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Year:2012
Continuityandanomalousuctuationsinrandomwalksindynamicrandom
environments:numerics,phasediagramsandconjectures
Avena,L;Thomann,P
Abstract:Weperformsimulationsforonedimensionalcontinuous-timerandomwalksintwodynamic
randomenvironmentswithfast(independentspin-ips)andslow(simplesymmetricexclusion)decayof
space-timecorrelations,respectively.Wefocusontheasymptoticspeedsandthescalinglimitsofsuch
randomwalks.Weobservedierentbehaviorsdependingonthedynamicsoftheunderlyingrandom
environmentandtheratiobetweenthejumprateoftherandomwalkandtheoneoftheenvironment.
Wecompare ourdatawith wellknownresults forstaticrandom environment.Weobservethatthe
non-diusiveregimeknownsofaronlyforthestaticcasecanoccurinthedynamicalsetuptoo.Such
anomalousuctuationsgiverisetoanewphasediagram.Furtherwediscusspossibleconsequencesfor
moregeneralstaticanddynamicrandomenvironments
DOI:https://doi.org/10.1007/s10955-012-0502-1
PostedattheZurichOpenRepositoryandArchive,UniversityofZurich
ZORAURL:https://doi.org/10.5167/uzh-156513
JournalArticle
PublishedVersion
Originallypublishedat:
Avena,L;Thomann,P(2012).Continuityandanomalousuctuationsinrandomwalksindynamicran-
domenvironments:numerics,phasediagramsandconjectures.JournalofStatisticalPhysics,147(6):1041-
1067.
DOI:https://doi.org/10.1007/s10955-012-0502-1
J Stat Phys (2012) 147:1041–1067
DOI 10.1007/s10955-012-0502-1
Continuity and Anomalous Fluctuations in Random
Walks in Dynamic Random Environments: Numerics,
Phase Diagrams and Conjectures
L. Avena · P. Thomann
Received: 16 January 2012 / Accepted: 18 May 2012 / Published online: 6 June 2012
© Springer Science+Business Media, LLC 2012
Abstract We perform simulations for one dimensional continuous-time random walks in
two dynamic random environments with fast (independent spin-flips) and slow (simple sym-
metric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic
speeds and the scaling limits of such random walks. We observe different behaviors de-
pending on the dynamics of the underlying random environment and the ratio between the
jump rate of the random walk and the one of the environment. We compare our data with
well known results for static random environment. We observe that the non-diffusive regime
known so far only for the static case can occur in the dynamical setup too. Such anomalous
fluctuations give rise to a new phase diagram. Further we discuss possible consequences for
more general static and dynamic random environments.
Keywords Random environments · Random walks · Law of large numbers · Scaling
limits · Particle systems · Numerics
1 Introduction
1.1 Random Walk in Static and Dynamic Random Environments
Random Walks in Random Environments (RWRE) on the integer lattice are RWs on Z
d
evolving according to random transition kernels, i.e., their transition probabilities/rates de-
pend on a random field (static case) or a stochastic process (dynamic case) called Random
Environment (RE). Such models play a central role in the field of disordered systems of
particles. The idea is to model the motion of a particle in an inhomogeneous medium. In
contrast with standard homogeneous RW, RWRE may show several unusual phenomena as
non-ballistic transience, non-diffusive scalings, sub-exponential decay for large deviation
probabilities. All these features are due to impurities in the medium that produce trapping
L. Avena (
) · P. Thomann
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Zürich 8057, Switzerland
e-mail:
luca.avena@math.uzh.ch
1042 L. Avena, P. Thomann
effects. Although they have been intensively studied by the physics and mathematics com-
munities since the early 70’s, except for the one-dimensional static case and few other par-
ticular situations, most of the results are of qualitative nature, and their behavior is far from
being completely understood. We refer the reader to [
23, 26]and[1, 14] for recent overviews
of the state of the art in static and dynamic REs, respectively.
In this paper we focus on two one-dimensional models in dynamic RE. In particular, the
RW will evolve in continuous time in (two-states) REs given b y two well known interacting
particle systems: independent spin flip and simple symmetric exclusion dynamics. Several
classical questions regarding these types of dynamical models are still open while the be-
havior of the analogous RW in a i.i.d. static case is completely understood. We perform
simulations focusing on their long term behavior. We s ee how such asymptotics are influ-
enced as a function of the jump rate γ of the dynamic REs. The idea is that by tuning the
speed of the REs, we get close to the static RE (γ close to 0) or to the averaged medium
(γ approaching infinity). We observe different surprising phases which allow us to set some
new challenging conjectures and open problems.
Although the choice of the models could appear too restrictive and of limited interest,
as it will be clear in the sequel, such particular examples present all the main features and
the rich behavior of the general models usually considered in the RWRE literature. The
conjectures we state can be extended to a more general setup (see Sect.
4.3).
The paper is organized as follows. In this section we define the models and give some
motivation. In Sect.
2 we review the results known for the analogous RW in a static RE.
Section
3 represents the main novel. We present therein the results of our simulations which
shine a light on the behavior of the asymptotic speed (Sect. 3.1) and on the scaling limits
of such processes (Sects.
3.2 and 3.3). When discussing each question we list several con-
jectures. In the last Sect. 4 we present a brief description of the algorithms, we discuss the
robustness of our numerics and possible consequences for more general RE.
1.2 The Model
We consider a one-dimensional RW whose transition rates depend on a dynamic RE given by
a particle system. In Sect.
1.2.1, we first g ive a rather general definition of particle systems
and then we introduce the two explicit examples we will focus on. In Sect. 1.2.2 we define
the RW in such dynamic REs.
1.2.1 Random Environment: Particle Systems
Let Ω ={0, 1}
Z
. Denote by D
Ω
[0, ∞) the set of paths in Ω that are right-continuous and
have left limits. Let {P
η
,η ∈ Ω} be a collection of probability measures on D
Ω
[0, ∞).
A particle system
ξ = (ξ
t
)
t≥0
with ξ
t
=
ξ
t
(x) :x ∈ Z
, (1.1)
is a Markov process on Ω with law P
η
,whenξ
0
= η ∈ Ω is the starting configuration. Given
a probability measure μ on Ω, we denote by P
μ
(·) :=
Ω
P
η
(·)μ(dη) the law of ξ when ξ
0
is drawn from μ. We say that site x is occupied by a particle (resp. vacant) at time t when
ξ
t
(x) = 1 (resp. 0).
Informally, a particle system is a collection of particles (1’s) on the integer lattice evolv-
ing in a Markovian way. Depending on the specific transition rates between the different
configurations, one obtains several types of particle systems. Each particle may interact
with the others: the evolution of each particle is defined in terms of local transition rates
Continuity and Anomalous Fluctuations in RWRE 1043
that may depend on the state of the system in a neighborhood of the particle. For a formal
construction, we refer the reader to Liggett [
18], Chap. I.
In the sequel we focus on two well known examples with strong and weak mixing prop-
erties, respectively.
(1) Independent Spin Flip (ISF)
Let ξ = (ξ
t
)
t≥0
be a one-dimensional independent spin-flip system, i.e., a Markov process
on state space Ω with generator L
ISF
given by
(L
ISF
f )(η) =
x∈Z
λη(x) +γ
1 −η(x)
f
η
x
−f(η)
,η∈ Ω, (1.2)
where λ,γ ≥ 0, f is any cylinder function on Ω, η
x
is the configuration obtained from η by
flipping the state at site x.
In words, this process is an example of a non-interacting particle system on {0, 1}
Z
where
the coordinates η
t
(x) are independent two-state Markov chains, namely, at each site (inde-
pendently with respect to the other coordinates) particles flip into holes at rate λ and holes
into particles at rate γ . This particle system has a unique ergodic measure given by the
Bernoulli product measure with density ρ = γ /(γ + λ) which we denote by ν
ρ
(see e.g.
[
18], Chap. IV).
(2) Simple Symmetric Exclusion (SSE)
The SSE is an interacting particle system ξ in which particles perform a simple symmetric
random walk at a certain rate γ>0 with the restriction that only jumps on vacant sites are
allowed. Formally, its generator L
SSE
, acting on cylinder functions f ,isgivenby
(L
SSE
f )(η) = γ
x,y∈Z
x∼y
f
η
x,y
−f(η)
,η∈ Ω, (1.3)
where the sum runs over unordered neighboring pairs of sites in Z,andη
x,y
is the configu-
ration obtained from η by interchanging the states at sites x and y.
It is known (see [
18], Chap. VIII) that the family of Bernoulli product measures ν
ρ
, with
density ρ ∈ (0, 1), characterizes the set of equilibrium measures for this dynamics.
Remark 1.1 Note that the ISF and the SSE are completely different types of dynamics. They
are both Markovian in time but while the ISF has no spatial correlations, the SSE has space-
time correlations. The ISF has very good mixing properties due to the fact that once γ and
λ are given, no matter of the starting configuration, it will converge exponentially fast to the
unique equilibrium given by ν
ρ
with ρ = γ /(γ +λ). On the contrary, the SSE dynamics is
strongly dependent on the starting configuration and therefore does not satisfy any uniform
mixing condition. In fact, it is a conservative type of dynamics with a family of equilibria
given by {ν
ρ
: ρ ∈ (0, 1)}. Because of these substantial d ifferences, in the sequel we will
informally say that the ISF and the SSE are examples of fast and slowly mixing dynamics,
respectively.
1.2.2 RW on Particle Systems
Conditional on the particle system ξ ,let
X = (X
t
)
t≥0
(1.4)
1044 L. Avena, P. Thomann
be the continuous time random walk jumping at rate 1 with local transition probabilities
x → x + 1atratepξ
t
(x) +(1 −p)
1 −ξ
t
(x)
,
x → x − 1atrate(1 −p)ξ
t
(x) +p
1 −ξ
t
(x)
,
(1.5)
with
p ∈[1/2, 1). (1.6)
In words, the RW X jumps according to an exponential clock with rate 1, if X is on
occupied sites (i.e. ξ
t
(X
t
) = 1), it goes to the right with probability p and to the left with
probability 1 −p, while at vacant sites it does the opposite.
We write P
ξ
0
to denote the law of X starting from X(0) =0 conditional on ξ ,and
P
μ,0
(·) =
D
Ω
[0,∞)
P
ξ
0
(·)P
μ
(dξ) (1.7)
to denote the law of X averaged over ξ . We refer to P
ξ
0
as the quenched law and to P
μ,0
as
the annealed law. In what follows, when needed, we w ill denote by
X(p, γ , ρ), (1.8)
the RW X just defined either in the ISF or in the SSE environment starting from ν
ρ
and
jumping at rate γ . Note that in the ISF case, the parameter λ is uniquely determined once
we fix γ and ρ.
From now on we assume w.l.o.g. ρ ∈[1/2, 1). The choice of p, ρ ∈[1/2, 1) is not re-
strictive, indeed, due to symmetry, it is easy to see the following equalities in distribution
X(p, ρ, γ )
P
= X(1 − p,1 − ρ, γ)
P
=−X(p, 1 −ρ, γ). (1.9)
1.3 On Mixing Dynamics
In our models, the REs at each site have only two possible states (0 or 1), in most of the
literature on RWRE, the models are defined in a more general framework where infinitely
many states are allowed. The first paper dealing w ith RW in dynamic RE goes back to [
9].
Since then, there has been intensive activity and several advances have recently been made
showing mostly LLN, invariance principles and LDP under different assumptions on the
REs or o n the transition probabilities of the walker. See for example [
3, 5, 7, 8, 10, 14, 19]
(most of these references are in a discrete-time setting). For an extensive list of reference we
refer the reader to [
1, 14].
One of the main difficulties in the analysis of random media arises when space-time
correlations in the RE are allowed. Both models presented in Sect.
1.2.1 fit in this class
but, as mentioned in Remark
1.1, their mixing properties are substantially different. The ISF
dynamics belongs to the class of fast mixing environments which is known to be qualitatively
similar to a homogeneous environment. In fact, a RW on this type of RE exhibits always
diffusive scaling.
The SSE is an example of what we called slowly mixing dynamics. For a RW driven by
these latter types of dynamics, we are not aware of any results other than [
2, 4, 12]. One of
the main result of our simulations is that the RW in (1.4) on the SSE, similarly to the RW in a
static RE (see Sect. 2.2), may exhibit non-diffusive behavior (see Sect. 3.2). This latter result
