Absolute Continuity and Weak Uniform Mixing of Random Walk in Dynamic Random Environment

0.41 MB·

by  Stein Andreas Bethuelsen

Absolute continuity and weak uniform
mixing of random walk in dynamic
random environment
Stein Andreas Bethuelsen Florian V¨ollering
6 ∗ †
1
0 October 6, 2016
2

t
c
O

5 We prove results for random walks in dynamic random environments
whichdonotrequirethestronguniformmixingassumptionspresentinthe

] literature. We focus on the “environment seen from the walker”-process
R
and in particular its invariant law. Under general conditions it exists and
P
is mutually absolutely continuous to the environment law. With stronger
.
h assumptions we obtain for example uniform control on the density or a
t
a quenchedCLT.Thegeneralconditionsaremademoreexplicitbylookingat
m hidden Markovmodels orMarkovchains as environmentandby providing
[ simple examples.

2
v
MSC2010. Primary 82C41;Secondary 82C43, 60F17, 60K37.
0
1 Key words and phrases. Random walk,dynamic randomenvironment,absolute conti-
7 nuity,stability,centrallimittheorem,hiddenMarkovmodels,disagreementpercolation
7
0
.
1
0
6 1 Introduction and main results
1
:
v 1.1 Background and motivation
i
X We study the asymptotic behaviour of a class of random walks (X ) on Zd whose
t
r transition probabilities depend on another process, the random environment. Such
a
∗TechnischeUniversit¨atMu¨nchen,Fakult¨atfu¨rMathematik,Boltzmannstr.3,85748Garching,Ger-
many. Email: [email protected]
†Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY,
UnitedKingdom
1
models play an important role in the understanding of disordered systems and serve
as natural generalisations of the classical simple random walk model for describing
transport processes in inhomogeneous media.
These types of random walks, which are called random walks in random environ-
ment, can be split into two broad areas, static and dynamic environments. In static
environments the environmentis created initially and then stays fixed in time. In dy-
namic environments the environment instead evolves over time. Note that a dynamic
environment in Zd can always be reinterpreted as a static environment in Zd+1 by
turning time into an additional space dimension.
A major interest in dynamic environments are their often complicated space-time
dependency structure. Typically, in order to show that the random walk is diffusive,
one looks for some way to guarantee that the environment is “forgetful” and random
walk increments are sufficiently independent on large time scales.
One approach to this are various types of mixing assumptions on the environment.
By now, general results are known for Markovian environments which are uniformly
mixingwithrespecttothestartingconfiguration(seeAvena,denHollander,andRedig
[3]andRedigandVo¨llering[25]). Forthis,therateatwhichthedynamicenvironment
converges towards its equilibrium state plays an important role.
Onthe otherhand,modelswherethedynamicalenvironmenthasnon-uniformmix-
ing properties serve as a major challenge and are still not well understood. Oppo-
site to the diffusive behaviour known for uniform mixing environments, it has been
conjectured that (X ) may be sub- or super-diffusive for certain non-uniform mixing
t
environments, see Avena and Thomann [2]. Though some particular examples yield-
ing diffusive behaviourhaverecentlybeen studiedby rigorousmethods, e.g.Deuschel,
Guo, and Ramirez [12], Hil´ario, den Hollander, Sidoravicius, dos Santos, and Teix-
eira [17], Huveneers and Simenhaus [19] and Mountford and Vares [23], these results
are model specific and/or perturbative in nature. No general theory has so far been
developed.
In this article we provide a new approach for determining limiting properties of
random walks in dynamic random environment, in particular about the invariant law
of the “environmentas seen from the walker”-process. Under generalmixing assump-
tions, we prove the existence of an invariant measure mutually absolutely continuous
with respect to the random environment (Theorems 1.1 and 1.3). Our mixing as-
sumptions are considerably weaker than the uniform mixing conditions present in the
literature (e.g. cone mixing) and do not require the environment to be Markovian.
Animportantfeatureofourapproachisthatitcanalsobeappliedtodynamicswith
non-uniform mixing properties. Examples include an environment given by Ornstein-
Uhlenbeck processes and the supercritical contact process.
Knowledge about the invariant measures for the “environment as seen from the
walker”-process yield limit laws for the random walk itself. One immediate applica-
tion of our approach is a strong law of large numbers for the random walk. Further
applications include a quenched CLT based on Dolgopyat, Keller, and Liverani [15],
Theorem 1, considerably relaxing its requirements.
Our key observation is an expansion of the “environment as seen from the walker”-
process (Theorem 3.1). This expansion enables us to separate the contribution of the
2
randomenvironmentto the law of the “environmentas seen from the walker”-process
from that of the transition probabilities of the random walk.
Wealsoshowstabilityunderperturbationsoftheenvironmentorofthejumpkernel
of the random walk. Under a strong uniform mixing assumption, we obtain uniform
controlonthe Radon-Nikodymderivativeofthe lawofthe “environmentasseenfrom
the walker”-process with respect to the environment, irrespective of the choice of the
jump kernel of the random walker.
Outline
In the next two subsections we give a precise definition of our model and present
our main results, Theorem 1.1 and Theorem 1.3. Section 2 is devoted to examples
and applications thereof. In Section 3 we derive the aforementioned expansion, and
present results on stability and control on the Radon-Nikodym derivative. Proofs are
postponed until Section 4.
1.2 The model
In this subsection we give a formal definition of our model. In short, (X ) is a ran-
t
dom walk in a translation invariant random field with a deterministic drift in a fixed
coordinate direction.
The environment
Let d N and let Ω := EZd+1 where E is assumed to be a finite set. We assign to
∈
the space Ω the standard product σ-algebra generated by the cylinder events. For
F
Λ Zd+1, we denote by the sub-σ-algebra generated by the cylinders of Λ. For
Λ
⊂ F
the forward half-space H:=Zd Z we write for .
≥0 ≥0 H
× F F
By (Ω) we denote the set of probability measures on (Ω, ). We call η Ω the
1
M F ∈
environment and denote by P (Ω) its law. A particular class of environments
1
∈ M
containedinoursetuparepathmeasuresofastochasticprocess(η )whosestatespace
t
is Ω := EZd. To emphasise this, for η Ω and (x,t) Zd Z, we often write η (x)
0 t
∈ ∈ ×
for the value of η at (x,t).
We assume throughout that P is measure preserving with respect to translations,
that is, for any x Zd, t Z,
∈ ∈
P()=P(θ ),
x,t
· ·
whereθ denotesthe shiftoperatorθ η (y)=η (y+x). Furthermore,weassume
x,t x,t s s+t
that P is ergodic in the time direction, that is, all events B for which θ B :=
o,1
∈ F
ω Ω: θ ω B =B are assumed to satisfy P(B) 0,1 . Here, o Zd denotes
o,−1
{ ∈ ∈ } ∈{ } ∈
the origin.
Remark 1.1. We present in Subsection 2.2-2.4 an approach where E is allowed to be
a general Polish space, by considering the environment as a hidden Markov model.
3
The random walk
The random walk (X ) is a process on Zd. We assume w.l.o.g. that X = o. The
t 0
transition probabilities of (X ) is assumed to depend on the state of the environment
t
as seen from the random walk. That is, given η Ω, then the evolution of (X ) is
t
∈
given by
P (X =o)=1
η 0
P (X =y+z X =y)=α(θ η,z),
η t+1 t y,t
|
where α: Ω Zd [0,1]satisfies α(η,z)=1 for all η Ω. The lawof the ran-
× → z∈Zd ∈
dom walk, Pη ∈M1((Zd)Z≥0), whPere we have conditioned on the entire environment,
is called the quenched law. We denote its σ-algebra by . Further, for P (Ω),
1
G ∈ M
we denote by PP 1 Ω (Zd)Z≥0 the joint law of (η,X), that is,
∈M ×
(cid:0) (cid:1)
P (B A)= P (A)dP(η), B ,A .
P η
× ∈F ∈G
ZB
The marginal law of PP on (Zd)Z≥0 is the annealed (or averaged) law of (Xt).
Weassumethatthetransitionprobabilitiesof(X )onlydependontheenvironment
t
within a finite region around its location. That is, there exist R N such that for all
∈
z Zd
∈
α(η,z) α(σ,z)=0 whenever σ η on [ R,R]d 0 .
− ≡ − ×{ }
Further, define
:= y Zd: supα(η,y)>0 (1.1)
R { ∈ }
η∈Ω
as the jump range of the random walker, which we assume to be finite and to contain
o. By possibly enlarging R we can guarantee that
sup y : y R.
{k k1 ∈R}≤
y∈Zd
Lastly, we say that (X ) is elliptic in the time direction if
t
α(η,o)>0, η Ω. (1.2)
∀ ∈
If, after replacing o with y , (1.2) holds for all y , then we say that (X ) is
t
∈ R ∈ R
elliptic.
The environment process
“Theenvironmentasseenfromthewalker”-processisofimportanceforunderstanding
the asymptotic behaviour of the random walk itself, but it is also of independent
interest. This process, which is given by
(ηEP):=(θ η), t Z ,
t Xt,t ∈ ≥0
is called the environment process. Note that (ηEP) is a Markov process on Ω under
t
P , η Ω, with initial distribution P.
η
∈
4
1.3 Main results
Inthissubsectionwepresentourmainresultsabouttheasymptoticbehaviourof(X )
t
and(ηEP). However,beforestatingourfirsttheoremweneedtointroduce somemore
t
notation.
Recall (1.1) and let
Γ := (γ ,γ ,...,γ ): γ Zd,γ γ , k i<0,γ =o
k −k −k+1 0 i i i−1 0
∈ − ∈R − ≤
be the set o(cid:8)f all possible backwards trajectories from (o,0) of length k. F(cid:9)or γ Γ
k
∈
and σ Ω, denote by
∈
−m
A−m(γ,σ):= θ η σ on [ R,R]d 0 , 1 m k,
−k γi,−i ≡ i − ×{ } ≤ ≤
i=−k
\ (cid:8) (cid:9)
the event that an element η Ω equals σ in the R-neighbourhood along the path
(γ ,...,γ ). A−1(γ,σ) is th∈e event that the path of the environment observed by
k −m −k
the random walk equals σ if the random walk moves along the path γ. Given γ Γ ,
k
∈
denote by
−m(γ):= A−m(γ,σ): σ Ω and P(A−m(γ,σ))>0
A−k −k ∈ −k
the set of all poss(cid:8)ible observations along the path (γ ,...,γ(cid:9) ). We write −m for
the set of events −m(γ). If m=1 we simkply wr−itme . A−∞
k≥m,γ∈ΓkA−k A−∞
Further,denoteby := (x,t) H: x (R+1)t theforwardconewithcentre
S C { ∈ k k1 ≤ }
at (o,0) and slope proportional to R+1. For j Z, denote by (j):= θ H and
o,j
∈ C C∩
let ∞ := be the tail-σ-algebra with respect to .
F∞ j∈NFC(j) FC
TheoremT1.1(Existenceofanergodicmeasurefortheenvironmentprocess). Assume
that P (Ω) satisfies
1
∈M
lim sup sup P(B A) P(B) =0. (1.3)
l→∞B∈FC(l)A∈A−∞| | − |
Then there exists PEP (Ω) invariant under (ηEP) satisfying PEP =P on ∞.
∈M1 t F∞
If (X ) is elliptic in the time direction and P is ergodic in the time direction, then
t
PEP is ergodic with respect to (ηEP). Moreover, for any Q P on ∞,
t ≪ F∞
t−1
1
P (ηEP ) converges weakly towards PEP as t . (1.4)
t Q s ∈· →∞
s=0
X
Remark 1.2. There is a certain freedom in the ellipticity and the ergodicity assump-
tions in Theorem 1.1. For instance, the statement still holds if, for some k N, the
∈
walker has a positive probability to return to o after k time steps, uniformly in the
environment. The definitions canalsobe modified to require ellipticity and ergodicity
with respect other directions (y,1) Zd+1, with y (instead of in the direction
∈ ∈ R
(o,1). On the other hand, both ellipticity and ergodicity in the time direction are
natural assumptions if P is the path measure of some stochastic process.
5
Corollary 1.2 (Law of large numbers). Assume that P (Ω) is ergodic in the
1
∈ M
time direction and satisfies (1.3), and that (X ) is elliptic in the time direction. Then
t
there exists v Rd such that lim 1X =v, P a.s.
∈ t→∞ t t P−
Condition (1.3) is a considerably weaker mixing assumption than the cone mixing
conditionintroducedbyCometsandZeitouni[11](seeCondition therein)andused
1
A
in Avena, den Hollander, and Redig [3] in the context of random walks in dynamic
random environment. For comparison, note that cone mixing is equivalent to taking
the supremum over events A := in (1.3). That Condition (1.3) is
∈ F<0 FZd+1\H
strictlyweakercanalreadybeseeninthecasewherePisi.i.d.withrespecttospace;see
Theorem 2.1. Further examples where Condition (1.3) improve on the classical cone
mixing condition are given in Section 2 and include dynamic random environments
with non-uniform mixing properties.
Under a slightly stronger mixing assumption on the environment we obtain more
information about PEP. For this, denote by Λ(l) := x H: x l , l N,
{ ∈ k k1 ≥ } ∈
where denotes the l distance from (o,0), and let ∞ := be the
k·k1 1 F≥0 l∈NFΛ(l)
tail-σ-algebra with respect to .
≥0
F T
Theorem 1.3 (Absolute continuity). Let φ: N [0,1] be such that
→
sup sup P(B A) P(B) φ(l), (1.5)
| | − |≤
B∈FΛ(l)A∈A−∞
with lim φ(l)=0. Then PEP =P on ∞ (with PEP as in Theorem 1.1) and
l→∞ F≥0
sup P(B) PEP(B) φ(l). (1.6)
| − |≤
B∈FΛ(l)
Furthermore, if (X ) in addition is elliptic, then P and PEP are mutually absolutely
t
continuous on (Ω, ).
≥0
F
Knowing that the environment process converges toward an ergodic measure, it is
wellknownhowtoapplymartingaletechnicsinordertodeduceanannealedfunctional
central limit theorem. However, it may happen that the covariance matrix is trivial.
In Redig andVo¨llering[25] itwas shownthat the covariancematrix is non-trivialin a
rathergeneralsetting whenthe environmentis givenbya Markovprocesssatisfyinga
certainuniformmixingassumption. Itisaninterestingquestionwhether(X )satisfies
t
anannealedfunctionalcentrallimit theoremwith non-trivialcovariancematrix under
the weaker mixing assumption of (1.3).
To obtain a quenched central limit theorem is a much harder problem and is only
known in a few cases for random walks in dynamic random environment, see e.g.
BricmontandKupiainen[9],Deuschel,Guo,andRamirez[12],DolgopyatandLiverani
[14] and Dolgopyat, Keller, and Liverani [15]. In [15], Theorem 1, a quenched central
limittheoremwasprovenundertechnicalconditionsonboththe environmentandthe
environmentprocess. Oneimportantconditiontherewasthattheenvironmentprocess
has an invariant measure mutually continuous with respect to the invariant measure
of the environment. By Theorem 1.3 above this condition is fulfilled. Combining this
result with rate of convergence estimates obtained in [25], we conclude a quenched
central limit theorem for a large class of uniformly mixing environments.
6
Corollary 1.4(Quenchedcentrallimittheorem). Assumethat (η )is aMarkov chain
t
on EZd. For σ,ω Ω let P be a coupling of (η ) started from σ,ω Ω respectively
0 σ,ω t 0
∈ ∈
and such that, for some c,C >0,
b
sup P (η(1)(o)=η(2)(o)) Ce−ct. (1.7)
σ,ω t 6 t ≤
σ,ω∈Ω
b
Furthermore, assume that (η ) satisfies Conditions (A3)-(A4) in [15] and that (X ) is
t t
elliptic. Then, there is a non-trivial d d matrix Σ such that for P -a.e. environment
µ
×
history (η )
t
X Nv
N − converges weakly towards (0,Σ) P a.s.,
√N N (ηt)−
where µ (Ω ) is the unique ergodic measure with respect to (η ).
1 0 t
∈M
Conditions (A3)-(A4) in [15] are mixing assumptions on the dynamic random envi-
ronment(η ). Condition(A3)isa(weak)mixingassumptiononµ,whereasCondition
t
(A4)ensuresthat(η )is“local”. Fortheprecisedefinitionswereferto[15],page1681.
t
In [15], Theorem 2, the statement of Corollary 1.4 was proven in a perturbative
regime. Corollary 1.4 extends their result as there are no restrictions (other than el-
lipticity)onthetransitionprobabilitiesoftherandomwalk. WeexpectthatCorollary
1.4canbe further improvedto afunctional CLT assumingonly a polynomialdecayin
(1.7).
2 Examples and applications
In this section we present examples of environments which satisfy the conditions of
Theorem 1.1 and Theorem 1.3. Particular emphasise is put on environments associ-
ated to a hidden Markov model for which we can improve on the necessary mixing
assumptions.
2.1 Environments i.i.d. in space
The influence ofthe dimensionon requiredmixing speeds is somewhatsubtle. Onthe
one hand,the randomwalkobservesonly alocalarea,and,inthe caseofconservative
particle systems like the exclusion process, one can expect that in high dimensions
informationaboutobservedparticlesinthepastdiffusesaway. Ontheotherhand,the
higherdimension,themoresitesthe randomwalkcanpotentiallyvisitinafixedtime.
Furthermore, a comparison with a contact process or directed percolation gives an
argumentthat informationcanspreadeasierinhigher dimensions,hence observations
along the path of the random walk could have more influence on future observations
if the dimension increases.
This problem becomes significantly easier when the environment is assumed to be
i.i.d. in space, that is P = P , and P (EZ) is the law of (η (x)) for
×x∈Zd o o ∈ M1 t t∈Z
any x Zd.
∈
7
Theorem 2.1. Assume that P= P and that
×x∈Zd o
sup P (B A) P (B) < , (2.1)
o o
| | − | ∞
t≥1B∈G≥t,A∈G<0
X
where ( ) is the σ-algebra of EZ generated by the values after time t (before
≥t <0
G G
time 0) with respect to P . Then (1.5) holds.
o
Observe that (2.1) does not depend on the dimension. This is in contrast to the
conemixingconditionofCometsandZeitouni[11],whereanadditionalfactortd inside
the sum of (2.1) is required. In Subsection 2.3 we present a class of environments
which have arbitrary slow polynomial mixing, thus showing that Theorem 2.1 yields
an essential improvement.
2.2 Hidden Markov models
When P is the path measure of a stochastic process (η ) evolving onΩ , the results of
t 0
Subsection1.3canbe improved. Inthis subsectionwediscuss indetailthe casewhere
the random environment is governedby a hidden Markov model.
The environment (η ) is a hidden Markov model if it is given via a function of a
t
Markovchain (ξ ). To be more precise,let E be a Polishspace, Ω =EZde with d d,
t 0
≥
and Ω=ΩZ. Denote by ˜ the corresponding σ-algebra. We assume that the Markov
0 F
chain (ξ ) is defined on Ω with law P andeis ergodic with laweµ e (Ω ).eHere
t ξ 1 0
∈ M
ξ Ωe deenotes the starting configuration. Let Φ : Ω Ω = EZd be a translation
0 0 0
∈ →
invariant map and let ηte= Φ(ξt). Wee call (ηt) a hidden Markovemodel, wheich has µ
as thee induced measure on Ω as invariant measure.eWe assume throughout that Φ is
0
of finite range, that is, the function Φ()(o) is ˜ -measurable for some Λ Zd˜finite.
Λ
· F ⊂
Remark 2.1. WhenE˜ isfinite,thecanonicalchoiceofΦistheidentitymap. However,
our setup opens for more sophisticated choices. One example is the projection map.
For instance, if d˜>1 and d=1, one can consider the hidden Markov model given by
η (x)=ξ (x,0,...,0). Inotherwords,therandomwalkonlyobservestheenvironment
t t
in one coordinate.
Condition (1.5) in Theorem 1.3 is an infinite volume condition which can be hard
to verify by direct computation. The next result yields a sufficient condition which
only needs to be checked for single site events. For its statement, we first introduce
the concept of P (Ω) having finite speed of propagation.
1
∈M
Definition2.1. WesaythatP (Ω)hasfinitespeedofpropagationifthefollowing
1
∈M
holds: for some α > 0, and for each A and A′ , where Λ(αt,t) :=
<0 Λ(αt,t)
∈ F ∈ F
(x,s) H: x αt,0 < s t , there is a coupling P of P( A,A′) and
{ ∈ k k1 ≥ ≤ } A,A′ · |
P( A) such that
·|
b
td sup P η1(o)=η2(o) < . (2.2)
A,A′ t 6 t ∞
t≥1 A∈F<0,A′∈FΛ(αt,t)
X (cid:0) (cid:1)
b
8
Furthermore, any such coupling satisfies P () = P (θ ) for all (x,s)
A,A′ · θx,sA,θx,sA′ x,s· ∈
Zd+1, where ω θ A if and only if θ ω A.
x,s −x,−s
∈ ∈
b b
Finite speed of propagationis a naturalassumptionfor many physicalapplications.
Note that, for many interacting particle systems there is a canonical coupling given
by the so-called graphical representationcoupling.
Corollary 2.2. Assume that (ξ ) has finite speed of propagation and that
t
td sup P Φ(ξ1) (o)=Φ(ξ2) (o) < . (2.3)
Ω,A 0 0
6 ∞
Xt≥1 A∈A−−∞t (cid:0) (cid:1)
b
Then (1.5) holds for (η )=Φ(ξ ).
t t
Remark 2.2. The measure P denotes the coupling of P() and P( A).
Ω,A
· ·|
Corollary 2.2 follows by a slightly more general statement, see Theorem 4.4. This
b
approachcanalsobeusedincaseswhere(ξ )doesnothavefinitespeedofpropagation.
t
In such cases, (2.3) is sufficient for (1.3) to hold. Observe also that, by applying the
projectionmapintroducedinRemark2.1,thedimensionalitydependenceinCondition
(2.3) can be replaced by the dimensionality of the range of the random walk.
Markovian environment
IfPisthepathmeasureofaMarkovchain(η ),wecanweakenthemixingassumption.
t
Insuchcases,weconsiderαasafunctionfromΩ Zd. BecausetheMarkovproperty
0
×
allows us to look at the invariant measure of the environment process just at a time
0 instead of in the entire upper half-space H, we have the following mixing condition.
Here we denote by ∞ the tail-σ-algebra of := .
F=0 F=0 FZd×{0}
Theorem 2.3. Assume that (η ) is a Markov chain with ergodic invariant measure
t
µ (Ω ). Further, assume that the path measure P (Ω) has finite speed of
1 0 µ 1
∈ M ∈ M
propagation and that
td−1 sup P η1(o)=η2(o) < . (2.4)
Ω,A 0 6 0 ∞
Xt≥1 A∈A−−∞t (cid:0) (cid:1)
b
Then thereexists µEP (Ω ) invariant for theMarkov chain (ηEP) suchthat µEP
∈M1 0 t
agrees with µ on ∞. If in addition (X ) is elliptic, then µEP and µ are mutually
F=0 t
absolutely continuous and µEP is ergodic with respect to (ηEP).
t
It is important to note that (2.4) (as well as (2.3)) does not require (η ) to be
t
uniquely ergodic. However, if for every σ,ξ Ω , there is a coupling P of P and
0 σ,ξ σ
∈
P which satisfies the finite speed of propagationproperty and
ξ
b
∞
td−1 sup P (η1(o)=η2(o))< , (2.5)
σ,ξ t 6 t ∞
Xt=1 σ,ξ∈Ωe0
b
9
thenitfollows,undertheassumptionsofTheorem2.3,that(ηEP)isuniquelyergodic.
t
Equation (2.5) should be compared with Assumption 1a in [25], that is;
∞
t(d)supEˆ ρ(η1(o),η2(o))dt< , (2.6)
η,ξ t t ∞
Z0 η,ξ
where ρ: E E [0,1] is the distance function. Their assumption was used to show
× →
(among others) the existence of µEP (Ω ) invariant and ergodic for the Markov
1 0
∈M
chain(ηEP),seeLemma3.2therein. NoteinparticularthatAssumption(2.6)hast(d)
t
inside the integral, whereas (2.5) only requires t(d−1).
2.3 Polynomially mixing environments
As example of environments which fully utilise the polynomial mixing assumption of
Theorem2.1andCorollary2.2,weconsiderlayeredenvironments. Thesewerealready
considered in [25] for the same purpose, but since we are in a different setting we use
the setting of hidden Markov models.
The idea of layered environments is that, given a summable sequence (b ) (0,1),
n
⊂
foreachlayern,the process(ξ (,n)) isanuniformexponentiallymixingMarkov
t · t∈Z≥0
chain on [ 1,1] with an exponential relaxation rate b , and independent layers. For
n
−
simplicity, in this example, we choose ξ (,n) to be i.i.d. spin flips, that is, for each
t
·
x Zd˜,
∈
ξ (x,n), with probability 1 b ;
t n
ξ (x,n)= −
t+1
(Unif[ 1,1], with probability bn;
−
independent for all x,n,t. In other words, at each time step the spin retains its old
value with probability 1 b and chooses uniformly on [ 1,1] with probability b .
n n
− −
In the context of the previous subsection we thus have E := [ 1,1]N. We further
−
choose d=d 1, E = 0,1 and set, for a summable sequence (a ) (0,1),
n
≥ { } ⊂
e
Φ(eξ)(x)=1P∞n=1anξ(x,n)>0.
The behaviour of this kind of processes is then determined by the two sequences (a )
n
and (b ). When a = 1n−α, b = 1n−β for some α,β > 1, we have the following
n n 2 n 2
bound on the mixing of (η ).
t
Theorem 2.4. There are constants 0<c <c < so that
1 2
∞
−α+1 −α+1 α−1
c1t β ≤supkPξ(ηt(0)∈·)−Pσ(ηt(0)∈·)kTV ≤c2t β (logt) β .
ξ,σ
Here is thetotalvariation distance betweenthetwo distributions. In particular,
k·kTV
if α>β+1, then (1.5) holds.
10