Invariance principle for the random conductance model with unbounded conductances

0.42 MB·English

by  M. T. Barlow

TheAnnalsofProbability
2010,Vol.38,No.1,234–276
DOI:10.1214/09-AOP481
(cid:13)c InstituteofMathematicalStatistics,2010
INVARIANCE PRINCIPLE FOR THE RANDOM CONDUCTANCE
MODEL WITH UNBOUNDED CONDUCTANCES
0
1
0 By M. T. Barlow1 and J.-D. Deuschel2
2
University of British Columbia and Technische Universit¨at Berlin

n
a WestudyacontinuoustimerandomwalkX inanenvironmentof
J i.i.d.randomconductancesµ ∈[1,∞).Weobtainheatkernelbounds
e
6 and prove a quenched invariance principle for X. This holds even
2 when Eµ =∞.
e

]
R 1. Introduction. We consider the Euclidean lattice Zd with d 2. Let
P ≥
E ,thesetofnonorientednearestneighbourbonds,and,writinge= x,y
d
h. E ,let(µ ,e E )benonnegativer.v.,definedonaprobabilityspace{(Ω,}P)∈.
d e d
at Throughout ∈this paper we will assume that (µe) is stationary and ergodic,
m andthatitslawisinvariantundersymmetriesofZd.Wewriteµ =µ =
xy {x,y}
[ µyx, and let µxy=0 if x y. Set
6∼
µ
1 xy
(1.1) µ = µ , P(x,y)= .
v x xy µ
x
2 y
X
0
There are two natural continuous time random walks associated with µ.
7
Both jumpaccording tothetransitions P(x,y).Thefirst(theconstant speed
4
. random walk or CSRW) waits at x for an exponential time with mean 1
1
while the second (the variable speed random walk or VSRW) waits at x for
0
0 an exponential time with mean 1/µ . Write and for their generators,
x C V
1 given by L L
:
v
Xi (1.2) LCf(x)=µ−x1 µxy(f(y)−f(x)),
y
X
r
a
(1.3) f(x)= µ (f(y) f(x)).
V xy
L −
y
X
Received October 2008.
1Supportedin part by NSERC(Canada) and EPSRC (UK).
2Supportedin part by DFG-Forschergruppe718.
AMS 2000 subject classifications. 60K37, 60F17, 82C41.
Key words and phrases. Randomconductancemodel,heatkernel,invarianceprinciple,
ergodic, corrector.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Probability,
2010,Vol. 38, No. 1, 234–276. This reprint differs from the original in pagination
and typographic detail.
1
2 M. T. BARLOW ANDJ.-D. DEUSCHEL
Set
1
(f,g)= µ (f(x) f(y))(g(x) g(y)).
xy
E 2 − −
xX∈ZdyX∈Zd
Let ν =1, x Zd. It is easy to check that if f, g have finite support, then
x
∈
(1.4) (f,g)= g(x) µ (f(y) f(x)),
xy
E − −
x y
X X
and so
(1.5) (f,g)= f,g = f,g .
V ν C µ
E −hL i −hL i
Thus the VSRW is the Markov process associated with the Dirichlet form
( , ( )) on L2(ν) and has stationary measure ν while the CSRW is the
E D E
Markov process associated with the Dirichlet form ( , ( )) on L2(µ) and
E D E
has stationary measure µ.
Let X =(X ,t 0,Px,x Zd) be either the CSRW or the VSRW. Write
t ≥ ω ∈
for its generator, θ for its invariant measure (so either θ=ν or θ=µ) and
L
let
Px(X =y)
(1.6) qω(x,y)= ω t
t θ
y
bethetransitiondensityofX (orheatkernelassociatedwith ).Thismodel,
L
ofareversible(orsymmetric)randomwalkinarandomenvironment,isoften
called the random conductance model or RCM, particularly in the special
case when (µ ) are i.i.d. We are interested in the long-range behavior of
e
X and, in particular, in obtaining heat kernel bounds for qω(x,y) and a
t
quenched or P-a.s. invariance principle for X. When Eµ < , an averaged
e
∞
invariance principle is obtained in [17].
We begin by discussing the case when (µ ) are i.i.d. If µ =0 then X
e e
never jumps across e. So if p =P(µ >0) is less than p (d), the critical
+ e c
probability for bond percolation in Zd, then X is P Px-a.s. confined to
× ω
a finite set. Thus we restrict to the case p >p . A number of different
+ c
authors have studied this model under various restrictions on the support
of µ . If µ 0,1 then this problem reduces to that of a random walk
e e
∈{ }
on (supercritical) percolation clusters (see [1] for heat kernel bounds, and
[10, 29, 34] for quenched invariance principles). More generally it is useful
to consider the following special cases:
Case 0. c−1 µ c for some c 1;
e
≤ ≤ ≥
Case 1. 0 µ 1;
e
≤ ≤
Case 2. 1 µ < .
e
≤ ∞
For case 0, heat kernel bounds follow from the results in [18, 19], and a
quenched invariance principle is proved in [34]. Case 1 is treated in [11, 12,
INVARIANCEPRINCIPLE FOR THE RANDOMCONDUCTANCEMODEL 3
30].(Thepapers[11,12]consideradiscretetimerandomwalk.)Thesepapers
prove an invariance principle, with a strictly positive diffusion constant σ2.
Further, [11] shows that Gaussian heat kernel boundsdo not hold in general
in this case.
Inthispaperwewilllookatcase2.Thereisnotagreatdifferencebetween
the CSRW and VSRW in case 1, but in case 2, and in particular when
Eµ = , the VSRW and CRSW do have different behaviors. Also, while
e
∞
thediscretetimerandomwalkwithjumpprobabilities P(x,y)givenby(1.1)
behaves in a similar fashion to the CSRW, there is no simple discrete time
analogue of the VSRW in case 2. We remark that our result for the CSRW
also gives an invariance principle for the discrete time random walk with
jump probabilities P(x,y).
Let
(ε)
(1.7) X =εX , t 0.
t t/ε2 ≥
Our first main result is the following quenched functional central limit the-
orem (QFCLT):
Theorem 1.1. Let d 2. Suppose that (µ ) are i.i.d., and µ 1 P-a.s.
e e
(a) Let X be the VSRW≥. Then P-a.s. X(ε) converges (under P≥0) in law
ω
to a Brownian motion on Rd with covariance matrix σ2I where σ >0 is
V V
nonrandom.
(b) Let X be the CSRW. Then P-a.s. X(ε) converges (under P0) in law
ω
to a Brownian motion on Rd with covariance matrix σ2I where
C
σ2/(2dEµ ), if Eµ < ,
σC2 = 0,V e if Eµe=∞.
e
(cid:26) ∞
We also have heat kernel bounds for the VSRW.
Theorem 1.2. Let d 2. Suppose that (µ ) are i.i.d. and µ 1 P-a.s.
e e
Let qω(x,y) be the heat ke≥rnel for the VSRW. Let η (0,1). There≥exist r.v.
U ,xt Zd, such that ∈
x
∈
(1.8) P(U (ω) n) c exp( c nη)
x 1 2
≥ ≤ −
and constants c (depending on d and the distribution of µ ) such that the
i e
following hold.
(a) For all x, y, t
(1.9) qω(x,y) c t−d/2.
t ≤ 3
(b) If x y t1/2 U , then
x
| − |∨ ≥
(1.10) qω(x,y) c t−d/2e−c4|x−y|2/t when t x y ,
t ≤ 3 ≥| − |
qω(x,y) c exp( c x y (1 log(x y /t)))
(1.11) t ≤ 3 − 4| − | ∨ | − |
when t x y .
≤| − |
4 M. T. BARLOW ANDJ.-D. DEUSCHEL
(c) Let x,y Zd, t>0. Then
∈
(1.12) qω(x,y) c t−d/2e−c6|x−y|2/t if t U2 x y 1+η.
t ≥ 5 ≥ x ∨| − |
(d) Let x,y Zd and t c x y 1+η. Then
7
∈ ≥ ∨| − |
(1.13) c t−d/2e−c9|x−y|2/t Eqω(x,y) c t−d/2e−c11|x−y|2/t.
8 ≤ t ≤ 10
UsingTheorems1.1and1.2 wecan obtainaparabolicHarnack inequality
(PHI) for qω (for the VSRW) by the same arguments as in [3] (see Theorem
t
4.7).SincetheCSRWisatimechangeoftheVSRW,harmonicfunctionsand
Green’s functions are the same for both processes. The PHI for the VSRW
implies an elliptic Harnack inequality (see Corollary 4.8) which therefore
holds for both CSRW and VSRW. Combining the invariance principle and
the PHI, we obtain, using the methods of [3], a local limit theorem for the
VSRW (see Theorem 5.14).
When d 3 the calculations in Section 6 of [3] then give bounds on the
≥
Green’s function g (x,y) defined by
ω
∞
(1.14) g (x,y)= qω(x,y)dt.
ω t
Z0
Theorem 1.3. Let d 3, and suppose that (µ ) are i.i.d. and µ 1
e e
P-a.s. ≥ ≥
(a) There exist constants c ,...,c and r.v. U , x Zd such that
1 4 x
∈
(1.15) P(U n) c exp( c n1/3)
x 1 2
≥ ≤ −
and
c c
3 4
(1.16) g (x,y) if x y U U .
x y d−2 ≤ ω ≤ x y d−2 | − |≥ x∧ y
| − | | − |
(b) Let C =Γ(d 1)/(2πd/2σ2). For any ε>0 there exists M =M(ε,ω)
with P(M < )=21−such that V
∞
(1 ε)C (1+ε)C
(1.17) − g (0,x) for x >M(ω).
x d−2 ≤ ω ≤ x d−2 | |
| | | |
(c) We have, P-a.s.,
(1.18) lim x 2−dg (0,x)= lim x 2−dEg (0,x)=C.
ω ω
|x|→∞| | |x|→∞| |
Remark 1.4. (b)and(c) inTheorem1.3usetheQFCLT,whichinturn
uses the ergodic theorem. As we do not have any rate of convergence in the
QFCLT this means that [unlike the r.v. U in (a)] we have no control on the
x
tail of the r.v. M in (b).
INVARIANCEPRINCIPLE FOR THE RANDOMCONDUCTANCEMODEL 5
The main difference between the RCM and the percolation case is the
possibility of traps. Suppose e= x,y is a bond with µ =K 1, and that
e
{ } ≫
all the other bonds e′ adjacent to x and y have µ 1. Then P(x,y)
e′
≃ ≃
1 c/K, so X will jump between x and y O(K) times before leaving x,y ,
− { }
andthustheCSRWwillbetrappedforatimeoforderK in x,y .However,
{ }
fortheVSRWeachjumptakesonlyO(K−1),sothetotaltimespentin x,y
{ }
is only O(1). A similar effect will arise from finite clusters of bonds of high
conductivity.
Thepresenceoftrapsofthiskindiswhywehave,whenEµ = ,thatthe
e
∞
diffusion constant σ2 for the CSRW is zero. In this case it is natural to ask
C
if a different scaling will give a nontrivial limit. There is a connection here
with “aging” (see [6–8]), and in [2] it is proved that if the tail distribution
P(µ >t) t−α, then εαX converges to the “fractional kinetic” motion
e t/ε2
∼
with parameter α (see [8]).
While we have written this paper for the case of i.i.d. conductances µ ,
e
our arguments do not require the full strength of this. In case 0, when the
conductances are bounded(both above and below), then uniformupperand
lower Gaussianheatkernelestimates, asinTheorem1.2,arewellknown(see
[18]). It follows (see Remark 6.3) that Theorem 1.1. holds for any stationary
ergodic environment. On the other hand, in the unbounded case 2, there
exist stationary ergodic environments such that the VSRW can explode in
finite time; for an example, see Remark 6.6 below.
FortheGaussianboundsinTheorem1.2weneedtocontrolthesizesofthe
clustersofhighconductivitywhichisdonebycomparisonbetweenthegraph
metric d(x,y) and a new metric d (introduced by Davies in [15, 16]) which
−1/2
is adapted to the structure of the random walk and satisfies d(x,y) µ
xy
≤
when x y. This new metric is ceonstructed by a first passage percolation
∼
procedure, and in this paper we have used first passage perecolation argu-
ments to compare the two metrics (see Lemma 4.2). These arguments use
estimates from [25] which in turn use the fact that µ are i.i.d. However,
e
we could also have used a direct argument as in [12], Lemma 3.1 or [30],
Lemma 5.3. Once we have the Gaussian bounds (with sufficiently good con-
trol on the tails of the r.v. U in Theorem 1.2), the quenched invariance
x
principle follows with no further hypotheses on µ ,e E other than that
e d
{ ∈ }
itisstationary, symmetricandergodic.Theorem6.1summarizesthegeneral
situation.
The structure of this paper is as follows. In Sections 2 and 3 we study a
deterministic graph Γ=(G,E) with edge weights µ . Under certain con-
xy
ditions (which are P-a.s. satisfied by the VSRW on the i.i.d. RCM) we ob-
tain heat kernel bounds in this setup. Our approach uses similar methods
to those used in [1] for percolation clusters. However, in [1] the Carne–
Varopoulos “long-range” bounds played an essential role at various points.
6 M. T. BARLOW ANDJ.-D. DEUSCHEL
These bounds do not hold for the VSRW, and instead we use more general
upper bounds obtained by Davies [15, 16], which are in terms of the metric
d(x,y). The same metric d is also needed to control Px(d(X ,x) λt1/2)
ω t ≥
which is the key step in obtaining general Gaussian upper bounds. Similar
beounds,inthe context of weeighted Laplacians on manifolds,eare obtained by
Grigor’yan [24]; here the metric d is the Riemannian metric. Very recently,
and independently, Mourrat [31] has obtained upper bounds for the VSRW
which in certain cases improve onethose in Theorem 1.2.
Once one has upper bounds, lower bounds follow by the same arguments
as in [1], Section 5 (see Section 3). In Section 4 we then prove Theorem 1.2.
Section 5 proves the invariance principle. We begin with the VSRW. The
basic technique in the proof (as in many previous papers such as [10, 12,
Ed
17, 27–29]) is to associate with X a process Z on Ω=[1, ] which is the
t t
environment seen from the random walk. More precisely, fo∞r each x Zd, let
∈
T :Ω Ω be given by
x
→
T (ω)(z,w)=ω(z+x,w+x).
x
Assuming that X =0 we define
0
(1.19) Z (ω)=T (ω).
t Xt(ω)
One seeks to use the process Z to construct the “corrector,” that is, a map
χ:Ω Zd Rd such that
× →
(1.20) M (ω)=X (ω) χ(ω,X (ω)), t 0,
t t t
− ≥
is a P0-martingale. Once one has constructed the corrector, showing the
ω
invarianceprinciplefortherescaledmartingaleεM isstandard,andusing
t/ε2
resultsfrom[12,34],theheatkernelestimates inTheorem1.2,together with
the sublinear growth of χ, imply that
(1.21) limεχ(ω,X )=0 in P0-probability.
t/ε2 ω
ε→0
However, the standard construction of the corrector is based on L2(P)
calculus, which requires finiteness of the first moment of the conductance
(see [17], page 816). In our case we wish to handle the case when Eµ = ,
e
and so we need an alternative approach. (We remark that if Eµ = th∞en
e
∞
it is not easy to find suitable function spaces on Ω which give a core for the
Dirichlet form associated with Z.)
Our solution relies on discretization. We define X =X , n Z , and
n n +
∈
consider the process
b
(ε)
(1.22) X =εX .
t ⌊t/ε2⌋
(ε) (ε)
We can control sup X X (see Lemma4.12), soan invariance prin-
t≤T| t − bt | b
ciple for X(ε) will follow from one for X(ε). The process X does not have
b
b b
INVARIANCEPRINCIPLE FOR THE RANDOMCONDUCTANCEMODEL 7
bounded jumps—in fact it jumps anywhere in Zd with positive probability.
However, the long-range bounds on qω(x,y) in (1.11) give good control on
t
these jumps, and, in particular, the bounds on qω(x,y) imply that
1
(1.23) EE0 X 2< ,
ω| 1| ∞
which is the key L2 condition on X for the construction of the corrector. As
we will see in Section 5, looking at the discrete time process does actually
introduce some simplifications in tbhe construction of the corrector χ. In the
end (see Remark 5.15) it will turn out that the “discrete time” corrector χ
also satisfies (1.20).
Finally, a short Section 6 makes some remarks on more general environ-
ments, and gives an example (a one-sided spanning tree) where the process
X fails to be conservative, and so the invariance principle fails.
2. Transition density upper bounds on a fixed graph. Let Γ=(G,E) be
an infinite (deterministic) graph, µ ,e E, be bond conductances and ν be
e
∈
a measure on G. We make the following assumptions on (G,E), µ and ν.
Assumption 2.1. (1) Γ is connected.
(2) The vertex degree is uniformly bounded,
(2.1) y:y x C for all x G.
D
|{ ∼ }|≤ ∈
(3) µ >0 for all e E.
e
∈
(4) There exists C 1 such that
M
≥
(2.2) C−1 ν =ν( x ) C for all x G.
M ≤ x { } ≤ M ∈
The results of this section do not explicitly require a strictly positive
lower boundon µ ;however, a later assumption [see Assumption 2.6(2)] will
e
impose some control on the edges e with µ small.
e
We write µ for µ , and set µ =0 if x y. Let d(x,y) be the usual
xy {x,y} xy
6∼
graph distance on G, and write
(2.3) B(x,r)= y:d(x,y)<r .
{ }
Let C < . We now construct, by a first passage percolation procedure, a
A
∞
second metric d on G satisfying
(2.4) (C−2 µ )d(x,y) d(x,y′)2 1 for every x G,y y′.
A ∨ eyy′ | − | ≤ ∈ ∼
We write B(x,r)= y:d(x,y)<r for balls in the metric d. (In this paper
e e
{ }
we can take C =1, but for possible future extensions we treat the general
A
case.) To ceonstruct d weeassign waiting times e
(2.5) t(e)=C µ−1/2, e E,
e A∧ e ∈
8 M. T. BARLOW ANDJ.-D. DEUSCHEL
and then take d(x,y) to bethe shortestjourney timebetween x and y.More
precisely,
e
n
(2.6) d(x,y)=inf t(e ) ,
i
( )
i=1
X
e
where the infimum is taken over paths (e ,...,e ) from x to y. Since we do
1 n
not have a strictly positive uniform lower bound on t(e), in general there
may notbeaminimizingpath.However, suchpathswilla.s.existwhent(e )
i
are i.i.d. positive random variables.
Lemma 2.2. The metric d constructed above satisfies (2.4).
Proof. Lete= y,z ;theend(x,z) d(x,y)+t(e),andusing(2.5)gives
(2.4). (cid:3) { } ≤
e e
Recall that
1
(f,g)= µ (f(y) f(x))(g(y) g(x)).
xy
E 2 − −
x y∼x
XX
Let µ = µ ,andextend µ toameasureon G.Then (f,g) isdefined
x y∼x xy E
for f,g L2(G,µ).
∈P
Let X =(X ,t [0, ),Px,x G) be the continuous time Markov chain
t
∈ ∞ ∈
on G with generator
f(x)=ν−1 µ (f(y) f(x)).
L x xy −
y
X
At this point we cannot exclude the possibility that X may explode, and we
write ζ for the lifetime of X. Let
f 2 = (f,f)+ f 2
k kE1 E k kL2(ν)
and be the closure of the set of functions on G with finite support with
F
respectto f .ThenX istheMarkovprocessassociatedwiththeDirichlet
k kE1
form( , )onL2(G,ν)(see[22]).Letq (x,y)bethetransitiondensity(heat
t
E F
kernel) of X with respect to ν:
Px(X =y)
t
q (x,y)= .
t
ν
y
We begin by using the results of Davies [15, 16] to obtain long-range
bounds on q . By Proposition 5 of [15], we have
t
(2.7) q (x,y) (ν ν )−1/2 inf exp(ψ(y) ψ(x)+Λ(ψ)t),
t x y
≤ ψ∈L∞(G) −
INVARIANCEPRINCIPLE FOR THE RANDOMCONDUCTANCEMODEL 9
where Λ(ψ)=sup b(ψ,x), and
x
1
(2.8) b(ψ,x)= µ (eψ(x)−ψ(y)+eψ(y)−ψ(x) 2).
xy
2ν −
x
y∼x
X
Theorem 2.3. Assume (G,E) and µ satisfy Assumption 2.1. There
exist constants c ,...,c (depending only on C ,C ,C ) such that the fol-
1 4 A D M
lowing hold.
(a) If x,y G and D=d(x,y) c t, then
1
∈ ≤
(2.9) q (x,y) c exp( c D2/t).
e t e 2 3
≤ −
(b) If x,y G and D=d(x,y) c t, then
1 e
∈ ≥
(2.10) q (x,y) c exp( c D(1 log(D/t))).
t e e 2 4
≤ − ∨
Proof. Fix x ,y G, let t>0, aned write D=e d(x ,y ). Let λ>0 and
0 0 0 0
∈
set
e e
ψ (x)= λ(D d(x ,x)), b(λ)=supb(ψ ,x).
λ 0 λ
− ∧
x
Let x G, y x and writeeµ e=C−2 µ ,
∈ ∼ xy A ∨ xy
J =µ (eψλ(x)−ψλ(y)+eψλ(y)−ψλ(x) 2).
xy xy e −
−1/2
Then as ψ (x) ψ (y) λµ , and cosh(x) is increasing on [0, ),
λ λ xy
| − |≤ ∞
(2.11) J 2µ (cosh(λµ−1/2) 1) 2µ (cosh(λµ−1/2) 1).
xy ≤ xy exy − ≤ xy xy −
Using the power series for cosh we have that the right-hand side of (2.11) is
e e e
decreasing in µ , so
xy
J C−2(eCAλ+e−CAλ 2).
e xy ≤ A −
Hence
b(ψ ,x) 1C C C−2(eCAλ+e−CAλ 2).
λ ≤ 2 M D A −
Let f(x)=ex+e−x 2; then b(λ) c f(C λ). Thus by (2.7), and writing
7 A
− ≤
y=C λ,
A
q (x ,y ) C infexp( λD+c tf(C λ))
t 0 0 M 7 A
≤ λ −
(2.12)
D e C c t
A 7
C exp inf y+ f(y) .
M
≤ (cid:18)CA(cid:18)y>0(cid:18)− D (cid:19)(cid:19)(cid:19)
e
e
10 M. T. BARLOW ANDJ.-D. DEUSCHEL
So if
F(s)= inf( y+(2s)−1(ey +e−y 2)),
y>0 − −
then
D D
(2.13) q (x ,y ) C exp F
t 0 0 M
≤ C 2C c t
(cid:18) A (cid:18) A 7 (cid:19)(cid:19)
e e
and it remains to bound F.
We have (see [15], page 70) that
F(s)=s−1((1+ss)1/2 1) log(s+(1+s2)1/2)
− −
and also that F(s) s/2(1 s2/10) for s>0. Hence, if s 3, then F(s)
≤− − ≤ ≤
s/20 while if s e, then
− ≥
F(s) 1 log(2s)= log(2s/e).
≤ − −
Substituting in (2.13) completes the proof. (cid:3)
Remark 2.4. Note that if D =ct then both (2.9) and (2.10) give a
bound of the form ce−t/c.
e
Since µ are not bounded above, the process X may explode. The follow-
e
ing condition is enough to exclude this.
Lemma 2.5. Suppose there exists x G and θ>0 such that
∈
(2.14) exp( θd(x,y))ν < .
y
− ∞
y∈G
X
e
Then X is conservative.
Proof. Let ζ be the lifetime of X. Then as Γ is connected it is easy
to see that either Py(ζ = )=1 for all y G, or else Py(ζ <t)>0 for all
∞ ∈
y G, t>0.
∈
For n C−2 let µ(n)=n µ , X(n) be the process associated with the
≥ A xy ∧ xy
conductances µ(n), and q(n)(x,y) be the transition density of X(n) with re-
t
spect to ν. We have q (x,y) = lim q(n)(x,y). Note that each X(n) is
t n→∞ t
conservative, and that the bounds in Theorem 2.3 hold (with the same con-
stants c ) for each q(n). With (2.2) the condition (2.14) implies that B(x,R)
i
is finite for each R>0.
Let t>0. With constants ci as in Theorem 2.3, choose r large enoeugh so
that r>c t and c (1 log(r/t)) θ. So, if R r, using (2.10),
1 4
∨ ≥ ≥
(n)
(2.15) q (x,y)ν c exp( θd(x,y))ν < .
t y ≤ 2 − y ∞
e e
y∈BX(x,R)c y∈BX(x,R)c
e