Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

81 Pages·2017·1.42 MB·English

by  Marek Biskup

Probab.TheoryRelat.Fields
DOI10.1007/s00440-017-0777-x
Mass concentration and aging in the parabolic
Anderson model with doubly-exponential tails
MarekBiskup1,2 · WolfgangKönig3,4 ·
RenatoS.dosSantos3
Received:4September2016/Revised:19March2017
©Springer-VerlagBerlinHeidelberg2017
Abstract Westudythenon-negativesolutionu =u(x,t)totheCauchyproblemfor
theparabolicequation∂ u =Δu+ξuonZd×[0,∞)withinitialdatau(x,0)=1 (x).
t 0
HereΔisthediscreteLaplacianonZdandξ =(ξ(z))z∈Zd isani.i.d.randomfieldwith
doubly-exponentialuppertails.We(cid:2)provethat,forlarget andwithlargeprobability,
most of the total mass U(t) := u(x,t) of the solution resides in a bounded
x
neighborhood of a site Z that achieves an optimal compromise between the local
t
DirichleteigenvalueoftheAndersonHamiltonianΔ+ξandthedistancetotheorigin.
Theprocessest (cid:4)→ Z andt (cid:4)→ 1logU(t)areshowntoconvergeindistributionunder
t t
suitable scaling of space and time. Aging results for Z , as well as for the solution
t
totheparabolicproblem,arealsoestablished.Theproofusesthecharacterizationof
eigenvalue order statistics for Δ+ξ in large sets recently proved by the first two
authors.
MathematicsSubjectClassification 60H25·82B44
©2017 MarekBiskup,WolfgangKönigandRenatoS.dosSantos.Reproduction,byanymeans,ofthe
entirearticlefornon-commercialpurposesispermittedwithoutcharge.
B
MarekBiskup
[email protected]
1 DepartmentofMathematics,UCLA,LosAngeles,CA,USA
2 CenterforTheoreticalStudy,CharlesUniversity,Prague,CzechRepublic
3 Weierstraß-InstitutfürAngewandteAnalysisundStochastik,Berlin,Germany
4 InstitutfürMathematik,TechnischeUniversitätBerlin,Berlin,Germany
123
M.Biskupetal.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Results:Massconcentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Results:Scalinglimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Results:Aging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Results:Limitprofiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Connectionsandheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Relationstoearlierwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Someheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Mainresultsfromkeypropositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Definitionofthelocalizationprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Propertiesofthecostfunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Massdecompositionandnegligiblecontributions . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Proofofmassconcentrationresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Proofofagingandlimitprofiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Potentialsandeigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Connectivitypropertiesofthepotentialfield . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Spectralbounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Pathexpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Keypropositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Massofthesolutionalongexcursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Equivalenceclassesofpaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 ProofofPropositions6.1–6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Analysisofthecostfunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Apointprocessapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Orderstatistics:ProofofPropositions7.1and4.5andTheorem2.6 . . . . . . . . . . . . . . .
8 Massdecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Lowerboundforthetotalmass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Macroboxtruncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Negligiblecontributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 UpperboundforthetotalmassandproofofTheorem2.5 . . . . . . . . . . . . . . . . . . . .
9 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Pathlocalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Fastapproachtothelocalizationcenter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Localconcentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Localprofiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Contributionofu(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Contributionofu(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Appendix:Atailestimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Appendix:Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Appendix:Propertiesofthecostfunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
RandomSchrödingeroperators—mostnotably,theAndersonHamiltonian H =Δ+
ξ—havebeenasubjectofintenseresearchoverseveraldecades.Mostoftheattention
hasbeenpaidtothecharacterofthespectrumandtheensuingphysicalconsequences
forthequantumevolution.However,theassociatedparabolicproblem—characterized
123
MassconcentrationandagingintheparabolicAnderson…
bythePDE∂ u =Δu+ξu—isofasmuchinterestbothfortheoryandapplications.
t
Herewestudythelatterfacetofthisproblemforaspecificclassofrandompotentials.
OurmainresultistheproofoflocalizationofthesolutiontotheabovePDEforlarge
timeinaneighborhoodofaprocessdeterminedsolelybytherandompotential.
A standard way to describe the parabolic Anderson model (PAM) is via a non-
negativesolutionu: Zd ×[0,∞)→[0,∞)oftheCauchyproblem
∂ u(z,t)=Δu(z,t)+ξ(z)u(z,t), z ∈Zd, t ∈(0,∞), (1.1)
t
u(z,0)=1 (z), z ∈Zd. (1.2)
0
Hereξ =(ξ(z))z∈Zd isani.i.d.randompotentialtakingvaluesin[−∞,∞),1x isthe
indicatorfunctionofapoint x ∈ Zd,∂ abbreviatesthederivativewithrespecttot,
t
andΔisthediscreteLaplacianactingon f : Zd →Ras
(cid:3) (cid:4) (cid:5)
Δf(z):= f(y)− f(z) , (1.3)
y:|y−z|=1
where|·|denotesthe(cid:5)1normonZd.
The interest in(1.1–1.2) for mathematics as wellas applications comes fromthe
competingeffectofthetwotermsontheright-handsideof(1.1).Indeed,theLaplacian
tendstomakethesolutionsmootherovertime,whilethefieldmakesitrougher.The
problem(1.1)appearsinthestudiesofchemicalkinetics[13],hydrodynamics[8],and
magneticphenomena[23].Werefertothereviews[8,19]formorebackground,and
to[13]forthefundamentalmathematicalpropertiesofthemodel.Arecentcompre-
hensivesurveyofmathematicalresultsonthePAMandrelatedmodelscanbefound
in[15];therelatedspectralorder-statisticsquestionsarereviewedin[3].
A non-negative solution to the Cauchy problem (1.1–1.2) exists and is unique as
soon as the upper tail of [ξ(0)/logξ(0)]d is integrable [13]. Under this condition,
thereisalsoarepresentationintermsofthechanged-pathmeasure,
(cid:6)(cid:7) (cid:8)
1 t
Q(ξ)(dX):= exp ξ(X )ds P (dX), (1.4)
t U(t) s 0
0
on nearest-neighbor paths X = (Xs)s≥0 on Zd, where P0 stands for the law of a
continuous-time random walk on Zd (with generator Δ) started at zero. Indeed, the
Feynman–Kacformulashows
(cid:9) (cid:11)
(cid:10)
u(z,t)=U(t)Q(tξ)(Xt =z)=E0 e 0tξ(Xs)ds1{Xt=z} , (1.5)
wherebythenormalizationconstantU(t)obtainsthemeaning
(cid:12) (cid:7) (cid:13)
(cid:3) t
U(t)= u(x,t)=E exp ξ(X )ds . (1.6)
0 s
x∈Zd 0
The aforementioned competitio(cid:10)n is now obvious probabilistically: the walk would
liketomaximizethe“energy” tξ(X )ds,byspendingitstimeatthesiteswhereξ
0 s
islarge,againstthe“entropy”ofsuchtrajectoriesunderthepathmeasureP .
0
123
M.Biskupetal.
Analternativeandequallyusefulwaytoview(1.1)isasthedefinitionofasemigroup
t (cid:4)→et(Δ+ξ)on(cid:5)2(Zd).Thesolutionto(1.1–1.2)isthengivenby
(cid:14) (cid:15)
u(x,t)= 1 ,et(Δ+ξ)1 . (1.7)
x 0 (cid:5)2(Zd)
Thisopensupthepossibilitytocontrolthelarge-t behaviorthroughspectralanalysis
of the Anderson Hamiltonian. To this end, it is useful to restrict the problem to a
sufficiently large (in t-dependent fashion) finite volume Λ ⊂ Zd (with 0 ∈ Λ) as
follows.DenotebyHΛtheAndersonHamiltonianinΛwith(zero)Dirichletboundary
conditions,i.e.,forφ ∈RΛ,HΛφ = Hφ˜whereH =Δ+ξandφ˜istheextensionofφ
toRZd thatisequaltozeroonΛc.LetuΛbethesolutionto(1.1–1.2)restrictedtoΛ
andwiththeright-handsideof(1.1)substitutedbyHΛu.Thentheaboveinterpretation
yields
(cid:3)|Λ|
uΛ(x,t)= etλ(Λk)φΛ(k)(x)φΛ(k)(0), (1.8)
k=1
whereλ(Λk) aretheeigenvaluesandφΛ(k) thecorrespondingeigenvectorsof HΛ,which
weassumetobeorthonormalin(cid:5)2(Λ).Hereafter,weextendboththesolutionuΛ(·,t)
andtheeigenfunctionsof HΛtoZd bysettingthemtobeequalto0onΛc.
The competition we described in the context of the changed-path measure (1.4)
nowmanifestsitselfasfollows.Theterminthesumin(1.8)thatgrowsthefastestint
isthatwiththelargesteigenvalue.However,thereisnoapriorireasonforittobethe
dominanttermatafixedtime.Indeed,aneigenvaluewillonlycontributeto(1.8)when
itseigenvectorputsnon-trivialmassonboth0andx.Sincetheleadingeigenvectors
decay exponentially away from their localization centers (Anderson localization),
|φ(k)(0)|willinfactbetypicallyextremelysmall.Itisthusthecombinedeffectofboth
Λ
etλ(Λk) andφΛ(k)(x)φΛ(k)(0)thatdecideswhichindexk willgivethemaincontributionto
thesum.
In the present paper, we analyze these competing effects for a class of random
potentialswithuppertailsclosetothedoubly-exponentialdistribution,characterized
by (cid:16) (cid:17) (cid:18) (cid:19)
Prob ξ(0)>r =exp −er/ρ , r ∈R, (1.9)
whereρ ∈(0,∞).(PrecisedefinitionswillappearinSect.2.)Forthesepotentialswe
showthat,atalllarget,mostofthetotalmassU(t)ofthesolutionresidesinabounded
neighborhood of a random point Z determined entirely by ξ. This point marks the
t
optimal local peak of ξ for the strategy where the random walk in (1.4) traverses
to Z intimeo(t),andthereafter“sticksaround” Z inordertoenjoythebenefitsofa
t t
“strong”localDirichleteigenvalue.Wealsocharacterizethescalinglimitsof Z and
t
1logU(t),andobtainagingresultsforboth Z andu(x,t).
t t
OurresultsbuildonalargebodyofliteratureonthePAMwhosefullaccounthere
woulddivertfromthemainmessageofthepaper.Fornowletusjustsaythatweextend
resultsfrom[9,17,21,26],dealingwithlocalizationononelatticesite,toabenchmark
classofrandompotentialsexemplifiedby(1.9),wherethelocalizationtakesplacein
large domains, albeit not growing with t. An important technical input for us is the
123
MassconcentrationandagingintheparabolicAnderson…
recentwork[7],whereeigenvalueorderstatisticsfortheAndersonHamiltonianH =
Δ + ξ was characterized for this class of ξ. Further connections will be given in
Sect.3.1.
2 Mainresults
Wenowmovetothestatementsofourmainresults.Throughoutthepaper,lnxdenotes
the natural logarithm of x, and ln x := lnlnx, ln x := lnlnlnx, etc denote its
2 3
iterates.Wewilluse“Prob”todenotetheprobabilitylawofthei.i.d.randomfieldξ.
2.1 Assumptions
We begin by identifying the class of potentials that we will consider in the sequel.
Besidessomeregularity,thefollowingensuresthattheuppertailsofξ(0)areinthe
vicinityofthedoubly-exponentialdistribution(1.9).
Assumption2.1 (Uppertails)Supposethatesssupξ(0)=∞andlet
1
F(r):=ln , r >essinfξ(0). (2.1)
2 Prob(ξ(0)>r)
Weassumethat F isdifferentiableonitsdomainandthat
1
lim F(cid:8)(r)= forsome ρ ∈(0,∞). (2.2)
r→∞ ρ
Theassumptionabove isexactly asAssumption1.1in[7],and impliesAssump-
tion (F) of [14]. While the latter would be enough for most of our needs, the extra
requirements of Assumption 2.1 are used in the crucial step, performed in [7], of
identifying the max-order class of the local principal eigenvalues of the Anderson
Hamiltonian. In order to avoid technical inconveniences, we will also assume the
followingconditiononthelowertailofξ.
Assumption2.2 (Lowertails)Letξ−(x):=max{0,−ξ(x)}.Weassumethat
(cid:7)
∞ (cid:16) (cid:17)
Prob ξ−(0)>es d1ds <∞. (2.3)
0
Assumption2.2isonlyusedintheproofofLemma8.1,whichisusedinProposi-
tion4.6togivealowerboundforthetotalmassU(t).Notethat(2.3)holdswhenever
ln(1+ξ−(0))hasa(d+ε)-thfinitemoment(cf.[18]).Webelievethat,withtheuseof
percolationarguments,thisassumptioncanberelaxedtoξ(0) > −∞almostsurely
ind ≥2.Ind =1,(2.3)isequivalenttoln(1+ξ−(0))havingthefirstmoment,which
isknowninthecaseofboundedpotentialstobe“essentiallynecessary”inthesense
that,when|ln(1+ξ−(0))|δ isnotintegrableforsomeδ ∈ (0,1),thesolutionmight
scaledifferently.See[6],inparticularRemarks3and4therein.
123
M.Biskupetal.
WewillassumethevalidityofAssumptions2.1–2.2throughouttherestofthepaper
withoutexplicitlystatingthisineachinstance.
2.2 Results:Massconcentration
Recallthat|x|denotesthe(cid:5)1-normof x.Ourfirstresultconcernstheconcentration
ofthetotalmassofthesolutiontotheCauchyproblem(1.1–1.2):
Theorem2.3 (Mass concentration) There is a Zd-valued càdlàg stochastic process
(Zt)t>0 dependingonlyonξ suchthatt (cid:4)→|Zt|isnon-decreasingandsuchthatthe
followingholds:Foreachδ >0,thereexistsR ∈Nsuchthat,foranyl >0satisfying
t
limt→∞ 1tlt =0,
⎛ ⎞
lim Prob ⎝ sup (cid:3) u(x,s) >δ⎠=0. (2.4)
t→∞ s∈[t−lt,t+lt] x: |x−Zt|>R U(s)
Inwords,(2.4)meansthatthesolutionattimet iswithlargeprobabilityconcentrated
nearasinglepointZ ,andthecontrolinfactextendstosublinearly-growingintervals
t
of time around t. This cannot be extended to linearly growing time-intervals due to
the jumps of the process s (cid:4)→ Z (cf. Theorem 2.6 below), but a refinement of our
s
methodswouldshowthat,inthiscase,twoislandswouldsuffice,i.e.,(2.4)wouldstill
holdifthesumistakenoverboxesofradius R centeredaroundtwoprocesses Z(1),
s
Z(2)[see(4.9)].Wealsobelievethatthealmost-sureversionofthisstatement,dubbed
s
asa“two-citiestheorem”andprovedin[16]forthecaseofParetopotentials,could
beobtainedwithmoreworkbutprefernottopursuethishere.
IntermsofthepathmeasureQ(ξ),Theorem2.3canbeinterpretedasconcentration
t
forthelawofthepositionofthepathattimet.Bylettingtheradius Rgrowslowlyto
infinity,thiscanbeimprovedtoincludeamajorityoftherandomwalkpath:
Theorem2.4 (Pathlocalization)Foranyεt ∈(0,1)withlimt→∞εtln3t =∞,
(cid:24) (cid:25)
lim Q(ξ) sup |X −Z |>ε lnt =0 inprobability, (2.5)
t→∞ t s∈[εtt,t] s t t
where(Zt)t>0 isthestochasticprocessinTheorem2.3.
To the best of our knowledge, statements about path localization such as Theo-
rem2.4werenotyetavailableintheliteratureoftheParabolicAndersonModel.The
scalesabovecomeoutofourmethodsandmaybeartificial;inparticular,wedonot
knowiflnt/ln3t isthecorrectscalingforsupεtt≤s≤t|Xs −Zt|.
2.3 Results:Scalinglimit
Ournexttheoremidentifiesthelarge-t behaviorofthepairofprocessest (cid:4)→ Z and
t
t (cid:4)→ 1lnU(t).WhileU(t)iscontinuous, Z isonlycàdlàgandthusitisnaturalto
t t
123
MassconcentrationandagingintheparabolicAnderson…
usetheSkorohodtopologytodiscussdistributionalconvergence.Tworelevantscales
are
ρ td ρ t
d := and r := t = , (2.6)
t t
dlnt ln t dlnt ln t
3 3
markingthesizeoffluctuationsof 1lnU(t),andthetypicalsizeof|Z |.
t t
Todescribethescalinglimit,considerasample{(λ ,z ): i ∈N}fromthePoisson
i i
pointprocessonR×Rd withintensitymeasuree−λdλ⊗dz.Forθ >0,define
|z|
ψθ(λ,z):=λ− , (λ,z)∈R×Rd. (2.7)
θ
Itcanbechecked that,foreveryθ > 0,theset{ψθ(λi,zi): i ∈ N}isbounded and
locallyfinite.Moreover,themaximizingpointisuniqueatallbutatmostacountable
setofθ’sandwecanthusdefine(Λθ,Zθ)tobethecàdlàgmaximizerofψθ overthe
samplepointsoftheprocess(cf.Sect.7.2).Weset
Ψθ :=ψθ(Λθ,Zθ). (2.8)
Thenwehave:
Theorem2.5 (Scalinglimitofthelocalizationprocessandthetotalmass)Thereisa
non-decreasingscalefunctiona >0obeying
t
a
lim t =ρ (2.9)
t→∞ln2t
suchthatthefollowingholds:Thestochasticprocess(Zt)t>0inTheorems2.3and2.4
canbechosensuchthat,foralls ∈(0,∞)andrelativetotheSkorohodtopologyon
D([s,∞),R×Rd),
(cid:24) (cid:25)
θ1t lnU(dθtt)−art, Zrθtt θ∈[s,∞) t−→la→w∞ (cid:16)Ψθ,Zθ(cid:17)θ∈[s,∞). (2.10)
Inparticular,foreachθ >0,thepair([θ1t lnU(θt)−art]/dt,Zθt/rt)convergesinlaw
tothepair(Ψθ,Zθ)∈R×Rd whosecoordinatesareindependentanddistributedas
follows:Ψθ followsaGumbeldistributionwithscale1andlocationdln(2θ),while
Zθ has i.i.d. coordinates, each of which is Laplace-distributed with location 0 and
scaleθ.
Thescalingfunctiona characterizestheleading-orderscaleoftheprincipalDirich-
t
leteigenvalueoftheAndersonHamiltonianinaboxofradiust,asidentifiedin[7].
See(7.3)belowforaprecisedefinition.
123
M.Biskupetal.
2.4 Results:Aging
Thetechniquesusedtoprovetheabovetheoremsalsopermitustoaddressthephe-
nomenon of aging in the problem under consideration. The term “aging” usually
referstothefactthatcertaindecisivechangesinthesystemoccurattimescalesthat
increase proportionally to the age of the system. Our next result addresses aging in
theprocess(Zt)t>0:
Theorem2.6 (Aging for the localization process) For each s > 0, and for (Zt)t>0
and(Zt)t>0 asinTheorems2.3,2.4and2.5,
(cid:16) (cid:17) (cid:16) (cid:17)
tl→im∞Prob Zt+θt = Zt ∀θ ∈[0,s] =tl→im∞(cid:16)Prob Zt+st =(cid:17) Zt (2.11)
=Prob Z1+s = Z1 =Prob(Θ >s),
wheretherandomvariable
Θ :=inf{θ >0: Z1+θ (cid:12)= Z1} (2.12)
ispositiveandfinitealmostsurely.Moreover,
sd dd
lim Prob(Θ >s)= . (2.13)
s→∞(logs)d d!
InlightofTheorem2.5,Theorem2.6canbeseenasareflectionofthefactthatthe
functionalconvergencestatedinTheorem2.5isnotachievedthroughalargenumber
ofmicroscopicjumps,butratherthroughsporadicmacroscopicjumps.
Oursecondagingresultdealswiththejumpsintheprofileofthenormalizedsolution
u(·,t)/U(t).Itcomesasaconsequenceofthemassconcentrationofthenormalized
solutionaround Z togetherwithTheorem2.6.
t
Theorem2.7 (Agingforthesolution)Foranyε ∈(0,1),therandomvariable
⎧ ⎫
1t inf⎨⎩s >0: (cid:3) (cid:29)(cid:29)(cid:29)(cid:29)uU(x(,tt++ss)) − uU(x(,t)t)(cid:29)(cid:29)(cid:29)(cid:29)>ε⎬⎭ (2.14)
x∈Zd
convergesindistributionast →∞totherandomvariableΘ definedin(2.12).
AkeypointtonoteaboutTheorem2.7isthatthelimitingrandomvariabledoesnot
dependonε.Thissuggeststhat,infact,thesumin(2.14)jumpsfromvaluesnear0
tovaluesnear1ass variesinatimeintervalofsublinearlengthint.
2.5 Results:Limitprofiles
ThelocalizationstatedinTheorem2.3canbegiveninamorepreciseformprovided
thatwemakeanadditionaluniquenessassumption.Inordertostatethisassumption,
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MassconcentrationandagingintheparabolicAnderson…
weneedfurtherdefinitions.GivenapotentialV: Zd →R,let
(cid:3)
V(x)
L(V):= e ρ . (2.15)
x∈Zd
ThefunctionalLplaystheroleofalargedeviationratefunctionforrandompotentialsξ
withdoubly-exponentialtails.WheneverL(V)<∞(infact,wheneverV(x)→−∞
as|x|→∞),Δ+V hasacompactresolventasanoperatoron(cid:5)2(Zd),anditslargest
eigenvalueλ(1)(V)iswell-definedandsimple.Theconstant
(cid:18) (cid:19)
χ =χ(ρ):=−sup λ(1)(V): V ∈RZd, L(V)≤1 ∈ [0,2d] (2.16)
iskeyintheanalysisoftheasymptoticgrowthofU(t).Thesetofcenteredmaximizers
! "
M∗ρ := V ∈RZd: 0∈argmax(V),L(V)≤1 and λ(1)(V)=−χ (2.17)
isknowntobenon-empty.Theassumptionbelowdealswithuniqueness:
Assumption2.8 (Uniqueness of maximizer) We assume that M∗ρ = {Vρ}, i.e., the
variationalproblem(2.16)admitsauniquecenteredsolutionVρ.
Theuniquenessofthecenteredminimizerisconjecturedtoholdforallρ >0,but
hassofaronlybeenprovedforρ largeenough;see[11].Inthelatterpaperitisalso
shownthat,foranyV ∈M∗ρ,thenon-negativeprincipaleigenfunctionoftheoperator
Δ+V isstrictlypositiveandliesin(cid:5)1(Zd).UnderAssumption(2.8),wewilldenote
henceforthbyvρ theprincipaleigenfunctionofΔ+Vρ,normalizedsothat
vρ >0 and (cid:14)vρ(cid:14)(cid:5)1(Zd) =1. (2.18)
Thenwehave:
Theorem2.9 (Limiting profiles) Suppose Assumption 2.8 and let (Zt)t>0 be the
process from Theorems 2.3, 2.4 and 2.5. There exist μ ∈ N and#a > 0 satisfy-
t t
inglimt→∞μt =∞andlimt→∞#at/(ρln2t)=1suchthat,forallε ∈(0,1),
(cid:29) (cid:29)
s∈[εstu,εp−1t] x∈Zds:u|px|≤μt(cid:29)ξ(x +Zs)−#at −Vρ(x)(cid:29) t−→→∞ 0 inprobability. (2.19)
Moreover,foranylt >0satisfyinglimt→∞ 1tlt =0,
(cid:29) (cid:29)
sup (cid:3) (cid:29)(cid:29)(cid:29)u(Zt +x,s) −vρ(x)(cid:29)(cid:29)(cid:29) −→ 0 inprobability. (2.20)
s∈[t−lt,t+lt]x∈Zd U(s) t→∞
Thescale#a in(2.19)coincides(uptotermsthatvanishast →∞)withthemaximum
t
ofξinsideaboxofradiust[see(5.1)forthedefinition,andalsoLemma5.1].Moreover,
the scales at and#at (with at as in Theorem 2.5) satisfy limt→∞#at −at = χ. The
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M.Biskupetal.
scaleμ providedintheproofofTheorem2.9satisfiesμ (cid:15)(lnt)κforsomearbitrary
t t
κ <1/d,butitsactualrateofgrowthisnotcontrolledexplicitly.
Therestofthepaperisorganizedasfollows.InSect.3belowwediscussconnec-
tions to the literature and provide some heuristics. Section 4 contains an extensive
overview of our proofs including the definition of the localization process Z . The
t
technicalcoreofthepaperisformedbySect.5(propertiesofthepotentialandspec-
tral bounds), Sect. 6 (path expansions) and Sect. 7 (a point process approach). The
bulkoftheproofsrelatedtoourmainresultsiscarriedoutinSects.8–11,concern-
ingrespectivelynegligiblecontributionstotheFeynman–Kacformula,localizationof
relevanteigenfunctions,pathlocalizationpropertiesandtheanalysisoflocalprofiles.
TheproofsofsometechnicalresultsaregiveninAppendices12–14.
3 Connectionsandheuristics
Inthissection,wemakeconnectionstoearlierworkonthisproblem,andalsoprovide
ashortheuristicargumentmotivatingthedefinitionofthescalesin(2.6).
3.1 Relationstoearlierwork
Letusgiveaquicksurveyonearlierworksontheparticularquestionthatweconsider;
wereferto[15]foracomprehensiveaccountontheparabolicAndersonmodel,and
to[20]forasurveyoncertainaspectscloselyrelatedtothepresentpaper.
Since1990,muchoftheeffortwentintodevelopingacharacterizationofthelog-
arithmicasymptotics oft (cid:4)→ U(t)anditsmoments,whichareallfiniteifandonly
ifallthepositiveexponentialmomentsofξ(0)arefinite.Forthiscase,underamild
regularityassumption,[27]identifiedfouruniversalityclassesofasymptoticbehav-
iors: potentials with tails heavier than (1.9) (corresponding formally to ρ = ∞),
double-exponentialtailsoftheform(1.9),theso-called“almostbounded”potentials
(correspondingformallytoρ =0),andboundedpotentials.Thefirsttwocaseswere
treatedin[14],andthelasttwoin[27]and[5],respectively.Potentialswithinfinite
exponentialmomentswereanalysedin[28](moreprecisely,ParetoandWeibulltails),
whereweaklimitsandalmostsureasymptoticsforU(t)wereobtained.
Inalloftheclassesmentionedabove,theasymptoticsofU(t)isexpressedinterms
of a variational principle for the local time of the path in Q(ξ) and/or the “profile”
t
of ξ that maximizes a local eigenvalue. The picture that emerges is that a typical
path sampled from Q(ξ) for t large will spend an overwhelming majority of time in
t
a relatively small volume whose location is characterized by a favourable value of
thelocalDirichleteigenvalue.Proofsofsuchstatementshavefirstbeenavailablefor
a related version of the model using the method of enlargement of obstacles [25]
andlateralsoforthedouble-exponentialclassbyprobabilisticpathexpansions[12].
However, neither of these approaches was sharp enough to distinguish among the
many “favourable eigenvalues.” In fact, while the expectation was that only a finite
numberofsucheigenvaluesneedstobeconsidered,thebestavailableboundontheir
numberwasto(1).
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