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Journal ArticleDOI

A Comparison of Random Walks in Dependent Random Environments

01 Mar 2016-Advances in Applied Probability (Applied Probability Trust)-Vol. 48, Iss: 1, pp 199-214
TL;DR: Comparing random walks in various dependent environments, it is demonstrated that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.
Abstract: Although the theoretical behavior of one-dimensional random walks in random environments is well understood, the actual evaluation of various characteristics of such processes has received relatively little attention. This paper develops new methodology for the exact computation of the drift in such models. Focusing on random walks in dependent random environments, including $k$-dependent and moving average environments, we show how the drift can be characterized and found using Perron-Frobenius theory. We compare random walks in various dependent environments and show that their drift behavior can differ signicantly.

Summary (2 min read)

1 Introduction

  • Random walks in environments are well-known mathematical models for motion through disorganized 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.
  • From an applied and computational point of view significant gaps in their understanding remain.
  • Exact drift computations and comparisons (as oposed to comparisons using simulation) between dependent random environments seem to be entirely missing from the literature.
  • In Section 3 the authors prove explicit results for the drift for each of these models, and compare their behaviors.

2.1 General theory

  • (2.1) The theoretical behavior of {Xn} is well understood, as set out in the seminal work of Solomon [14].
  • In particular, Theorems 2.1 and 2.2 below completely describe the transience/recurrence behavior and the Law of Large Numbers behavior of {Xn}.
  • The authors follow the notation of Alili [1] and first give the key quantities that appear in these theorems.
  • These follow directly from the stationarity of U.

3 Evaluating the drift

  • And then further specify the transience/recurrence and drift results to the Markov environment, the 2-dependent environment, and the moving average environments.the authors.
  • The authors 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.

3.1 General solution for swap models

  • Consider now the RWRE swap model with a random environment generated by a Markov chain {Yi, i ∈ Z}, as specified in Section 2.2.
  • The authors summarize these findings in the following theorem.

3.2.1 Comparison with the iid environment

  • Substitution into the expression for V (here in the case of positive drift only, see (3.7)) and rewriting yields V = (2p−.
  • This enables us not only to immediately quickly obtain the drift for the iid case (take ̺ = 0), but also to study the dependence of the drift V on ̺.
  • Figures 2 illustrates various aspects of the difference between iid and Markov cases.
  • Clearly, compared to the iid case (for the same value of α), the Markov case with positive correlation coefficient has lower drift, but also a lower ‘cutoff value’ of p at which the drift becomes zero.
  • For negative correlation coefficients the authors see a higher cutoff value, but not all values of α are possible (since they should have a < 1).

3.3 2-dependent environment

  • Unfortunately, the eigenvalues of PD are now the roots of a 4-degree polynomial, which are hard to find explicitly.
  • Using Perron–Frobenius theory and the implicit function theorem it is possible to prove the following lemma, which has the same structure as in the Markovian case.
  • Now, moving σ from 1 to any other positive value, λ0(σ) must continue to play the role of the Perron–Frobenius eigenvalue; i.e., none of the other λi(σ) can at some point take over this role.
  • Including the transience/recurrence result from the first part of this section, and including the cases with negative drift, the authors obtain the following analogon to Proposition 3.1.

3.4 Moving average environment

  • The proof is similar to that of Lemma 3.2; the authors only give an outline, leaving details for the reader to verify.
  • The cutoff value for p is now easily found as (1+σcutoff) −1, which can be numerically evaluated.
  • It is interesting to note that the cutoff points (where V becomes 0) are significantly lower in the moving average case than the iid case, using the same α, while at the same time the maximal drift that can be achieved is higher for the moving average case than for the iid case.
  • This is different behavior from the Markovian case; see also Figure 2.

4 Conclusions

  • Random walks in random environments can exhibit interesting and unusual behavior due to the trapping phenomenon.
  • The dependency structure of the random environment can significantly affect the drift of the process.
  • For the wellknown swap RWRE model, this approach allows for easy computation of drift, as well as explicit conditions under which the drift is positive, negative, or zero.
  • The cutoff values where the drift becomes zero, are determined via Perron–Frobenius theory.
  • Various generalizations of the above environments can be considered in the same (swap model) framework, and can be analyzed along the same lines, e.g., replacing iid by Markovian {Ûi} in the moving average model, or taking moving averages of more than 3 neighboring states.

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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 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
i1
= 1) = P(Y
i
= 1 | Y
i1
= 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
i1
= u
i1
, U
i2
= u
i2
, . . .) = (2.5)
P(U
i
= u
i
| U
i1
= u
i1
, U
i2
= u
i2
, . . . , U
ik+1
= u
ik+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
ik
, . . . , u
i1
) only
two possible transitions can ta ke place, given by
(u
ik
, . . . , u
i1
) (u
ik+1
, . . . , u
i1
, u
i
), u
i
{− 1, 1},
4

with corresponding probabilities 1a
(u
ik
,...,u
i2
)
, a
(u
ik
,...,u
i2
)
, b
(u
ik
,...,u
i2
)
, and
1 b
(u
ik
,...,u
i2
)
, for (u
i1
, 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
ik+1
, . . . , U
i
). In the special case k = 1
(Markovian environment), we omit the subindices of a (transition probability from
U
i1
= 1 to U
i
= +1) and b (from U
i1
= +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

Citations
More filters
01 Mar 2010
TL;DR: In this article, the authors considered the asymptotic variance of the departure counting process of the GI/G/1 queue and showed that the departures variability has a singularity in case the system load is 1.
Abstract: We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case that the system load $\rho$ equals 1, and prove that the asymptotic variance rate satisfies \[ \lim_{t \rightarrow \infty} \frac{Var D(t)}{t} = \lambda (1-\frac{2}{\pi})(c^2_a+c^2_s) \] , where $\lambda$ is the arrival rate and $c^2_a$, $c^2_s$ are squared coefficients of variation of the inter-arrival and service times respectively. As a consequence, the departures variability has a remarkable singularity in case $\rho$ equals 1, in line with the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) which was previously encountered in the finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multi-server queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue we present an explicit expression of the variance of D(t) for any t. Keywords: GI/G/1 queues, critically loaded systems, uniform integrability, departure processes, renewal theory, Brownian bridge, multi-server queues.

30 citations

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TL;DR: The transition from diffusion growth to diffusionless growth is traced on the basis of the derived relationship between the degree of short-range order in the new phase and the magnitude of the coefficient of diffusion as discussed by the authors.

24 citations


"A Comparison of Random Walks in Dep..." refers background in this paper

  • ...Early applications can be found in Chernov [4] and Temkin [16]; see also Kozlov [9] and references therein....

    [...]

Proceedings ArticleDOI
20 Oct 2008
TL;DR: In this paper, the authors consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times.
Abstract: We consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull system was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Thus each server can either process jobs of one of the types, which it pulls from the other server, or jobs of the other type which it pushes out of the infinite supply towards the other server. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform an asymptotic analysis of the push pull network under these policies to quantify its behavior: We show that under fluid scaling the fluid model of the network is stable. We adapt the proofs of Dai, to show that as a result the queues of jobs waiting for pull operation are positive Harris recurrent. Finally we obtain the diffusion scale behavior of the network, in which we show that the queues are zero under diffusion scaling, and calculate the Brownian approximation of the output processes of the two types of jobs. The approximation shows that the two output streams are highly negatively correlated.

23 citations

Journal ArticleDOI
Sunkyo Kim1
TL;DR: Both the cross correlation and the autocorrelation can be modeled in parametric decomposition approximations of queueing networks by integrating the MMPP(2) approximation of the arrival/departure process and the innovations method.
Abstract: In two-moment decomposition approximations of queueing networks, the arrival process is modeled as a renewal process, and each station is approximated as a GI/G/1 queue whose mean waiting time is approximated based on the first two moments of the interarrival times and the service times The departure process is also approximated as a renewal process even though the autocorrelation of this process may significantly affect the performance of the subsequent queue depending on the traffic intensity When the departure process is split into substreams by Markovian random routing, the split processes typically are modeled as independent renewal processes even though they are correlated with each other This cross correlation might also have a serious impact on the queueing performance In this paper, we propose an approach for modeling both the cross correlation and the autocorrelation by a three-moment four-parameter decomposition approximation of queueing networks The arrival process is modeled as a nonrenewal process by a two-state Markov-modulated Poisson process, viz, MMPP(2) The cross correlation between randomly split streams is accounted for in the second and third moments of the merged process by the innovations method The main contribution of the present research is that both the cross correlation and the autocorrelation can be modeled in parametric decomposition approximations of queueing networks by integrating the MMPP(2) approximation of the arrival/departure process and the innovations method We also present numerical results that strongly support our refinements

16 citations

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TL;DR: A general framework is proposed for the study of the charge transport properties of materials via random walks in random environments (RWRE), which combines a model for the fast generation of random environments that realistically mimic materials morphology with an algorithm for efficient estimation of key properties of the resulting random walk.
Abstract: A general framework is proposed for the study of the charge transport properties of materials via random walks in random environments (RWRE). The material of interest is modeled by a random environment, and the charge carrier is modeled by a random walker. The framework combines a model for the fast generation of random environments that realistically mimic materials morphology with an algorithm for efficient estimation of key properties of the resulting random walk. The model of the environment makes use of tools from spatial statistics and the theory of random geometric graphs. More precisely, the disordered medium is represented by a random spatial graph with directed edge weights, where the edge weights represent the transition rates of a Markov jump process (MJP) modeling the motion of the random walker. This MJP is a multiscale stochastic process. In the long term, it explores all vertices of the random graph model. In the short term, however, it becomes trapped in small subsets of the state space a...

15 citations

01 Jan 2013
TL;DR: The lecture notes arose out of a mini-course I taught in January 2013 at the Instituto Nacional de Matematica pura e Aplicada (IMPA) in Rio de Janeiro, Brazil as mentioned in this paper.
Abstract: This lecture notes arose out of a mini-course I taught in January 2013 at Instituto Nacional de Matematica pura e Aplicada (IMPA) in Rio de Janeiro, Brazil. In these lecture notes I do not always give all the details of the proofs, nor do I prove all the results in their greatest generality. A more detailed treatment of most of these topics can be found in Zeitouni’s lecture notes on RWRE [Zei04].

11 citations


"A Comparison of Random Walks in Dep..." refers background in this paper

  • ...Overviews of the current state of the art, with a focus on higher-dimensional RWREs, can be found, for example, in Hughes [7], Sznitman [15], Zeitouni [17, 18], and Révész [11]....

    [...]

  • ...Overviews of the current state of the art, with a focus on higher-dimensional RWREs, can be found, for example, in Hughes [7], Sznitman [15], Zeitouni [17, 18], and Révész [11]....

    [...]