A Comparison of Random Walks in

Dependent Random Environments

Werner R.W. Scheinhardt

University of Twente

w.r.w.scheinhardt@utwente.nl

Dirk P. Kroese

The University of Queensland

kroese@maths.uq.edu.au

April 13, 2015

Abstract

We provide exact computations for the drift of random walks in depen-

dent random environments, including k-dependent and moving average envi-

ronments. We show how the drift can be characterized and evaluated using

Perron-Frobenius theory. Comparing random walks in various dependent en-

vironments, we demonstrate that their drifts can exhibit interesting behavior

that depends signiﬁcantly on the dependency structure of the random envi-

ronment.

MSC subject classiﬁcations: Primary 60K37, 60G50; secondary 82B41

Keywords: random walk, depen dent random environment, drift, Perron–

Frobenius eigenvalue

1 Introduction

Random walks in random environments (RWREs) are well-known mathematical

models for motion through disorganized (random) media. They generalize ordinary

random walks whereby the transition probabilities from any position are determined

by the random state of the environment at that position. RWREs exhibit interesting

and unusual behavior that is not seen in ordinary random walks. For example, the

walk can tend to inﬁnity almost surely, while its overall drift is 0. The reason for

such surprising behavior is that RWREs can spend a long time in (rare) re gions

from which it is diﬃcult to escape — in eﬀect, the walker becomes “trapped” for a

long time.

Since the late 1960s a vast body of knowledge has been built up on the behav-

ior of RWREs. Early applications can be found in Chernov [

4] and Temkin [17];

see also Kozlov [

9] and references therein. Recent applications to charge transport

in designed materials are given in Brereton et al. [3] and Stenze l et al. [15]. The

mathematical framework for one-dimensional RWREs in independent environments

was laid by So lomon [14], and was further ex tended by K esten et al. [8], Sinai [13],

1

Greven and Den Hollander [6]. Markovian environments were investigated in Dol-

gopyat [5] and Mayer-Wolf et al. [10]. Alili [1] showed that in the one-dimensional

case much of the theory for independent env ironments could be generalized to the

case where the environment process is stationary and ergodic. Overviews of the

current state of the art, with a foc us on higher-dimensional RWREs, can be found,

for example, in Hughes [

7], Sznitman [16], Zeitouni [18, 19], and R´ev´esz [1 1].

Although from a theoretical perspective the behavior of one-dimensional RWREs

is well understood, from an applied and computational point of view signiﬁcant

gaps in our understanding remain. For e xample, exact drift computatio ns and com-

parisons (as oposed to comparisons using simulation) be twee n dependent random

environments seem to be entirely missing from the literature. The reason is that

such exact computations are not trivial and require additional insights.

The contribution of this paper is twofold. First, we provide new methodology

and explicit expressions for the c omputation of the drift of one-dimensiona l random

walks in various dependent environments, focusing on so-called ‘swap models’. In

particular, our approach is based on Perron–Frobenius theory, which allows ea sy

computation of the drift and as well a s various cutoﬀ points for transient/recurrent

behavior. Second, we compare the drift behavior between various dependent envi-

ronments, including moving avera ge and k-dependent e nvironments. We show that

this behavior c an deviate considerably from that of the (known) independent c ase.

The rest of the pape r is organized as follows. In Section

2 we formulate the

model for a one-dimensional RWRE in a stationary and ergodic environment and

review some of the key results fro m [

1]. We then formulate a ﬂexible mechanism

for constructing depe ndent random environment that includes the iid, Markovian,

k-dependent, and moving average environments. In Section

3 we prove explicit

(computable) results for the drift for each of these models, and compare their be-

haviors. Conclusions and directions for future research are given in Section

4.

2 Model and preliminaries

In this section we review so me key results on one-dimens ional RWREs and introduce

the class of ‘swap-models’ that we will study in more detail.

2.1 General theory

Consider a stochastic process {X

n

, n = 0, 1, 2, . . .} with state space Z, and a sto chas-

tic “Underlying” environment U taking value s in some set U

Z

, where U is the set

of possible environment states for each site in Z. We assume that U is statio nary

2

(under P) as well as ergodic (under the natural shift operator on Z). The e volution

of {X

n

} depends on the rea liz ation of U, which is random but ﬁxed in time. For

any re alization u of U the process {X

n

} behaves a s a simple random walk with

transition probabilities

P(X

n+1

= i + 1 | X

n

= i, U = u) = α

i

(u)

P(X

n+1

= i − 1 | X

n

= i, U = u) = β

i

(u) = 1 − α

i

(u).

(2.1)

The theor e tical behavior of {X

n

} is well understood, as set out in the seminal

work of Solomon [

14]. In particular, Theorems 2.1 and 2.2 below c ompletely de-

scribe the transience/recurrence behavior and the Law of Large Numbers behavior

of {X

n

}. We follow the notation of Alili [

1] and ﬁrst give the key q uantities that

appear in these theorems. Deﬁne

σ

i

= σ

i

(u) =

β

i

(u)

α

i

(u)

, i ∈ Z , (2.2)

and let

S = 1 + σ

1

+ σ

1

σ

2

+ σ

1

σ

2

σ

3

+ · · · (2.3)

and

F = 1 +

1

σ

−1

+

1

σ

−1

σ

−2

+

1

σ

−1

σ

−2

σ

−3

+ · · · (2.4)

Theorem 2.1. (Theorem 2.1 in [

1])

1. If E[ln σ

0

] < 0, then almost surely lim

n→∞

X

n

= ∞ .

2. If E[ln σ

0

] > 0, then almost surely lim

n→∞

X

n

= −∞ .

3. If E[ln σ

0

] = 0, then almost surely lim inf

n→∞

X

n

= −∞ and lim sup

n→∞

X

n

= ∞ .

Theorem 2.2. (Theorem 4.1 in [1])

1. If E[S] < ∞, then almost surely lim

n→∞

X

n

n

=

1

E[(1 + σ

0

)S]

=

1

2E[S] − 1

.

2. If E[F ] < ∞, then almost surely lim

n→∞

X

n

n

=

−1

E[(1 + σ

−1

0

)F ]

=

−1

2E[F ] − 1

.

3. If E[S] = ∞ and E[F ] = ∞, then almost surely lim

n→∞

X

n

n

= 0.

Note that we have added the second equalities in statements 1. and 2. of Theo-

rem

2.2. These follow directly from the stationarity of U.

We will call lim

n→∞

X

n

/n the drift of the proce ss {X

n

}, and denote it by V .

Note that, as mentioned in the introduction, it is possible for the chain to be

transient with drift 0 (namely when E[ln σ

0

] 6= 0, E[S] = ∞ and E[F] = ∞).

3

2.2 Swap model

We next focus on what we will c all swap models, as studied by Sinai [

13]. Here,

U = {−1, 1}; that is , we assume that all elements U

i

of the process U ta ke value

either −1 or +1. We assume that the transition probabilities in state i only depends

on U

i

and not on other e lements of U, as follows. When U

i

= −1, the transition

probabilities of {X

n

} fr om state i to states i + 1 a nd i − 1 are swapped with r esp ect

to the values they have when U

i

= + 1. Thus, for some ﬁxed value p in (0, 1) we

let α

i

(u) = p (and β

i

(u) = 1 − p) if u

i

= 1, and α

i

(u) = 1 − p (and β

i

(u) = p) if

u

i

= −1. Thus, (2.1) becomes

P(X

n+1

= i + 1 | X

n

= i, U = u) =

(

p if u

i

= 1

1 − p if u

i

= −1

and

P(X

n+1

= i − 1 | X

n

= i, U = u) =

(

1 − p if u

i

= 1

p if u

i

= −1 .

Next, we choose a dependence structure fo r U using the following s imple, but

novel, construction. Let {Y

i

, i ∈ Z} be a sta tionary and ergodic Markov chain

taking values in some ﬁnite set M and let g : M → {−1, 1} be a given function.

Now deﬁne the environment at state i as U

i

= g(Y

i

), i ∈ Z. Despite its simplicity,

this formalism covers a number of interesting dependence structures on U, discussed

next.

iid environment. In this case the {U

i

} are i.i.d. random variables, with α

def

=

P(U

i

= 1 ) = 1 − P(U

i

= − 1). Formally, this ﬁts the framework above by choosing

g the identity function on M = {−1, 1} and {Y

i

} the Markov chain with one-step

transition probabilities P(Y

i

= 1 | Y

i−1

= −1) = P(Y

i

= 1 | Y

i−1

= 1) = α for all i.

k-dependent environment. Deﬁne a k-dependent environment as an environ-

ment {U

i

} for which

P(U

i

= u

i

| U

i−1

= u

i−1

, U

i−2

= u

i−2

, . . .) = (2.5)

P(U

i

= u

i

| U

i−1

= u

i−1

, U

i−2

= u

i−2

, . . . , U

i−k+1

= u

i−k+1

), u

j

∈ {− 1, 1}. (2.6)

Spec ial cases are the independent environment (k = 0; see above) and the so-called

Markovian environment (k = 1). For k > 1, let {Y

i

, i ∈ Z} be a Markov chain

that takes values in M = {−1, 1}

k

such that from any state (u

i−k

, . . . , u

i−1

) only

two possible transitions can ta ke place, given by

(u

i−k

, . . . , u

i−1

) → (u

i−k+1

, . . . , u

i−1

, u

i

), u

i

∈ {− 1, 1},

4

with corresponding probabilities 1−a

(u

i−k

,...,u

i−2

)

, a

(u

i−k

,...,u

i−2

)

, b

(u

i−k

,...,u

i−2

)

, and

1 − b

(u

i−k

,...,u

i−2

)

, for (u

i−1

, u

i

) equal to (−1, −1), (−1, 1), (1, −1), and (1, 1), re-

spectively. Now deﬁne U

i

as the last component of Y

i

. Then {U

i

, i ∈ Z} is a

k-dependent environment, and Y

i

= (U

i−k+1

, . . . , U

i

). In the special case k = 1

(Markovian environment), we omit the subindices of a (transition probability from

U

i−1

= −1 to U

i

= +1) and b (from U

i−1

= +1 to U

i

= −1).

Moving average environment. Cons ide r a “ moving average” environment, which

is built up in two phases as follows. First, start with an iid environment {

b

U

i

} as

in the iid case, with P(

b

U

i

= 1) = α. Let Y

i

= (

b

U

i

,

b

U

i+1

,

b

U

i+2

). Hence, {Y

i

} is

a Markov process with states 1 = (−1, −1, −1), 2 = (−1, −1, 1), . . . , 8 = (1, 1, 1)

(lexicographical o rder). The corresponding transition matrix is given by

P =

1 − α α 0 0 0 0 0 0

0 0 1 − α α 0 0 0 0

0 0 0 0 1 − α α 0 0

0 0 0 0 0 0 1 − α α

1 − α α 0 0 0 0 0 0

0 0 1 − α α 0 0 0 0

0 0 0 0 1 − α α 0 0

0 0 0 0 0 0 1 − α α

. (2.7)

Now deﬁne U

i

= g(Y

i

), where g(Y

i

) = 1 if at le ast two of the three random var iables

b

U

i

,

b

U

i+1

and

b

U

i+2

are 1, and g(Y

i

) = −1 otherwise. Thus,

(g(1), . . . , g(8)) = (−1, −1, −1, 1, −1, 1, 1, 1) , (2.8)

and we see that ea ch U

i

is obtained by taking the moving average of

b

U

i

,

b

U

i+1

and

b

U

i+2

, as illustrated in Fig ure 2.2.

Figure 1: Moving average environment.

3 Evaluating the drift

In this section we ﬁrst give the gene ral solution approach for the Markov-based

swap model, and then further specify the transience/recurrence and drift results to

the Markov environment, the 2-dependent environment, and the moving average

environments. We omit a separate derivation for the i.i.d. environment, which can

be viewed as a special case of the Markovian environment, see Remark

3.1.

5