32 ± 0.01 to ω = 0.29 ± 0.01. The number of

backward bonds in the NDOP model remains negligibly

small. It is B ≈ 15 for L = 1024 if ρ = 0.40 where δ ≈ 1.0.

In that case almost every path has at least one such bond.

According to these results we obtain l ∼ L.

Although the roughness exponent increases significantly

for ρ ≥ 0.3 and might even come closer to 1 for ρ ≈ 0.6 it

stays still smaller than one. Also the length of the short-

est path scales linearly with L for the values of ρ that we

could study. Both observations imply a non-fractal short-

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Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials5

0.6

0.7

0.8

0.9

1.0

-0.6-0.4 -0.2 0 0.2 0.4

ν

ρ

NDOP

-0.6 -0.4 -0.2 00.2 0.4

-0.10

0

0.10

νNDOP-νDOP

ρ

0.2

0.4

0.6

0.8

-0.6 -0.4-0.2 0 0.2 0.4

ω

ρ

NDOP

DOP

-0.6 -0.4 -0.2 00.2 0.4

-0.10

0

0.10

ωNDOP-ωDOP

ρ

Fig. 4. The scaling exponents ν and ω in d = 2 depend on the

strength ρ of the correlation. Close to ρ = 0 correlations start

to affect them significantly. In either case the insets show the

difference of both the exponents of the undirected and directed

lattice respectively. The reference lines represent the values

ν = 1/3 and ω = 2/3 respectively.

est path. However, the properties of the shortest path clus-

ter change remarkably when coming closer and closer to

ρ = 0.40, where the fractal dimension of its surface be-

comes dc≈ 1.1 instead of dc≈ 1 if ρ = 0. In Fig.6 one can

see that for weak correlations the surface of the shortest

path cluster is a semicircle whereas for larger correlations

it becomes topologically more complicated.

Not only the disorder averaged roughness D scales

with Lνbut the whole probability distribution PL(D): It is

PL(D) = Lνp(D/Lν) as we show in Fig.7 for ρ = 0.40. For

the scaling function p(x) we can fit a log-normal distribu-

tion given by p(x) = (2πσ2)−1/2exp(−(lnD/D0)2/2σ2).

The parameters σ and D0do not depend on ρ for ρ ≤ 0

(where D0≈ 0.13, σ ≈ 0.58) and vary slightly with ρ for

ρ > 0.

Three dimensions (d = 3)

The results in d = 3 are qualitatively similar to those of

the preceeding section. For uncorrelated randomness we

obtain ν = 0.62 ± 0.02 and ω = 0.22 ± 0.01 in agreement

0.8

0.9

1.0

1.1

1.2

-0.6-0.4 -0.2 0 0.2 0.4

2 ν - ω

ρ

NDOP

DOP

Fig. 5. The exponent relation is depicted for the NDOP model

in d = 2. Both data sets diverge only close to ρ = 0.40 accord-

ing to the insets of Fig.4.

a)b)

d)c)

Fig. 6.

for H = 512 and L = 256. In each case the growth process

stops when the growth front reaches the base of the lattice. We

choose a) uncorrelated random numbers and correlated random

numbers with b) ρ = 0, c)+d) ρ = 0.40.

Realizations of the shortest path cluster (NDOP)

with [1,2,17]. Both ν and ω increase monotonuously with

ρ, i.e. increasing correlations. While ν does so without

any difference between directed and undirected paths, ω

differs from this behavior: the stronger the correlations the

more estimates for the exponents deviate from each other

(Fig.8). As ν < 1 the undirected path becomes directed

on large scales. We should emphasize that obtaining data

in the regime ρ > 0.25 for d = 3 is much more delicate

than in d = 2. The local slopes of the energy fluctuations

indicate that even for the maximal size L = 128 we are

not yet in the asymptotic regime for ρ > 0.3. Therefore we

explicitly restrict ourselves to values ρ ≤ 0.3. This result

refers to both kinds of lattices and, so is not a consequence

of saturation effects. The distinct behavior of both kinds

of paths can be demonstrated more obviously by plotting

the exponent relation 2ν − ω = 1 (see inset of Fig.8).

Even though the scaling regimes in d = 3 are not very

wide our estimates for ν and ω deviate significantly from

their values in the uncorrelated case. Significantly means

here that their estimated errors (obtained in the same way

as in our 2d-study before) are smaller than the deviation

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6 Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials

0

1

2

3

4

5

6

7

8

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

D / Lν

PL ( D )

p(x)

L=1024

L=512

L=256

L=128

L=64

Fig. 7. Scaling plot for the probability distribution PL(D) of

the roughness in case of directed paths for ρ = 0.40. The scaling

function p(x) is a log-normal distribution with D0 = 0.11 and

σ = 0.70.

from the uncorrelated values. Therefore we can infer that

for strong enough correlations (ρ ≥ 0.3) the DOP and

NDOP problems constitute new universality classes. The

precise determination of the critical value for ρ, beyond

which correlations become relevant, is, however, beyond

our numerical precision.

For completeness we mention that even in case of L =

128 and ρ = 0.40 the number of bonds turning backward

is negligibly small: B ≈ 1.6 compared to B ≈ 3 in d =

2. We do not expect a significant change in B for larger

lattices as this quantity scales by an exponent δ ≈ 1.0 but

the number of paths including such bonds should increase

from 60% to 100%.

In contrast to the difficulties above, the fractal dimen-

sion dc of the surface can be extracted quite clearly. It

is dc ≈ 2.35 ± 0.10 if we choose ρ = 0.40 in contrast

to dc → 2.0 for uncorrelated numbers. We illustrate the

change of the cluster by cutting the system along the

x − t−plane. Fig.9 shows such cuts for y = H/2 where

a) denotes random costs and b) corresponds to strongest

correlations.

4 Summary

In the present paper we have considered the effect of isotrop-

ically correlated bond weights on the scaling behavior of

both undirected and directed paths in two and three di-

mensions. We found that in d = 2 the algebraic correla-

tion of the disorder are relevant for a decay slower than

1/r (i.e. ρ = 0) and the roughness exponent ν and the

energy fluctuation exponent ω increase monotonously for

ρ ≥ 0. The Flory estimate (8) for the roughness exponent

is fulfilled as a lower bound of our numerical estimates. A

precise value for ρc, above which correlations modify the

universality class, is hard to estimate numerically, but we

observe that it is close to and slightly smaller than zero in

0.60

0.65

0.70

0.75

0.80

-0.6-0.4 -0.2 0 0.2 0.4

ν

ρ

NDOP

1

10

10 100

D ( L )

L

ρ=0.40

ρ=0.25

ρ=-0.50

0.2

0.4

0.6

-0.6 -0.4-0.2 0 0.2 0.4

ω

ρ

NDOP

DOP

0.8

0.9

1.0

1.1

1.2

-0.6 -0.4-0.2 0 0.2 0.4

2 ν - ω

ρ

NDOP

DOP

Fig. 8. The exponents ν and ω in d = 3 are plotted versus

ρ. The insets show the scaling of the height-height correlation

D(L) for NDOP and the exponent relation in d = 3 respec-

tively. The reference lines belong to our results ν = 0.62 and

ω = 0.22 in case of uncorrelated numbers.

d = 2 and d = 3, in agreement with the Flory argument

(8).

Moreover, the results in d = 2 indicate that the scal-

ing exponents are independent from the directedness even

in case of very strong correlations. In contrast to this,

in d = 3 directed and undirected lattices yield different

results for ρ > ρc≈ 0 indicating that both cases consti-

tute different universality classes. In 2d the scaling relation

2ν − ω appears to remain valid even for stronge correla-

tions (although Galilean invariance is broken), whereas for

d = 3 we observe in the directed case significant deviations

from it (in the undirected case it appears to remain valid).

Finally we could exclude the possibility that optimal

paths become fractal for strong disorder correlations as

long as they decay algebraically. As a consequence ν stays

smaller than one and overhangs turn out to be irrelevant.

This behavior changes if we consider the following disor-

der correlations: ?(ηj− ηj+r)2? ∝ rα, with α > 0, which

increase with distance r = |r|. These type of correlators

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Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials7

a)

b)

Fig. 9. Cluster of shortest paths in d = 3 that is cut along

the x − t− plane with y = H/2. a) uncorrelated randomness

b) strong correlations (ρ = 0.40). It is seen that we pass again

from a semicircle like cluster to a topologically more compli-

cated one.

are relevant for an effective model of dislocations in a vor-

tex line lattice [28]. In this case we found that the optimal

paths are fractals with a fractal dimension significantly

larger than one and depending on the value of α [29].

This work has been supported by the Deutsche Forschungsge-

meinschaft (DFG).

A Appendix

Here we explain how to initialize the correlation function gρ(j)

and we show that N = 2dLHd−1correlated random numbers

have to be created for our purposes. We need to generate 2N

uncorrelated random numbers and have to perform a Fourier

transformation twice. If we consider the simple square lattice

of nodes with lattice spacing a we can immediately see that we

have to distinguish between positions of nodes and positions of

bonds. The latter build a simple lattice that is rotated towards

the node lattice by π/4 with a lattice spacing enlarged by a

factor√2 (Fig.10). But we have to create correlations accord-

ing to positions of bonds. Let us assume now this lattice to

be a two dimensional array. The most comfortable solution of

the generation of correlations due to bond positions is to keep

the original lattice structure, meaning we switch to a spacing

a/2 and finally focus on those positions within this array that

correspond to bonds in our lattice. Then we define the center

of that lattice (array) and initialize each position of the array

by |j|2ρ−1, where −H/2 ≤ ji < H/2 or −L/2 ≤ ji < L/2 is the

position vector. This array is called the correlator gρ(j). There

are two problems: by doing so, we generate twice the quantum

of correlated random numbers we really need (the information

of the black squared positions) and we still have to discuss how

to define gρ(0). As mentioned in section II, we already have to

take into account a factor 2 from the longitudinal expansion

so that N becomes N = 2 · 2dLHd−1, where again all sites in

the left part of Fig.10 would be occupied. In order to restrict

memory usage we compress the lattice along one axes (here

the x-axes), whereby we loose the information of all the po-

sitions indicated by open symbols. By doing so, we avoid the

necessity of defining gρ(0) corresponding to a virtual node at

the origin and, consequently, being not part of the right side of

Fig.10. More important, N is reduced by a factor 2. Note that

the Fourier transform on the compressed lattice also yields the

desired correlations.

L=2

t

x

H=4

pbc

pbc

a

Fig. 10. First we stretch the lattice of size (H ×2L) resulting

in (2H×4L) and finally compress it again. Here• denotes sites

that refer to nodes on a lattice of size L and H,◦ corresponds

to virtual nodes which have to be introduced in order to follow

the definition of the correlator (periodicity in each direction)

and ? refers to bonds. Squares denote positions that addition-

ally arise by switching to a lattice spacing a/2. In contrast to

black squares the entries of white squares do not play any role

concerning the generation of correlated random numbers.

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