Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

0.27 MB·English

by  Marek Biskup

SUBDIFFUSIVE HEAT-KERNEL DECAY IN FOUR-DIMENSIONAL
I.I.D. RANDOM CONDUCTANCE MODELS
2
M.BISKUP1,2AND O.BOUKHADRA3
1
0
1DepartmentofMathematics, UCLA,LosAngeles,California, USA
2
2SchoolofEconomics, UniversityofSouthBohemia,Cˇeske´ Budeˇjovice, CzechRepublic
n
a 3De´partementdeMathe´matiques, UMC,Constantine, Algeria
J

7 ABSTRACT. Westudythediagonalheat-kerneldecayforthefour-dimensionalnearest-neighbor
1 randomwalk(onZ4)amongi.i.d.randomconductancesthatarepositive,boundedfromabovebut
canhavearbitrarilyheavytailsat zero. Ithasbeen known thatthequenched returnprobability

R] Pw2n(0,0) after 2n steps is at most C(w )n−2logn, but the best lower bound till now has been
C(w )n 2. Here wewill show that the lognterm marks a real phenomenon by constructing an
P environ−ment,foreachsequencel n ¥ ,suchthat
. →
h Pw2n(0,0) C(w )log(n)n−2/l n,
t ≥
a withC(w )>0a.s.,alongadeterministicsubsequenceofn’s. Notably,thisholdssimultaneously
m
witha(non-degenerate)quenchedinvarianceprinciple. Asforthed 5casesstudiedearlier,the
≥
[ sourceoftheanomalousdecayisatrappingphenomenonalthoughthecontributionisinthiscase

collectedfromawholerangeofspatialscales.
2
v
2
4 1. INTRODUCTION AND RESULTS
5
5 Recent years have witnessed remarkable progress in the understanding of a class of reversible
. randomwalksinrandomenvironmentsthatgounderthenameRandomConductanceModel. The
0
1 settingofatypicalinstanceofthisproblemisasfollows: Considerthed-dimensionalhypercubic
0 lattice Zd and let Bd denote the set of unordered nearest-neighbor pairs. For a configuration
1
: w =(w b)b Bd (0,¥ )Bd,definetheMarkovchainX =(Xn)n 0withstatespaceZd andtransition
v probability∈ ∈ ≥
i
X w
xy , (x,y) Bd,
r Pw (x,y):= pw (x) ∈ (1.1)
a (0, otherwise,
where
p w (x):= (cid:229) w xy. (1.2)
y:(x,y) Bd
∈
Sometimes even w b =0 is permitted; the state space is then just x: p w (x)>0 or, when such
{ }
exists, an infinite connected component thereof. Let Px denote the distribution of X subject to
w
Pwx(X0=x)=1. Theprincipal itemsofinterestarevariousasymptotics ofthelawofX underPwx
inthesituation whenw isasamplefromaprobability distribution P.
(cid:13)c 2012M.BiskupandO.Boukhadra.Reproduction,byanymeans,oftheentirearticlefornon-commercialpurposes
ispermittedwithoutcharge.
1
2 M.BISKUPANDO.BOUKHADRA
Much of the early effort by probabilists concerned the validity of the (functional) Central
Limit Theorem. In a sequence of papers (Kipnis and Varadhan [27], De Masi, Ferarri, Gold-
stein and Wick [19, 20], Sidoravicius and Sznitman [34], Berger and Biskup [9], Mathieu and
Piatnitski [31], Mathieu [30], Biskup and Prescott [13], Barlow and Deuschel [5], Andres, Bar-
low, Deuschel and Hambly [1]), it has gradually been established that, as n ¥ , the law of
→
t X nt /√n under Pw0 scales to a non-degenerate Brownian motion for almost every environ-
m7→entw⌊ ,⌋provided that certain conditions are metby thelaw ofw . Fori.i.d.lawsP concentrated
on[0,¥ )Bd,ind 2theseconditions are
≥
E(w )<¥ and P(w >0)> p (d), (1.3)
b b c
where E denotes the expectation in P and p (d) is the critical threshold for bond percolation
c
onZd. Ind =1the second condition needs tobereplaced byE(w 1)<¥ ;independence isnot
b−
required (e.g.,BiskupandPrescott[13]). Thesameconditions asind =1aresufficienttoimply
thequenched CLTind =2forgeneralergodic environments (Biskup[11]).
While the proof of the functional CLT is remarkably soft for the law on path space that is
averaged over the environment — the so called annealed or averaged law — the almost-sure or
quenched lawgenerally requiresalsotheheatkernelupperbound,
c
Pnw (x,y) 1 e−c2|x−y|2/n, n N(w ,x,y). (1.4)
≤ nd/2 ≥
This is conceptually wrong as the CLTseems to require alocal-CLT type ofestimate. Notwith-
standing, for environments possessing some form of uniform ellipticity, these heat-kernel upper
bounds canbeobtained. Indeed, theyaretheresults ofthesequence ofpapersbyDelmotte[18],
Benjamini and Mossel [8], Heicklen and Hoffman [26], Mathieu and Remy [32] culminating in
Barlow’swork[3]forthesimplerandomwalkonthesupercritcalpercolationcluster. (Weregard
this case as uniformly elliptic too although on a spatially inhomogeneous graph.) However, in
the environments with heavy tails at zero, it was in fact discovered that (1.4) may fail (Fontes
and Mathieu [23], Berger, Biskup, Hoffman and Kozma [10]) and a coarse-graining procedure
wasrequired to overcome this difficulty and derive the quenched functional CLT(Mathieu [30],
Biskup and Prescott [13]). We note that when the left condition in (1.3) fails, the scaling limit
ofX maybenotbediffusiveatall(BarlowandCˇerny´ [4],BarlowandZheng[7],Cˇerny´ [17]).
The study [10] presents two types of results. First, for i.i.d. environment laws bounded from
above,itrestricts thediagonal heat-kernel decaybythefollowingestimates
n d/2, d =2,3,
−
Pwn(0,0) C(w )n−2logn, d =4, (1.5)
≤
n−2, d 5,
≥
whereC(w )<¥ almostsurely, withtheadditional observation,

n2P2wn(0,0) 0 P-a.s. ind 5. (1.6)
−n−−→¥ ≥
→
Second,foranysequence l ¥ ,ani.i.d.environment lawisconstructed sothat
n
↑
C(w )
P2wnk(0,0)≥ l n2, k≥1, (1.7)
nk k
HEAT-KERNELDECAYINRANDOMCONDUCTANCEMODELS 3
along a deterministic sequence n ¥ , whereC(w )>0almost surely. Since the Central Limit
k
→
TheoremforX holds,wealsohave
n
C(w )
Pw2n(0,0) , (1.8)
≥ nd/2
cf,e.g.,[13,Remark2.2].
Putting the bits and pieces together we conclude that the return probability Pw2n(0,0) always
decays diffusively in spatial dimensions d = 2,3, while, in dimensions d 5, it can decay as
≥
slowaso(n 2). (Ind=1,thedecaycanbearbitrarily slow.) Furtherprogress hasbeenmadeby
−
Boukhadra[14,15]whoshowedthatthetransitionfromregulardecayn d/2 toanomalousdecay
−
n 2+o(1) ind 5actually occurs intheclassofpower-law tails—withtheexponent g =d/2in
−
P(0<w <s)≥ sg beingpresumablythecriticalfortheanomalytoappear. Ind 5thismeshes
b
∼ ≥
nicelywiththeannealed estimatesobtained byFontesandMathieu[23].
Thecombinedresultsof[10,14,15]providedefinitiveanswersinallspatialdimensionsexcept
d = 4, where (1.5) and (1.7) differ by a logarithmic factor. Computations for time-dependent
environments (cf. Theorem 5.3 of [10]) suggested that (1.5) is presumably the one closer to the
truth, but any feasible method of proof seemed to require control of off-diagonal heat-kernel
lower bounds. This would seem in turn to demand — in order to avoid circuitous reasoning —
running a complicated induction along scales. We are thus pleased to report on a conceptually
straightforward, albeitstilltechnically complicated, proofofthefollowingtheorem:
Theorem1.1 Assumed=4. Foreverysequence l withl ¥ ,thereexistsani.i.d.environ-
n n
mentlawPwithP(0<w 1)=1,arandomvari{abl}eC(w )w↑ithP(C>0)=1andasequence
b
n ¥ suchthatforevery≤n n ,
k k
→ ∈{ }
logn
P2wn(0,0)≥C(w )l n2. (1.9)
n
The conclusion is that the possibility of anomalous heat-kernel decay in the random conduc-
tance model extends to all dimensions d 4; yet in d =4 the correction to diffusive behavior
≥
is only logarithmic. We actually believe, although cannot prove, that similarly to d 5 the up-
≥
per bound in (1.5) can be approached arbitrarily closely but will not be attained in any given
environment. Weformulate thisasaconjecture:
Conjecture1.2 Assumed=4. Foreveryi.i.d. environment PwithP(0<w 1)=1,
b
≤
n2
P2wn(0,0) 0, P-a.s. (1.10)
logn −n−−→¥
→
Wepointoutthattheproofoftheasymptotic(1.6)ind 5doesnotseemtocarryovertothis
≥
case and so presumably a new idea is needed here. Update in revised version: The conjecture
has in the meantime been proved in an upcoming preprint by Biskup, Louidor, Rozinov and
Vandenberg-Rodes [12]. AnovelinputistheuseoftheDominatedErgodicTheorem.
Here is the main idea underlying the proof of Theorem 1.1. In order to ensure that the chain
returns to its starting point at a required time, we can either let it arrive there more or less by
accident (thedominating strategy ind =2,3)ormakeitfallinto(andhideinside) aspecifictrap
nearbywhichmakesalaterreturntothestartingpointconsiderably lessdifficult(thedominating
4 M.BISKUPANDO.BOUKHADRA
strategy in d 5). However, in d =4 the difference between the two strategies is so subtle that
≥
wecannolonger force whichtrapthechain fallsinto; itneeds tofindonebyitself. Thisputsus
in the class of ideas underlying the approach taken by Boukhadra [14, 15] to control the d 5
≥
anomalyintheclassofpower-lawtails.
Remarks 1.3 (1) Should one desire to have (1.9) without a random constant, this can be done
bymakingtherangeofk’srandom.
(2) We did not try to optimize the tails of the distribution of w for which the anomalous
b
behavior occurs ind =4although wethinkitisunlikely tooccurintheclassofpower-law tails.
On the other hand, our method of proof — being closer to the approach of [14, 15] — could
conceivably be adapted to yield a proof of a sharp threshold in the exponents that would yield
anomalous behavior ind 5.
≥
(3)Asin[10], wemadeno attempt to derive off-diagonal estimates onthe heat kernel. Thus,
inallcasesd 4,aquestionremainshowonecanreconcilethesubdiffusivediagonalheat-kernel
≥
decaywiththestandard heat-kernel decaythatshouldresumevalidity (atleastonaverage) atthe
diffusivespace-time scale(bytheCLT).
(4) The proof of the upper bounds (1.5) in [10] applies even to non-i.i.d. environments that
possessa“strong” componentalmostsurely. (Thiscanbeguaranteed byrequiringthatthelawis
dominated frombelowbyani.i.d.lawinwhichedgeswithpositive conductances percolate.) On
theotherhand,itisnothardtoconstructcorrelatedenvironmentsthatwouldmaketheheatkernel
decay arbitrarily slowly(in anyd 1). Aquestion remains whether there aresome robust (e.g.,
≥
moment)conditions thatwouldimplya“standard” diffusivedecayregardlessofcorrelations.
(5)Aninteresting questioniswhatstrategydominatestheevent X =0=X whenanoma-
2n 0
{ }
lousdecayoccurs. Inparticular, doesthewalkvisitatmostonetrapduring itsrunorasequence
oftraps(withvarious “strengths” oftrapping), etc? Updateinrevisedversion: Thisquestion has
alsobeenresolvedintheaforementioned preprint[12];thepathdoesspendamajorityofitstime
inafewverylocalized places.
(6) It would be interesting to see whether and how the subdiffusivity of the random walk
among random conductances manifests itself in the behavior of its loop-erasure. Here we note
that, for instance, in d =2 the scaling limit of the loop-erasure of the simple random walk on
the supercritical percolation cluster coincides with that for the ordinary simple random walk
— namely SLE (Yadin and Yehudayoff [35]). The question is thus whether one can extend
2
this remarkable result to other dimensions (obviously, with a different scaling limit) and other
conductance laws.
The plan of the remainder of this paper is as follows. In the next section wecollect the ideas
entering the proof and structure the main steps into proper lemmas and propositions. The proof
of the main theorem then can be given subject to a Key Lemma (Lemma 2.6) that controls the
number of traps the chain typically sees along its path. The Key Lemma is then subsequently
reducedtomomentboundsonthecorresponding numberforacoarse-grained chain;thisisdone
in Section 3. In Section 4 we then invoke certain technical facts about the heat kernel for the
coarse-grained chain, and also thetrap density, tojustify these moment bounds. These technical
factsarethenprovedinSection5(trapdensitybounds) andSections6-7(heat-kernel estimates).
HEAT-KERNELDECAYINRANDOMCONDUCTANCEMODELS 5
2. KEY STEPS OF THE PROOF
A distinguished feature of the four-dimensional problem, and the reason why the heat-kernel
anomaly is manifested only by logarithmic corrections, is that the leading contribution to return
probability may come from a whole range of spatial scales. Anticipating some form of scale
invariance, wepartition Zd intoasequence of(disjoint) annuli
Bk:= x Zd: 2k−1 1< x¥ <2k , k 0. (2.1)
∈ − | | ≥
Let B denote the cardinality(cid:8)of B . An opening step of the(cid:9)proof is the following version of a
k k
| |
standard (deterministic) Cauchy-Schwarzestimate.
Lemma2.1 Suppose that0<w 1forallb. Then
b
≤
P2wn(0,0) p w (0) (cid:229) Pw0(Xn∈Bk)2. (2.2)
≥ 2d B
k 0 | k|
≥
Proof. BytheMarkovproperty andreversibility
Pw2n(0,0)= (cid:229) Pwn(0,x)Pwn(x,0)
x Zd
= (cid:229)∈ Pnw (0,x)2 pp ww ((0x)). (2.3)
x Zd
∈
Boundingp w (x) 2d andusingthat Bk k 1 formapartition ofZd,weget
≤ { } ≥
P2wn(0,0) p w (0) (cid:229) (cid:229) Pnw (0,x)2. (2.4)
≥ 2d
k 0x Bk
≥ ∈
ByCauchy-Schwarz, thesumoverxexceeds Bk −1Pw0(Xn Bk)2. (cid:3)
| | ∈
In order to motivate our next step, we recall a classic argument (cf, e.g., [13, Remark 2.2])
that shows how the CLT implies (1.8). Indeed, for k := 1log (n) and n 1 we have that
⌊2 2 ⌋ ≫
diam(B ) √nandsothequenched CLTgives
k
∼
Pw0(Xn Bk) C1(w )>0, P-a.s. (2.5)
∈ ≥
Retainingonlythecorresponding terminthesum,from B C nd/2 weget
k ′
| |≤
C (w ) C (w )
P2wn(0,0) 2 = 2 , (2.6)
≥ nd/2 d=4 n2
whereC >0a.s. Thisisthestandard diffusivedecay. Thekeyideaunderlying ourworkisthat,
2
in d =4 there are environments for which order-logn other k’s contribute a comparable amount
to(2.2)—thusproducing alognmultiplicative term.
In order to state the requisite lower bound on Pw0(Xn Bk) in dimension-independent form,
∈
consider theabbreviation
t :=22k, k 0, (2.7)
k
≥
andnotethatthisisthediffusivetimescaleassociated withthespatialscaleofB .
k
6 M.BISKUPANDO.BOUKHADRA
Proposition 2.2 Letd 4andconsider ani.i.d.lawPsatisfying P(0<w 1)=1forwhich
b
thereexistsasequence n≥ ¥ suchthatthequantity ≤
ℓ
→
r n:=P(w b 1/2)P(1/n w b 2/n)4d−2 (2.8)
≥ ≤ ≤
obeys
r logn ¥ . (2.9)
nℓ ℓ −ℓ−−→¥
TherearerandomvariablesC =C (w )andN =→N (w ),withC (w )>0andN (w )<¥ P-a.s.,
1 1 1 1 1 1
suchthatforalln n withn N (w )andallk 1satisfying
ℓ ℓ 1 1
∈{ } ≥ ≥ ≥
n
e(loglogn)2 t (2.10)
k
≤ ≤ logn
wehave
t
Pw0(Xn Bk) C1(w )r n k. (2.11)
∈ ≥ n
Remark2.3 Notethatthefactthatr issummableonn—whichisseen, e.g.,fromthebound
n
r n P(1/n w b 2/n) — forces us to work with subsequences in (2.9). On the other hand,
the≤requirem≤ent o≤fsubpolynomial decay ofr isconvenient, albeit perhaps unnecessary, for our
n
proofs. (Specifically, this assumption is used in Lemmas 4.2 and 4.5 and also in the proof of
(3.17) from Lemmas 4.4–4.5.) All of these steps will need to be reevaluated when studying the
questionforwhattailsdoestheanomalystarttooccurind=4. (Ind 5,thisquestionhasbeen
≥
addressed byBoukhadra[14,15].)
Proposition 2.2permits ustofinishtheproof ofourmainresult. Itisimportant tonotethatin
(and only in) d =4 we have that t2 B which puts all terms in the sum in (2.2) on the same
k ∼| k|
orderofmagnitude.
Proof of Theorem 1.1 from Proposition 2.2. Abbreviate q :=1/[2(4d 2)] and suppose d =4.
Wemayassumewithoutlossofgenerality thatl tendstoinfinitysosl−owlythat
n
l n−1/2logn−n−−→¥ ¥ . (2.12)
→
Let n beanincreasing sequence ofintegerswithn >1andsuchthat
ℓ ℓ 1 ℓ
{ } ≥
(cid:229) l q 1. (2.13)
n−ℓ ≤ 2
ℓ 1
≥
Wethen define the environment law P to be an i.i.d. measure whose one-dimensional marginals
areconcentrated on 1 n 1 withprobabilities
{ }∪{ −ℓ }ℓ≥1
P(w =n 1):=l q , ℓ 1, (2.14)
b −ℓ n−ℓ ≥
and
P(w =1):=1 (cid:229) l q . (2.15)
b n−
− ℓ
ℓ 1
≥
Notethat,forthisenvironment,
1
r 2 P(w =1)2P(w =n 1)2(4d 2) l 1, ℓ 1, (2.16)
nℓ ≥ b b −ℓ − ≥ 4 n−ℓ ≥
andso(2.9)isimpliedforthesubsequence n by(2.12)and(2.16).
ℓ
{ }
HEAT-KERNELDECAYINRANDOMCONDUCTANCEMODELS 7
Now pick n n with n N (w ), where N as in Proposition 2.2, and introduce the
ℓ ℓ 1 1 1
shorthand Z(n)∈:={ k} ≥N: e(loglo≥gn)2 t n/log(n) . Lemma2.1andProposition 2.2imply
k
{ ∈ ≤ ≤ }
Pw2n(0,0) p w (0) (cid:229) Pw0(Xn∈Bk)2
≥ 8 B
k Z(n) | k|
∈
p w (0)C (w )2 r n 22 4 Z(n) , (2.17)
1 −
≥ 8 n
(cid:16) (cid:17) (cid:12) (cid:12)
where we used that tk2/|Bk|≥2−4. For n≫1, we have |Z(n)|(cid:12)≥ 12log(cid:12)4n≥ 41logn. Hence, for
n n sufficientlylarge,
ℓ ℓ 1
∈{ } ≥
P2wn(0,0)≥ p w (05)C121(w )2 llognn2. (2.18)
n
ThisandthefactthatP2wn(0,0)>0foralln 1implytheclaim. (cid:3)
≥
It remains to construct the proof of Proposition 2.2. As in the examples showing anomalous
decay in d 5, a mechanism that could make Pw0(Xn Bk) large even when n tk (which is
≥ ∈ ≫
outsidethecentral-limitingscaling)istoletthewalkfallintoatrap. Inanalogywith[10,14,15],
weadoptthefollowing(somewhatarbitrary) definition:
Definition 2.4 Atrapatscalen isanedgeb=(y,z) suchthatw b 1/2 andsuchthatforany
≥
edgeb =bincidentwitheitheryorz,
′
6
1 2
w . (2.19)
b
n ≤ ′ ≤ n
LetA (x)betheeventonthespaceofenvironments thatxisavertexneighboring atrapedge
n
atscalen. Letusabbreviate
B◦k := x Zd: 2k−1+2< x¥ <2k 3 (2.20)
∈ | | −
andnotethatB B and,infact,(cid:8)dist(B ,Bc) 3. Inparticular, ifA(cid:9) (x)occursforx B ,then
◦k ⊂ k ◦k k ≥ n ∈ ◦k
the corresponding trap(s) and the edges incident therewith alllie in B . Theeffect of trapping is
k
captured bythenextestimate:
Lemma2.5 Letd 1. Thereisanabsolute constant c =c (d)>0suchthatforalln,k 1,
1 1
≥ ≥
Pw0(Xn∈Bk)≥ cn1Ew0 n/ℓ(cid:229)=2−011{Xℓ∈B◦k}1An(Xℓ)!. (2.21)
Proof. Forx Zd such that A (x)occurs, let(y,z) be the trap edge that makes A (x) occur. (In
n n
∈
the presence of more such edges next to x, we pick the one that is smallest in a fixed complete
orderonBd.) Weassumethatthisedgeislabeled sothatxandyareneighbors inZd. Forℓ 0,
≥
weuseD (x,ℓ)todenotetheevent
n
n
D (x,ℓ):= X =x X =y X y,z . (2.22)
n ℓ ℓ+1 m
{ }∩{ }∩ ∈{ }
m=ℓ+1
\ (cid:8) (cid:9)
8 M.BISKUPANDO.BOUKHADRA
Firstwenotethat
n/2 1
−
X B D (x,ℓ). (2.23)
n k n
{ ∈ }⊃
An(xx[∈)oBc◦kcurs ℓ[=0
Indeed,onD (x,ℓ)(withxandℓintheunionsabove)thewalkattimenisatoneoftheendpoints
n
ofthetrap,whicharebothinB bytherestriction x B .
k ∈ ◦k
Nextweclaim thattheunions in(2.23)aredisjoint, i.e.,D (x,ℓ) D (x,ℓ)=0/ foranypairs
n n ′ ′
∩
ofindices(x,ℓ)=(x,ℓ)contributingto(2.23). ThisisbecauseonD (x,ℓ),thewalkspendsmore
′ ′ n
6
thanhalfofitstimecrossingasingle(trap)edge—namely,(y,z)in(2.22). Thiswalkmusthave
entered thetrapfromvertexxattimeℓandsoif(x,ℓ)isdistinct from(x,ℓ),itcannot belongto
′ ′
D (x,ℓ). Weconclude
n ′ ′
n/2 1
Pw0(Xn∈Bk)≥ (cid:229) 1An(x) (cid:229) − Pw0 Dn(x,ℓ) . (2.24)
x∈B◦k ℓ=0 (cid:0) (cid:1)
TheMarkovproperty andasimplecalculation imply
Pw0 Dn(x,ℓ) Pw0(Xℓ=x) 1 1+4(2d−1) ℓ−n. (2.25)
≥ 2dn n
(cid:18) (cid:19)
(cid:0) (cid:1)
Thelast twoterms are atmost c /n for c :=(2d) 1e 4(2d 1). Once(2.25)is used for all terms
1 1 − − −
in(2.24),thesumscombineintothedesiredexpectation. (cid:3)
TheproofofProposition 2.2isnowreducedtothefollowingKeyLemma:
Lemma 2.6 (Key Lemma) Let d 4. For any i.i.d. law P satisfying P(0<w 1)=1 and
b
(2.9) for a sequence n , there≥are P-a.s. finite and positive random variables≤C :=C (w )
ℓ ℓ 1 2 2
andN =N (w )such{tha}t≥foralln n withn N andallkobeying (2.10)wehave
1 1 ℓ ℓ 1 1
∈{ } ≥ ≥
n/2 1
Ew0 ℓ(cid:229)=−0 1{Xℓ∈B◦k}1An(Xℓ)!≥C2(w )r ntk. (2.26)
ProofofProposition2.2fromKeyLemma. Theconditionsonkandnareidentical,andcombining
(2.26)with(2.21)weget(2.11)withC (w ):=c C (w ). (cid:3)
1 1 2
3. PROOF OF KEY LEMMA
Our proof of the Key Lemmawill require introduction of some technical tools that we will first
trytomotivatebygivingaheuristic argumentwhy(2.26)shouldholdtrue.
Recall the notation t from (2.7). By reducing the sum in (2.26) to t ℓ 2t — which is
k k k
≤ ≤
allowed because t n by the assumptions (2.10) — the expectation in (2.26) pertains to paths
k
≪
of the random walkon temporal scalet and spatial scale √t . This is adiffusive scaling so one
k k
mightexpectthatthelawofX willbealready closetothestationary distribution, andthusmore
ℓ
orlessuniformly distributed, overB . Theexpectation ofeachterminthe(reduced) sumshould
◦k
HEAT-KERNELDECAYINRANDOMCONDUCTANCEMODELS 9
therefore bebounded belowbyaconstanttimesP(A (0)). As
n
P(A (0)) r , (3.1)
n n
≥
andasthereareordert termsinthe(reduced) sum,thiswouldyield(2.26).
k
Afundamentalproblemwiththisreasoningisthat,duetothepresenceofveryweakbonds,the
lawofX inB fort ℓ 2t willnotbeclosetothestationarydistributionattherequiredlevel.
ℓ ◦k k≤ ≤ k
After all, the sole purpose of this note is to demonstrate the failure of a local-CLT scaling! As
in [13, 10, 30, 14, 15], we will circumvent this problem by observing the walk only on a strong
component; namely, theconnected component ofedgesbwithw a forsomesmallenough a
b
≥
to bechosen momentarily. This walk already has good mixing properties but, unfortunately, the
reductionoftheexpectationin(2.26)tothiswalkinvolvesatimechangethatwillnowneedtobe
controlled aswell. Andasthishappens onthebackground ofanexpectation ofa(large) random
variable, wewillhavetocontrolmomentsofthisrandom variableaswell.
We now begin to formulate the aforementioned technical aspects precisely. Following up on
earlierwork[13,10,14,15],wewillintroduceacutoffa andexaminetheconnectivityproperties
of the graph Gw with vertices Zd and edges b: w b a . The key facts we will need are as
{ ≥ }
follows:
Proposition 3.1 Assume d 2. Then there is p = p (d) (0,1) such that whenever P(w
0 0 b
a ) p ,thenthefollowing h≥oldsP-a.s.: ∈ ≥
0
≥
(1) ThegraphGw contains auniqueinfiniteconnected componentC¥ ,a =C¥ ,a (w ).
(2) ThecomplementZd C¥ ,a hasonlyfiniteconnected components.
\
IfFx denotestheconnectedcomponentofZd C¥ ,a containingx(withFx=0/ forx C¥ ,a )and
distw (x,y)istheshortest-path distance measu\redonC¥ ,a thenalso: ∈
(3) Almostsurelyon 0 C¥ ,a ,
{ ∈ }
limsupdistw (0,x) <¥ . (3.2)
x
x ¥
| |→ | |
(4) If diamw (Fx) denotes the maximum of distw (y,z) over all pairs of Zd-neighbors y,z
C¥ ,a ofFx,thendiamw (F0)hasallmoments. (Naturally,diamw (0/)=0.) ∈
Finally,letG denotetheunionofF foryrunningthrough neighbors ofxinZd. LetG denote
x y ′w
thegraphobtainedfromC¥ ,a byaddinganedgebetweenanyy,z C¥ ,a withGy Gz=0/ andlet
∈ ∩ 6
d′w (x,y)denotethegraph-theoretical distance measuredonG′w . Then:
(5) Forsomex >0,
1
limsup logP 0,x C¥ ,a &d′w (0,x) x x <0. (3.3)
x ¥ x ∈ ≤ | |
| |→ | | (cid:16) (cid:17)
Hereandhenceforth, x denotes theEuclidean normofx.
| |
Proof(Sketch). Firstnotethatallproperties(2-5)areproperlystochastically monotoneina . (An
exception is the uniqueness of C¥ ,a which holds for all a by an argument of, e.g., Burton and
Keane[16].) Ourproof isbestexplained byrunning acoarse-graining argument: Consider bond
percolation onZd withparameter pandcallaunitcubeof2d vertices inZd occupied ifallofits
edges are occupied. Twocubes are called adjacent ifthey share aside. Bythe result of Liggett,
10 M.BISKUPANDO.BOUKHADRA
StaceyandSchonmann[28],thefactthatthecubesmorethandistanceoneapartareindependent
permits us dominate the process of occupied cubes from below by site percolation on Zd with
a parameter h (p), where h (p) 1 when p 1. In particular, there is p (0,1) such that for
0
↑ ↑ ∈
all p p , the occupied cubes percolate and theremoval of the (a.s. unique) infinite component
0
≥
ofoccupied cubesresultsonlyinfinitecomponents whosediametershaveanexponential tail.
Properties (1,2) now follow immediately by standard facts about percolation on Zd while (4)
is the consequence of the fact that diamw (Fx) will be bounded by the number of unit cubes
adjacenttothefinitecomponentofthecube-process(necessarily)containingF . Property(3)isa
x
consequenceofTheorem1.1of[2]whileproperty(5)isarestatementofLemma3.1of[13]. (cid:3)
Nowletusfix p asinProposition 3.1andpicka (0,1)by
0 0
∈
P(w a ) p . (3.4)
b 0 0
≥ ≥
Wewillkeep a fixedthroughout therestofthepaper. Notethattheproperties (1-5) inProposi-
0
tion3.1applytoallcutoffsa (0,a ].
0
Consider now a path of the∈Markov chain X. Forany w with 0 C¥ ,a , wedefine a sequence
∈
T :=0,T ,T ,...via
0 1 2
Tj+1:=inf ℓ>T0+ +Tj :Xℓ C¥ ,a (T0+ +Tj) (3.5)
{ ··· ∈ }− ···
and
Xˆ :=X , ℓ 0. (3.6)
ℓ T1+···+Tℓ ≥
The sequence (Xˆℓ)ℓ 1 records the successive visits of (Xn) to the strong component C¥ ,a . Note
thatwehaveTj <¥ ≥forall j 0,Pw0-a.s. Infact, thereisa(deterministic) momentboundonthe
≥
timethewalkcan“hide”inacomponent ofZd C¥ ,a :
\
Lemma 3.2 (Hidding time estimate) For x Zd, let G = G (w ) be as in Proposition 3.1.
x x
∈
Setc :=4da 1. Thenforallw (0,1]Bd,
2 −
∈
Ewx(T1) c2 Gx . (3.7)
≤ | |
Proof. Thisisarestatement ofLemma3.8from[10]. (cid:3)
Afundamental concept inthestudyofrandom walksinrandom environments isthe“pointof
view of the particle.” The idea is that instead of recording the position of the walk relative to a
given environment, we follow the sequence of environments that the walker sees along its path.
Explicitly, lett denotethe“shiftbyx”whichisformallydefinedby
x
(t w ) :=w , x Zd,(y,z) Bd. (3.8)
x yz y+x,z+x
∈ ∈
Given a trajectory Xˆ =(Xˆ ) of the coarse-grained random walk in environment w with 0
n n 0
C¥ ,a (w ), the sequence (tXˆnw≥)n≥0 is itself a Markov chain on the space of environments wit∈h
stationary measure
Qa ( ):=Q( 0 C¥ ,a ), (3.9)
− −| ∈
where
Q(dw ):= p w (0)P(dw ) for Z :=Ep w (0). (3.10)
Z