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El e c t ro n ic

Jo ur

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f P

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b a b il i t y

Vol. 14 (2009), Paper no. 72, pages 2091–2129. Journal URL

http://www.math.washington.edu/~ejpecp/

Intermittency on catalysts:

three-dimensional simple symmetric exclusion

∗

Jürgen Gärtner

Institut für Mathematik

Technische Universität Berlin

Straße des 17. Juni 136

D-10623 Berlin, Germany

jg@math.tu-berlin.de

Frank den Hollander

Mathematical Institute

Leiden University, P.O. Box 9512

2300 RA Leiden, The Netherlands

and EURANDOM, P.O. Box 513

5600 MB Eindhoven, The Netherlands

denholla@math.leidenuniv.nl

Grégory Maillard

CMI-LATP

Université de Provence

39 rue F. Joliot-Curie

F-13453 Marseille Cedex 13, France

maillard@cmi.univ-mrs.fr

Abstract

We continue our study of intermittency for the parabolic Anderson model∂u/∂t=κ∆u+ξuin a space-time random mediumξ, whereκis a positive diffusion constant,∆is the lattice Laplacian onZd_{,d≥1, andξis a simple symmetric exclusion process on}Zd_{in Bernoulli equilibrium. This} model describes the evolution of areactant uunder the influence of acatalystξ.

In[3]we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t _{→ ∞} of the successive moments of the solution u. This led to an almost ∗_{The research of this paper was partially supported by the DFG Research Group 718 “Analysis and Stochastics in}

Complex Physical Systems”, the DFG-NWO Bilateral Research Group “Mathematical Models from Physics and Biology”, and the ANR-project MEMEMO

complete picture of intermittency as a function of d and κ. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents asκ_{→ ∞}in thecritical dimensiond=3, which was left open in[3]and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by aGreenterm, as in d_≥4, but also by apolaronterm. The presence of the latter implies intermittency ofallorders above a finite threshold forκ.

Key words: Parabolic Anderson model, catalytic random medium, exclusion process, graphical representation, Lyapunov exponents, intermittency, large deviation.

AMS 2000 Subject Classification:Primary 60H25, 82C44; Secondary: 60F10, 35B40. Submitted to EJP on December 17, 2008, final version accepted August 17, 2009.

1

Introduction and main result

1.1

Model

In this paper we consider theparabolic Anderson model(PAM) onZd_,d≥1,

  

∂u

∂t =κ∆u+ξu onZ

d_×[0,∞),

u(_·, 0) =1 onZd_,

(1.1)

whereκis a positive diffusion constant,∆is the lattice Laplacian acting onuas

∆u(x,t) = X

y∈Zd

ky−xk=1

[u(y,t)₋u(x,t)] (1.2)

(_{k · k}is the Euclidian norm), and

ξ= (ξt)t≥0, ξt={ξt(x): x∈Zd}, (1.3)

is a space-time random field that drives the evolution. If ξis given by an infinite particle system dynamics, then the solution uof the PAM may be interpreted as the concentration of a diffusing

reactantunder the influence of acatalystperforming such a dynamics.

In Gärtner, den Hollander and Maillard [3]we studied the PAM for ξSymmetric Exclusion (SE), and developed an almost complete qualitative picture. In the present paper we finish our study by focussing on the limiting behavior asκ→ ∞in thecriticaldimensiond=3, which was left open in

[3]and which is the most challenging. We restrict toSimple Symmetric Exclusion(SSE), i.e.,(ξt)t≥0 is the Markov dynamics on Ω = _{0, 1_}Z3

(0 = vacancy, 1 = particle) with generator L acting on cylinder functions f:Ω_→Ras

(L f)(η) = 1

6 X

{a,b}

h

f ηa,b−f(η)i, η∈Ω, (1.4)

where the sum is taken over all unoriented nearest-neighbor bonds{a,b} ofZ3_{, andη}a,b _denotes

the configuration obtained fromηby interchanging the states ataand b:

ηa,b(a) =η(b), ηa,b(b) =η(a), ηa,b(x) =η(x)forx∈ {/ a,b_}. (1.5) (See Liggett[7], Chapter VIII.) LetP_ηandE_{η denote probability and expectation forξgivenξ0=}

η∈Ω. Letξ0be drawn according to the Bernoulli product measureνρonΩwith densityρ∈(0, 1).

The probability measuresνρ,ρ∈(0, 1), are the only extremal equilibria of the SSE dynamics. (See

Liggett[7], Chapter VIII, Theorem 1.44.) We writeP_ν

ρ= R

Ωνρ(dη)PηandEνρ= R

Ωνρ(dη)Eη.

1.2

Lyapunov exponents

Forp_∈N_{, define thep-thannealed Lyapunov exponentof the PAM by}

λp(κ,ρ) =_tlim_→∞

1

ptlogEνρ([u(0,t)]

We are interested in the asymptotic behavior ofλp(κ,ρ)asκ→ ∞for fixedρand p. To this end,

letGdenote the value at 0 of theGreen functionof simple random walk onZ3with jump rate 1 (i.e., the Markov process with generator 1₆∆), and letP3be the value of thepolaron variational problem

P3= sup

f∈H1(R3)

kfk2=1

−∆R3

₋1/2

f2

2 2−

_∇R3f

2

2

, (1.7)

where_∇R3 and∆R3are thecontinuousgradient and Laplacian,k · k2 is theL2(R3)-norm,H1(R3) =

{f ∈L2(R3): _∇R3f ∈L2(R3)}, and

−∆R3

−1/2

f2

2 2=

Z

R3

d x f2(x)

Z

R3

d y f2(y) 1

4πkx₋ y_k. (1.8)

(See Donsker and Varadhan[1]for background on how_P3 arises in the context of a self-attracting Brownian motion referred to as the polaron model. See also Gärtner and den Hollander[2], Section 1.5.)

We are now ready to formulate our main result (which was already announced in Gärtner, den Hollander and Maillard[4]).

Theorem 1.1. Let d=3,ρ∈(0, 1)and p_∈N_{. Then}

lim

κ→∞κ[λp(κ,ρ)−ρ] =

1

6ρ(1−ρ)G+ [6ρ(1−ρ)p] 2

P3. (1.9)

Note that the expression in the r.h.s. of (1.9) is the sum of aGreenterm and apolaronterm. The existence, continuity, monotonicity and convexity ofκ7→λp(κ,ρ)were proved in[3]for alld ≥1

for all exclusion processes with an irreducible and symmetric random walk transition kernel. It was further proved thatλp(κ,ρ) =1 when the random walk is recurrent and ρ < λp(κ,ρ)< 1 when

the random walk is transient. Moreover, it was shown that for simple random walk in d _≥ 4 the asymptotics as κ→ ∞ ofλp(κ,ρ) is similar to (1.9), but without the polaron term. In fact, the

subtlety ind =3 is caused by the appearance of this extra term which, as we will see in Section 5, is related to the large deviation behavior of the occupation time measure of a rescaled random walk that lies deeply hidden in the problem. For the heuristics behind Theorem 1.1 we refer the reader to[3], Section 1.5.

1.3

Intermittency

The presence of the polaron term in Theorem 1.1 implies that, for eachρ ∈(0, 1), there exists a κ0(ρ)>0 such that thestrictinequality

λp(κ,ρ)> λp−1(κ,ρ) ∀κ > κ0(ρ) (1.10)

holds for p = 2 and, consequently, for all p ≥ 2 by the convexity of p 7→ pλp(κ,ρ). This means

that all moments of the solution u are intermittent for κ > κ0(ρ), i.e., for large t the random field u(_·,t) develops sparse high spatial peaks dominating the moments in such a way that each moment is dominated by its own collection of peaks (see Gärtner and König[5], Section 1.3, and den Hollander[6], Chapter 8, for more explanation).

In[3]it was shown that for alld_≥3 the PAM is intermittent forsmallκ. We conjecture that ind=3 it is in fact intermittent forallκ. Unfortunately, our analysis does not allow us to treat intermediate values ofκ(see the figure).

0 ρ 1

r r r p=3 p=2 p=1

?

κ λp(κ)

Qualitative picture ofκ_7→λ_p(κ)forp=1, 2, 3.

The formulation of Theorem 1.1 coincides with the corresponding result in Gärtner and den Hollan-der[2], where the random potential ξis given by independent simple random walks in a Poisson equilibrium in the so-called weakly catalytic regime. However, as we already pointed out in[3], the approach in[2]cannot be adapted to the exclusion process, since it relies on an explicit Feynman-Kac representation for the moments that is available only in the case ofindependentparticle motion. We must therefore proceed in a totally different way. Only at the end of Section 5 will we be able to use some of the ideas in[2].

1.4

Outline

Each of Sections 2–5 is devoted to a major step in the proof of Theorem 1.1 forp=1. The extension top_≥2 will be indicated in Section 6.

In Section 2 we start with the Feynman-Kac representation for the first moment of the solutionu, which involves a random walk sampling the exclusion process. After rescaling time, we transform the representation w.r.t. the old measure to a representation w.r.t. a new measure via an appropriate absolutely continuous transformation. This allows us to separate the parts responsible for, respec-tively, the Green term and the polaron term in the r.h.s. of (1.9). Since the Green term has already been handled in[3], we need only concentrate on the polaron term. In Section 3 we show that, in the limit asκ→ ∞, the new measure may be replaced by the old measure. The resulting represen-tation is used in Section 4 to prove thelower boundfor the polaron term. This is done analytically with the help of a Rayleigh-Ritz formula. In Section 5, which is technical and takes up almost half of the paper, we prove the correspondingupper bound. This is done by freezing and defreezing the exclusion process over long time intervals, allowing us to approximate the representation in terms of the occupation time measures of the random walk over these time intervals. After applying spectral estimates and using a large deviation principle for these occupation time measures, we arrive at the polaron variational formula.

2

Separation of the Green term and the polaron term

In Section 2.1 we formulate the Feynman-Kac representation for the first moment of uand show how to split this into two parts after an appropriate change of measure. In Section 2.2 we formulate two propositions for the asymptotics of these two parts, which lead to, respectively, the Green term and the polaron term in (1.9). These two propositions will be proved in Sections 3–5. In Section 2.3 we state and prove three elementary lemmas that will be needed along the way.

2.1

Key objects

The solutionuof the PAM in (1.1) admits the Feynman-Kac representation

u(x,t) =EX_x ‚

exp –Z t

0

dsξt−s Xκs

™Œ

, (2.1)

where X is simple random walk on Z3 _{with step rate 6 (i.e., with generator ∆) and P}X

x and E

X x

denote probability and expectation with respect to X given X₀ = x. Sinceξis reversible w.r.t.νρ,

we may reverse time in (2.1) to obtain

E_ν

ρ u(0,t)

=E_ν

ρ,0

exp Z t

0

dsξs Xκs

, (2.2)

whereE_ν

ρ,0is expectation w.r.t.Pνρ,0=Pνρ⊗P

X

0. As in[2]and[3], we rescale time and write

e−ρ(t/κ)E_ν

ρ u(0,t/κ)

=E_ν

ρ,0

exp ₁

κ Z t

0

dsφ(Z_s)

(2.3) with

φ(η,x) =η(x)₋ρ (2.4)

and

Zt= ξt/κ,Xt

. (2.5)

From (2.3) it is obvious that (1.9) in Theorem 1.1 (forp=1) reduces to lim

κ→∞κ

2_λ∗_{(κ) =} 1

6ρ(1−ρ)G+ [6ρ(1−ρ)] 2

P3, (2.6)

where

λ∗(κ) = lim

t→∞

1

t logEνρ,0 ‚

exp –

1 κ

Z t

0

dsφ(Z_s)

™Œ

. (2.7)

Here and in the rest of the paper we suppress the dependence on ρ ∈(0, 1) from the notation. UnderP_{η,x =}P_η⊗PX

x,(Zt)t≥0 is a Markov process with state spaceΩ×Z3and generator

A = 1

(acting on the Banach space of bounded continuous functions onΩ_×Z3_{, equipped with the}

supre-mum norm). Let(_St)_t≥0denote the semigroup generated by_A.

Our aim is to make an absolutely continuous transformation of the measureP_{η,x with the help of}

an exponential martingale, in such a way that, under the new measurePnew

η,x , (Zt)t≥0 is a Markov process with generator_Anewof the form

Anewf =e−1κψA

e1κψf

−e−1κψAe

1

κψ

f. (2.9)

This transformation leads to an interaction between the exclusion process part and the random walk part of(Zt)t≥0, controlled byψ:Ω×Z3→R. As explained in[3], Section 4.2, it will be expedient to chooseψas

ψ=

Z T

0

ds Ssφ

(2.10) withT a large constant (suppressed from the notation), implying that

− Aψ=φ− STφ. (2.11)

It was shown in[3], Lemma 4.3.1, that

Nt=exp

– 1 κ

ψ(Zt)−ψ(Z0)

−

Z t

0

ds

e−1κψAe

1

κψ

(Zs)

™

(2.12) is an exponentialP_{η,x-martingale for all(η,x)∈Ω×}Z3_{. Moreover, if we define}Pnew

η,x in such a way

that

Pnew

η,x(A) =Eη,x Nt11A

(2.13) for all eventsAin theσ-algebra generated by(Z_s)_s∈[0,t], then underPnew

η,x indeed(Zs)s≥0is a Markov process with generator_Anew. Using (2.11–2.13) andEnew

ν_ρ,0= R

Ωνρ(dη)E

new

η,0, it then follows that the expectation in (2.7) can be written in the form

E_ν

ρ,0 ‚

exp –

1 κ

Z t

0

dsφ(Z_s)

™Œ

=Enew

ν_ρ,0

exp ₁

κ

ψ(Z0)−ψ(Zt)

+

Z t

0

ds

e−κ1ψAe

1

κψ

− A

₁

κψ

(Zs)

+ 1

κ Z t

0

ds _STφ(Z_s)

.

(2.14)

The first term in the exponent in the r.h.s. of (2.14) stays bounded ast → ∞and can therefore be discarded when computingλ∗(κ)via (2.7). We will see later that the second term and the third term lead to the Green term and the polaron term in (2.6), respectively. These terms may be separated from each other with the help of Hölder’s inequality, as stated in Proposition 2.1 below.

2.2

Key propositions

Proposition 2.1. For anyκ >0,

λ∗(κ) ≤

≥ I

q

1(κ) +I

r

with

I₁q(κ) = 1

qtlim→∞

1

t logE

new

ν_ρ,0

exp

q

Z t

0

dse−1κψAe

1

κψ

− A

₁ κψ

(Z_s)

,

I₂r(κ) = 1

r tlim→∞

1

t logE

new

ν_ρ,0

exp _r

κ Z t

0

ds STφ

(Zs)

,

(2.16)

where 1/q+1/r = 1, with q >0, r > 1in the first inequality and q <0, 0< r < 1in the second inequality.

Proof. See[3], Proposition 4.4.1. The existence and finiteness of the limits in (2.16) follow from Lemma 3.1 below.

By choosingr arbitrarily close to 1, we see that the proof of our main statement in (2.6) reduces to the following two propositions, where we abbreviate

lim sup

t,κ,T→∞

=lim sup

T→∞

lim sup

κ→∞

lim sup

t→∞

and lim

t,κ,T→∞=Tlim→∞κlim→∞tlim→∞. (2.17)

In the next proposition we writeψT instead ofψto indicate the dependence on the parameterT.

Proposition 2.2. For anyα∈R_,

lim sup

t,κ,T→∞

κ2

t logE

new

ν_ρ,0 ‚

exp –

α Z t

0

dse−1κψT_Ae1_κψT

− A1

κψT

(Z_s)

™Œ

≤ α

6ρ(1−ρ)G. (2.18) Proposition 2.3. For anyα >0,

lim

t,κ,T→∞

κ2

t logE

new

ν_ρ,0

exp

α κ

Z t

0

ds STφ

(Z_s)

= [6α2ρ(1−ρ)]2_P3. (2.19)

These propositions will be proved in Sections 3–5.

2.3

Preparatory lemmas

This section contains three elementary lemmas that will be used frequently in Sections 3–5.

Let p(_t1)(x,y) and p_t(x,y) = p(_t3)(x,y) be the transition kernels of simple random walk in d =1 andd=3, respectively, with step rate 1.

Lemma 2.4. There exists C>0such that, for all t _≥0and x,y,e_∈Z3 _withkek=1,

p(_t1)(x,y)_≤ C (1+t)12

, pt(x,y)≤

C

(1+t)32

, pt(x+e,y)−pt(x,y)

_≤ C

(1+t)2. (2.20)

Proof. Standard.

From the graphical representation for SSE (Liggett[7], Chapter VIII, Theorem 1.1) it is immediate that

E_{η ξt}(x)= X y∈Zd

p_t(x,y)η(y). (2.21)

Recalling (2.4–2.5) and (2.10), we therefore have

Ssφ(η,x) =Eη,x

φ(Z_s)=E_η‚ X

y∈Z3

p_6s(x,y)ξs/κ(y)−ρ

Œ

= X

z∈Z3

p_6s1[κ](x,z)

η(z)₋ρ

(2.22)

and

ψ(η,x) =

Z T

0

ds X

z∈Z3

p_6s1[κ](x,z)

η(z)₋ρ, (2.23)

where we abbreviate

1[κ] =1+ 1

6κ. (2.24)

Lemma 2.5. For allκ,T>0,η∈Ω, a,b_∈Z3 _{withka−bk=1and x∈}Z3_,

|ψ(η,b)₋ψ(η,a)_{| ≤}2CpT for T_≥1, (2.25)

ψ ηa,b,x−ψ(η,x)

≤2G, (2.26)

X

{a,b}

ψ ηa,b,x₋ψ(η,x)

2

≤ 1₆G, (2.27)

where C >0is the same constant as in Lemma 2.4, and G is the value at0of the Green function of simple random walk onZ3_.

Proof. For a proof of (2.26–2.27), see[3], Lemma 4.5.1. To prove (2.25), we may without loss of generality consider b=a+e₁ withe₁= (1, 0, 0). Then, by (2.23), we have

|ψ(η,b)₋ψ(η,a)_{| ≤}

Z T

0

ds X

z∈Z3

_p6s1[κ](z+e1)−p6s1[κ](z)

=

Z T

0

ds X

z∈Z3

p₆(1_s1)_[κ](z1+e1)−p

(1)

6s1[κ](z1)

p(₆1_s)_1[κ](z2)p

(1)

6s1[κ](z3)

=

Z T

0

ds X

z1∈Z

p(₆1_s)_1[κ](z1+e1)−p

(1)

6s1[κ](z1)

=2 Z T

0

ds p(₆1_s)_1[κ](0)_≤2CpT.

(2.28)

Let_G be the Green operator acting on functionsV:Z3_→[0,∞)as

GV(x) = X y∈Z3

G(x₋y)V(y), x_∈Z3_{, (2.29)}

withG(z) =R₀∞d t p_t(z). Let_{k · k∞}denote the supremum norm. Lemma 2.6. For all V:Z3_{→[0,∞)and x ∈}Z3_,

EX_x

exp Z∞

0

d t V(X_t)

≤

1_{− kG}V_k∞

₋1

≤exp ‚

kGV_k∞

1_{− kG}Vk∞

Œ

, (2.30)

provided that

kGVk∞<1. (2.31)

Proof. See[2], Lemma 8.1.

3

Reduction to the original measure

In this section we show that the expectations in Propositions 2.2–2.3 w.r.t. the new measurePnew

νρ,0 are asymptotically the same as the expectations w.r.t. the old measureP_ν

ρ,0. In Section 3.1 we state a Rayleigh-Ritz formula from which we draw the desired comparison. In Section 3.2 we state the analogues of Propositions 2.2–2.3 whose proof will be the subject of Sections 4–5.

3.1

Rayleigh-Ritz formula

Recall the definition ofψin (2.10). Letmdenote the counting measure onZ3_{. It is easily checked}

that bothµρ=νρ⊗mandµnewρ given by

dµnew_ρ =e2κψdµ_ρ (3.1)

are reversible invariant measures of the Markov processes with generators _A defined in (2.8), respectively,_Anewdefined in (2.9). In particular,_A and_Aneware self-adjoint operators in L2(µρ)

andL2(µnew

ρ ). LetD(A)andD(Anew)denote their domains.

Lemma 3.1. For all bounded measurable V:Ω_×Z3_→R_,

lim

t→∞

1

t logE

new

ν_ρ,0

exp Z t

0

ds V(Zs)

= sup

F∈D(Anew)

kFk_L2_(µnew

ρ )=1

ZZ

Ω×Z3

dµnew_ρ V F2+FAnewF

. (3.2)

The same is true whenEnew

νρ,0,µ new

ρ ,A

new_{are replaced byE}

ν_ρ,0,µρ,A, respectively.

Proof. The limit in the l.h.s. of (3.2) coincides with the upper boundary of the spectrum of the operator_Anew+V onL2(µnew_ρ ), which may be represented by the Rayleigh-Ritz formula. The latter coincides with the expression in the r.h.s. of (3.2). The details are similar to[3], Section 2.2.

Lemma 3.1 can be used to express the limits ast_{→ ∞}in Propositions 2.2–2.3 as variational expres-sions involving the new measure. Lemma 3.2 below says that, for largeκ, these variational expres-sions are close to the corresponding variational expresexpres-sions for the old measure. Using Lemma 3.1 for the original measure, we may therefore arrive at the corresponding limit for the old measure. For later use, in the statement of Lemma 3.2 we do not assume thatψis given by (2.10). Instead, we only suppose thatη7→ψ(η)is bounded and measurable and that there is a constantK>0 such that for allη∈Ω, a,b∈Z3_withka−bk=1 andx ∈Z3_,

|ψ(η,b)₋ψ(η,a)_{| ≤}K and

ψ ηa,b,x₋ψ(η,x)

≤K, (3.3)

but retain thatAnewandµnew_ρ are given by (2.9) and (3.1), respectively. Lemma 3.2. Assume(3.3). Then, for all bounded measurable V:Ω_×Z3_→R_,

sup

F∈D(Anew)

kFk_L2_(µnew

ρ )

=1

ZZ

Ω×Z3 dµnew_ρ

V F2+F_AnewF

≤ ≥ e

∓K_{κ sup}

F∈D(A)

kFk_L2_(µρ)=1

ZZ

Ω×Z3 dµρ

e±KκV F2+FAF

,

(3.4)

where±means+in the first inequality and−in the second inequality, and∓means the reverse. Proof. Combining (1.2), (1.4) and (2.8–2.9), we have for all(η,x)_∈Ω_×Z3 _{and allF∈ D(A}new_),

V F2+F_AnewF(η,x) =V(η,x)F2(η,x) + 1

6κ X

{a,b}

F(η,x)eκ1[ψ(η

a,b_{,x)−ψ(η,x)]}h

F(ηa,b,x)₋F(η,x)i

+ X

y:ky−xk=1

F(η,x)e1κ[ψ(η,y)−ψ(η,x)]F(η,y)−F(η,x).

(3.5)

Therefore, taking into account (2.9), (3.1) and the exchangeability ofνρ, we find that

ZZ

Ω×Z3

dµnew_ρ V F2+F_AnewF=

ZZ

Ω×Z3

dµnew_ρ (η,x)

‚

V(η,x)F2(η,x)

− 1

12κ X

{a,b} e1κ[ψ(η

a,b_{,x)−ψ(η,x)]}h

F(ηa,b,x)₋F(η,x)i2

−1

2 X

y:ky−xk=1

e1κ[ψ(η,y)−ψ(η,x)]F(η,y)−F(η,x)2 Œ

. (3.6)

LetFe=eψ/κF. Then, by (3.1) and (3.3),

(3.6) ≤

≥

ZZ

Ω×Z3

dµnew_ρ (η,x)

‚

V(η,x)F2(η,x)

−e

∓K

κ 12κ

X

{a,b}

h

F(ηa,b,x)₋F(η,x)i2₋ e ∓K

κ 2

X

y:ky−xk=1

F(η,y)₋F(η,x)2

Œ

=

ZZ

Ω×Z3

dµρ(η,x)

‚

V(η,x)Fe2(η,x)

−e

∓K_κ

12κ X

{a,b}

h e

F(ηa,b,x)₋Fe(η,x)i2₋ e ∓K_κ

2

X

y:ky−xk=1 h

e

F(η,y)₋Fe(η,x)i2

Œ

= e∓Kκ ZZ

Ω×Z3 dµρ

e±Kκ VFe2+eFAFe

.

(3.7)

Taking further into account that eF2_L2_(µ

ρ)=kFk 2

L2_(µnew

ρ )

, (3.8)

and that eF_{∈ D}(_A)if and only ifF _{∈ D}(_Anew), we get the claim.

3.2

Reduced key propositions

At this point we may combine the assertions in Lemmas 3.1–3.2 for the potentials

V =αe−κ1ψAe

1

κψ

− A

₁

κψ

(3.9) and

V = α

κ STφ

(3.10) withψgiven by (2.10). Because of (2.25–2.26), the constant K in (3.3) may be chosen to be the maximum of 2G and 2CpT, resulting inK/κ→0 asκ→ ∞. Moreover, from (2.27) and a Taylor expansion of the r.h.s. of (3.9) we see that the potential in (3.9) is bounded for each κ and T, and the same is obviously true for the potential in (3.10) because of (2.4). In this way, using a moment inequality to replace the factor e±K/κα by a slightly larger, respectively, smaller factorα′ independent of T and κ, we see that the limits in Propositions 2.2–2.3 do not change when we replaceEnew

νρ,0by

E_ν

ρ,0. Hence it will be enough to prove the following two propositions. Proposition 3.3. For allα∈R_,

lim sup

t,κ,T→∞

κ2

t logEνρ,0 ‚

exp –

α Z t

0

ds

e−1κψAe

1

κψ

− A1

κψ

(Zs)

™Œ

≤ α

6ρ(1−ρ)G. (3.11) Proposition 3.4. For allα >0,

lim

t,κ,T→∞

κ2

t logEνρ,0

exp

α κ

Z t

0

ds STφ

(Zs)

=6α2ρ(1−ρ)2_P3. (3.12) Proposition 3.3 has already been proven in[3], Proposition 4.4.2. Sections 4–5 are dedicated to the proof of the lower, respectively, upper bound in Proposition 3.4.

4

Proof of Proposition 3.4: lower bound

In this section we derive the lower bound in Proposition 3.4. We fixα,κ,T>0 and use Lemma 3.1, to obtain

lim

t→∞

1

t logEνρ,0

exp _α

κ Z t

0

ds _STφ(Z_s)

= sup

F∈D(A)

kFk_L2(µρ)=1

ZZ

Ω×Z3 dµρ

_α κ STφ

F2+F_AF. (4.1) In Section 4.1 we choose a test function. In Section 4.2 we compute and estimate the resulting expression. In Section 4.3 we take the limitκ,T _{→ ∞}and show that this gives the desired lower bound.

4.1

Choice of test function

To get the desired lower bound, we use test functionsF of the form

F(η,x) =F1(η)F2(x). (4.2)

Before specifyingF1 andF2, we introduce some further notation. In addition to the counting mea-sure m on Z3_{, consider the discrete Lebesgue measure mκ on} Z3

κ = κ−1Z3 giving weightκ−3 to

each site inZ3

κ. Letl2(Z3)andl2(Z3κ)denote the correspondingl2-spaces. Let∆κdenote the lattice

Laplacian onZ3

κdefined by

∆_κf(x) =κ2 X

y∈Z3_κ

ky−xk=κ−1

f(y)₋f(x). (4.3)

Choose f ∈ Cc∞(R 3_)with

kfkL2₍R3₎=1 arbitrarily, whereC_c∞(R3) is the set of infinitely

differen-tiable functions onR3 _{with compact support. Define}

fκ(x) =κ−3/2f κ−1x

, x ∈Z3_{, (4.4)}

and note that

kfκkl2₍Z3₎=kfk_l2₍Z3

κ)→1 asκ→ ∞. (4.5)

ForF₂choose

F₂=_kf_κk−1

l2_(Z3₎fκ. (4.6)

To choose F₁, introduce the function e

φ(η) = α

kfκk2_l2_(Z3₎

X

x∈Z3

STφ

(η,x)f_κ2(x). (4.7)

GivenK>0, abbreviate

S=6T1[κ] and U =6Kκ21[κ] (4.8)

(recall (2.24)). Forκ >pT/K, defineψe:Ω_→R_by

e

ψ=

Z U−S

0

where(_Tt)_t≥0 is the semigroup generated by the operator Lin (1.4). Note that the construction of e

ψfromφein (4.9) is similar to the construction ofψfromφin (2.10). In particular,

−Lψe=φe− TU−Sφ.e (4.10)

Combining the probabilistic representations of the semigroups (_St)_t≥0 (generated by_A in (2.8)) and(_Tt)_t≥0 (generated byLin (1.4)) with the graphical representation formulas (2.21–2.22), and using (4.4–4.5), we find that

e

φ(η) = α

kf_k2

l2_(Z3

κ) Z

Z3

κ

mκ(d x)f2(x)

X

z∈Z3

pS(κx,z)[η(z)−ρ] (4.11)

and

e

ψ(η) = X z∈Z3

h(z)[η(z)₋ρ] (4.12)

with

h(z) = α

kfk2

l2₍Z3

κ) Z

Z3

κ

m_κ(d x)f2(x)

Z U S

ds p_s(κx,z). (4.13)

Using the second inequality in (2.20), we have 0_≤h(z)_{≤ p}Cα

T, z∈

Z3_{. (4.14)}

Now chooseF₁ as

F1= eψe−_L21_(ν

ρ)e

e

ψ_{. (4.15)}

For the above choice of F1 and F2, we have kF1kL2_(ν

ρ) = kF2kl2(Z3) = 1 and, consequently,

kF_kL2_(µ

ρ) = 1. With F1, F2 and φe as above, and A as in (2.8), after scaling space byκ we ar-rive at the following lemma.

Lemma 4.1. For F as in(4.2),(4.6)and(4.15), allα,T,K>0andκ >pT/K,

κ2 ZZ

Ω×Z3 dµρ

_α κ STφ

F2+F_AF

= 1

kfk2

l2_(Z3

κ) Z

Z3

κ

d mκf∆κf +

κ

keψek2

L2_(ν

ρ) Z

Ω dνρ

e

φe2ψe+eψeLeψe,

(4.16)

whereφeandψeare as in(4.7)and(4.9).

4.2

Computation of the r.h.s. of (4.16)

Clearly, asκ→ ∞the first summand in the r.h.s. of (4.16) converges to Z

R3

d x f(x) ∆f(x) =_−∇R3f

2

L2_(R3₎. (4.17)

Lemma 4.2. For allα >0and0< ε <K,

lim inf

κ,T→∞

κ

keψe_k2

L2_(ν

ρ) Z

Ω dνρ

e

φe2ψe+eψeLeψe

≥6α2ρ(1₋ρ)

Z

R3

d x f2(x)

Z

R3

d y f2(y)

‚ Z 6K

6ε

d t p(_tG)(x,y)₋

Z 12K

6K

d t p(_tG)(x,y)

Π,

(4.18)

where

p(_tG)(x,y) = (4πt)−3/2exp[_−kx₋y_k2/4t] (4.19)

denotes the Gaussian transition kernel associated with∆R3, the continuous Laplacian onR3. Proof. Using the probability measure

dν_ρnew=eψe−_L22_(ν

ρ)e 2ψe_dν

ρ (4.20)

in combination with (4.10), we may write the term under the lim inf in (4.18) in the form κ

Z

Ω

dν_ρnewe−ψeLeψe₋Lψe+_TU−Sφe. (4.21)

This expression can be handled by making a Taylor expansion of the L-terms and showing that the

TU−S-term is nonnegative. Indeed, by the definition of Lin (1.4), we have

e−ψeLeψe−Lψe(η) = 1

6 X

{a,b}

e[ψ(ηe a,b)−ψ(η)]e −1−hψ ηe a,b−ψe(η)i. (4.22)

Recalling the expressions forψein (4.12–4.13) and using (4.14), we get fora,b_∈Z3_withka−bk=

1,

eψ ηa,b−ψe(η)=_|h(a)₋h(b)_{| |}η(b)₋η(a)_{| ≤ p}Cα

T. (4.23)

Hence, a Taylor expansion of the exponent in the r.h.s. of (4.22) gives Z

Ω

dν_ρnewe−ψeLeψe₋Lψe≥ e

−Cα/pT

12 Z

Ω

dν_ρnewX

{a,b}

h e

ψ ηa,b−ψe(η)i2. (4.24) Using (4.12), we obtain

Z

Ω

ν_ρnew(dη)X {a,b}

h e

ψ ηa,b−ψe(η)i2= X {a,b}

h(a)₋h(b)2

Z

Ω

ν_ρnew(dη)η(b)₋η(a)2. (4.25) Using (4.20), we have (after cancellation of factors not depending onaor b)

Z

Ω

ν_ρnew(dη)η(b)₋η(a)2=

Z

Ω

νρ(dη)e2χa,b(η)

η(b)₋η(a)2

Z

Ω

νρ(dη)e2χa,b(η)

with

χa,b(η) =h(a)η(a) +h(b)η(b). (4.27)

Using (4.14), we obtain that Z

Ω

ν_ρnew(dη)η(b)₋η(a)2_≥e−4Cα/pT

Z

Ω

νρ(dη)

η(b)₋η(a)2=e−4Cα/pT2ρ(1₋ρ). (4.28) On the other hand, by (4.13),

X

{a,b}

h(a)₋h(b)2= α

2

kf_k4

l2₍Z3

κ) Z U

S

d t

Z U S

ds

Z

Z3

κ

m_κ(d x)f2(x)

Z

Z3

κ

m_κ(d y)f2(y)

× X

{a,b}

pt(κx,a)−pt(κx,b)

ps(κy,a)−ps(κy,b)

(4.29)

with X

{a,b}

p_t(κx,a)₋p_t(κx,b)p_s(κy,a)₋p_s(κy,b)=₋X a∈Z3

p_t(κx,a)∆p_s(κx,a)

=₋6X

a∈Z3

p_t(κx,a)

_∂

∂sps(κy,a)

,

(4.30)

where∆acts on the first spatial variable ofp_s(_·,_·)and∆p_s=6(∂p_s/∂s). Therefore,

(4.29) =6 Z U

S

d t

Z

Z3

κ

mκ(d x)f2(x)

Z

Z3

κ

mκ(d y)f2(y)

X

a∈Z3

p_t(κx,a)p_S(κy,a)₋p_U(κy,a)

=6 Z

Z3

κ

mκ(d x)f2(x)

Z

Z3

κ

mκ(d y)f2(y)

‚ Z S+U

2S

d t p_t(κx,κy)₋

Z 2U U+S

d t p_t(κx,κy)

Œ . (4.31) Combining (4.24–4.25) and (4.28–4.29) and (4.31), we arrive at

Z

Ω

dν_ρnewe−ψeLeψe−Lψe≥ e

−5Cα/pT_α2

kfk4_l2_(Z3

κ)

ρ(1−ρ)

Z

Z3

κ

mκ(d x)f2(x)

Z

Z3

κ

mκ(d y)f2(y)

×

‚ Z S+U

2S

d t p_t(κx,κy)₋

Z 2U U+S

d t p_t(κx,κy)

Π.

(4.32)

After replacing 2Sin the first integral by 6εκ21[κ], using a Gaussian approximation of the transition kernelp_t(x,y)and recalling the definitions ofSandU in (4.8), we get that, for anyε >0,

lim inf

κ,T→∞κ

Z

Ω dν_ρnew

e−ψeLeψe₋Lψe

≥6α2ρ(1−ρ)

Z

R3

d x f2(x)

Z

R3

d y f2(y)

‚ Z 6K

6ε

d t p(_tG)(x,y)₋

Z 12K

6K

d t p(_tG)(x,y)

Π.

At this point it only remains to check that the_TU−S-term in (4.21) is nonnegative. By (4.11) and the probabilistic representation of the semigroup(_Tt)_t≥0, we have

Z

Ω

dν_ρnewTU−Sφe=

α

kfk2

l2_(Z3

κ) Z

Z3

κ

mκ(d x)f2(x)

X

z∈Z3

p_U(κx,z)

Z

Ω

ν_ρnew(dη)[η(z)₋ρ] (4.34) and, by (4.20),

Z

Ω

ν_ρnew(dη)[η(z)₋ρ] =₋ρ+ ρe

2h(z)

ρe2h(z)_+1−ρ =−ρ+

ρ

1₋(1₋ρ) 1₋e−2h(z)

≥ −ρ+ρh1+ (1−ρ)1−e−2h(z)

i

=ρ(1−ρ)1−e−2h(z)

,

(4.35)

which proves the claim.

4.3

Proof of the lower bound in Proposition 3.4

We finish by using Lemma 4.2 to prove the lower bound in Proposition 3.4.

Proof. Combining (4.16–4.18), we get lim inf

κ,T→∞κ

2 ZZ

Ω×Z3 dµρ

_α κ STφ

F2₋F_AF

≥6α2ρ(1₋ρ)

Z

R3

d x f2(x)

Z

R3

d y f2(y)

‚ Z 6K

6ε

d t p(_tG)(x,y)₋

Z 12K

6K

d t p(_tG)(x,y)

Œ

−∇R3f

2

L2_(R3₎.

(4.36)

Lettingε ↓0, K → ∞, replacing f(x) byγ3/2_{f(γx)withγ= 6α}2_{ρ(1−ρ), taking the supremum} over all f _∈C_c∞(R3_{)such thatkfkL}2_(R3₎=1 and recalling (4.1), we arrive at

lim inf

t,κ,T→∞

κ2

t logEνρ,0

exp

α κ

Z t

0

ds STφ

(Z_s)

≥6α2ρ(1−ρ)2_P3, (4.37) which is the desired inequality.

5

Proof of Proposition 3.4: upper bound

In this section we prove the upper bound in Proposition 3.4. The proof is long and technical. In Sections 5.1 we “freeze” and “defreeze” the exclusion dynamics on long time intervals. This allows us to approximate the relevant functionals of the random walk in terms of its occupation time measures on those intervals. In Section 5.2 we use a spectral bound to reduce the study of the long-time asymptotics for the resulting dependent potentials to the investigation of time-independent potentials. In Section 5.3 we make a cut-off for small times, showing that these times are negligible in the limit as κ → ∞, perform a space-time scaling and compactification of the underlying random walk, and apply a large deviation principle for the occupation time measures, culminating in the appearance of the variational expression for the polaron termP3.

5.1

Freezing, defreezing and reduction to two key lemmas

5.1.1 Freezing

We begin by deriving a preliminary upper bound for the expectation in Proposition 3.4 given by

E_ν

ρ,0

exp Z t

0

ds V(Z_s)

(5.1) with

V(η,x) =α

κ STφ

(η,x) = α

κ X

y∈Z3

p6T1[κ](x,y)(η(y)−ρ), (5.2)

where, as before, T is a large constant. To this end, we divide the time interval[0,t]into⌊t/Rκ⌋

intervals of length

R_κ=Rκ2 (5.3)

withR a large constant, and “freeze” the exclusion dynamics(ξt/κ)t≥0 on each of these intervals. As will become clear later on, this procedure allows us to express the dependence of (5.1) on the random walkX in terms of objects that are close to integrals over occupation time measures ofX on time intervals of lengthRκ. We will see that the resulting expression can be estimated from above by

“defreezing” the exclusion dynamics. We will subsequently see that, after we have taken the limits

t _{→ ∞}, κ→ ∞and T _{→ ∞}, the resulting estimate can be handled by applying a large deviation principle for the space-time rescaled occupation time measures in the limit asR → ∞. The latter will lead us to the polaron term.

Ignoring the negligible final time interval[_⌊t/Rκ⌋Rκ,t], using Hölder’s inequality withp,q>1 and

1/p+1/q=1, and inserting (5.2), we see that (5.1) may be estimated from above as

E_ν

ρ,0

exp

Z⌊t/R_κ⌋R_κ

0

ds V(Zs)

=E_ν

ρ,0 ‚

exp –

α κ

⌊tX/Rκ⌋

k=1 Z kRκ

(k−1)R_κ

ds X

y∈Z3

p6T1[κ](Xs,y) ξs/κ(y)−ρ

™Œ

≤ER(1,α)q(t)

1/q

ER(2,α)p(t)

1/p

(5.4)

with

ER(1,α)(t) =E (1)

R,α(κ,T;t) =Eνρ,0 ‚

exp –

α κ

⌊t_X/R_κ⌋ k=1

Z kR_κ (k−1)Rκ

ds X

y∈Z3

p_6T1[κ](X_s,y)ξs

κ(y)

−p_6T1[κ]+s−(k−1)Rκ κ

(X_s,y)ξ(k−1)Rκ κ

(y)

™Œ (5.5)

and

ER(2,α)(t) =E (2)

R,α(κ,T;t) =E_ν

ρ,0 ‚

exp –

α κ

⌊t_X/R_κ⌋ k=1

Z kR_κ (k−1)R_κ

ds X

y∈Z3

p_6T1[κ]+s−(k−1)R_κ

κ

(Xs,y)

ξ(k−1)R_κ

κ

Therefore, by choosing pclose to 1, the proof of the upper bound in Proposition 3.4 reduces to the proof of the following two lemmas.

Lemma 5.1. For all R,α >0,

lim sup

t,κ,T→∞

κ2

t logE (1)

R,α(κ,T;t)≤0. (5.7)

Lemma 5.2. For allα >0,

lim sup

R→∞

lim sup

t,κ,T→∞

κ2

t logE (2)

R,α(κ,T;t)≤

6α2ρ(1₋ρ)2_P3. (5.8)

Lemma 5.1 will be proved in Section 5.1.2, Lemma 5.2 in Sections 5.1.3–5.3.3.

5.1.2 Proof of Lemma 5.1

Proof. Fix R,α > 0 arbitrarily. Given a pathX, an initial configurationη ∈Ω and k _∈N_{, we first}

derive an upper bound for

E_η

‚ exp

– α κ

Z Rκ 0

ds X

y∈Z3

p6T1[κ]

X_s(k,κ),y

ξs

κ(y)−p6T1[κ]+

s

κ

X_s(k,κ),y

η(y)

™Œ

, (5.9)

where

X_s(k,κ)=X(k−1)Rκ+s. (5.10)

To this end, we use the independent random walk approximationξeofξ(cf.[3], Proposition 1.2.1), to obtain

(5.9)_≤ Y y∈A_η

EY₀ ‚

exp –

α κ

Z Rκ 0

ds

p6T1[κ]

X_s(k,κ),y+Ys

κ

−p6T1[κ]+s

κ

X_s(k,κ),y

™Œ

, (5.11)

whereY is simple random walk onZ3 _{with jump rate 1 (i.e., with generator} 1

6∆), E

Y

0 is expectation w.r.t.Y starting from 0, and

Aη={x∈Z3:η(x) =1}. (5.12)

Observe that the expectation w.r.t.Y of the expression in the exponent is zero. Therefore, a Taylor expansion of the exponential function yields the bound

EY₀ ‚

exp –

α κ

Z Rκ 0

ds

p6T1[κ]

X_s(k,κ),y+Ys

κ

−p6T1[κ]+_κs

X_s(k,κ),y

™Œ

≤1+ ∞

X

n=2

n

Y

l=1 ‚

α κ

Z R_κ sl−1

ds_l X

yl∈Z3 psl−sl−1

κ

(y_l−1,y_l)

×

p_6T1[κ]X_s(k,κ) l ,y+yl

+p_6T1[κ]+sl

κ

X_s(k,κ) l ,y

Π,

wheres₀ =0, y₀ =0, and the product has to be understood in a noncommutative way. Using the Chapman-Kolmogorov equation and the inequalitypt(z)≤pt(0),z∈Z3, we find that

Z Rκ

sl−1

ds_l X

yl∈Z3 psl−sl−1

κ

(y_l−1,y_l)

p_6T1[κ]

X_s(k,κ) l ,y+ yl

+p_6T1[κ]+sl

κ

X_s(k,κ) l ,y

Œ

≤2 Z ∞

0

ds pT+_κs(0) =2κGT(0)

(5.14)

with

G_T(0) =

Z _∞

T

ds p_s(0) (5.15)

the cut-off Green function of simple random walk at 0 at time T. Substituting this into the above bound for l = n,n−1,· · ·, 3, computing the resulting geometric series, and using the inequality 1+x _≤ex, we obtain

(5.13)_≤exp –

CTα2

κ2 2 Y

l=1 Z R_κ

sl−1

ds_l X

yl∈Z3 psl−sl−1

κ

(y_l−1,y_l)

×

p_6T1[κ]

X_s(k,κ) l ,y+yl

+p_6T1[κ]+sl

κ

X_s(k,κ) l ,y

™ (5.16)

with

C_T = 1

1−2αG_T(0), (5.17)

provided that 2αGT(0) < 1, which is true for T large enough. Note that CT → 1 as T → ∞.

Substituting (5.16) into (5.11), we find that

(5.9)_≤exp –

CTα2

κ2 X

y∈Z3

2 Y

l=1 Z Rκ

sl−1

ds_l X

yl∈Z3 psl−sl−1

κ

(y_l−1,y_l)

×

p6T1[κ]

X_s(k,κ) l ,y+yl

+p_6T1[κ]+sl

κ

X_s(k,κ) l ,y

™ .

(5.18)

Using once more the Chapman-Kolmogorov equation and pt(x,y) = pt(x− y), we may compute

the sums in the exponent, to arrive at

(5.9)_≤exp –

C_Tα2 κ2

Z R_κ

0

ds1 Z R_κ

s1 ds2

p_12T1[κ]+s₂−s₁

κ

X_s(k,κ) 2 −X

(k,κ) s1

+3p_12T1[κ]+s₂+s₁

κ

X_s(k,κ) 2 −X

(k,κ) s1

™ .

(5.19)

Note that this bound does not depend on the initial configuration η and depends on the process

X only via its increments on the time interval [(k−1)Rκ,kRκ]. By (5.10), the increments over

the time intervals labelled k=1, 2,_{· · ·},_⌊t/Rκ⌋ are independent and identically distributed. Using

E_ν

ρ,0 = R

(ξt/κ)t≥0 at timesRκ, 2Rκ, · · ·, (⌊t/Rκ⌋ −1)Rκ to the expectation in the r.h.s. of (5.5), insert the

bound (5.19) and afterwards use that(Xt)t≥0 has independent increments, to arrive at log_ER(1_,α)(t)_≤ t

Rκ

log EX₀ ‚

exp –

CTα2

κ2 Z Rκ

0

ds₁

Z Rκ

s1 ds₂

p_12T1[κ]+s2−s1

κ

X_s2−X_s1

+3p_12T1[κ]+s2+s1

κ

X_s 2−Xs1

™Œ

.

(5.20)

Hence, recalling the definition ofR_κ in (5.3), we obtain lim sup

t→∞

κ2

t logE (1) R,α(t)

≤ _R1log EX₀ ‚

exp –

C_Tα2R R_κ

Z Rκ 0

ds₁

Z Rκ

s1 ds₂

p_12T1[κ]+s₂−s₁

κ

X_s2−X_s1

+3p_12T1[κ]+s2+s1

κ

X_s2−X_s1

™Œ .

(5.21)

Let

b

X_t=X_t+Y_t/κ, (5.22)

and let EX₀b =EX₀EY₀ be the expectation w.r.t.Xbstarting at 0. Observe that

p_t+s/κ(z) =EY₀p_t z+Y_s/κ. (5.23)

We next apply Jensen’s inequality w.r.t. the first integral in the r.h.s. of (5.21), substitutes₂=s₁+s, take into account thatX has independent increments, and afterwards apply Jensen’s inequality w.r.t. EY₀, to arrive at the following upper bound for the expectation in (5.21):

EX₀ ‚

exp –

CTα2R

Rκ

Z R_κ

0

ds₁

Z R_κ s1

ds₂

p_12T1[κ]+s2−s1

κ

X_s2−X_s1

+3p_12T1[κ]+s2+s1

κ

X_s 2−Xs1

™

≤ 1 Rκ

Z R_κ

0

ds₁E₀X ‚

exp –

C_Tα2R

Z ∞

0

dsEY₀

p_12T1[κ]X_s+Ys

κ

+3p_12T1[κ]+2s₁

κ

X_s+Ys

κ ™Œ

≤ 1 Rκ

Z R_κ

0

ds₁E₀Xb ‚

exp –

C_Tα2R

Z ∞

0

dsp_12T1[κ] Xb_s+3p_12T1[κ]+2s₁

κ b

X_s

™Œ .

(5.24)

Applying Lemma 2.6, we can bound the last expression from above by exp

–

4C_Tα2_RGb 2T(0)

1₋4C_Tα2_RGb 2T(0)

™

, (5.25)

where Gb2T(0) is the cut-off at time 2T of the Green function Gb at 0 for Xb (which has generator

1[κ]∆). Since Gb_2T(0)_→ 1

6G12T(0)asκ→ ∞, and since the latter converges to zero asT → ∞, a combination of the above estimates with (5.21) gives the claim.

where_M1(Q)is the space of probability measures onQ, andH_per1 (Q)denotes the space of functions inH1(Q)with periodic boundary conditions. By[2], Lemma 7.4, we have

lim sup Q↑R3

P3(Q)(θ;ε,K)≤ P3(θ;ε,K) (5.119)

with

P3(θ;ε,K)

= sup

f∈H1(R3)

kfk2=1

6θ2α2ρ(1₋ρ) Z

R3

d x f2(x) Z

R3

d y f2(y) Z K

ε

d r p(_rG)(x,y)_−∇R3f 2

L2_(R3₎

≤ sup

f∈H1(R3)

kfk2=1

6θ2α2ρ(1₋ρ) Z

R3

d x f2(x) Z

R3

d y f2(y) Z ∞

0

d r p(_rG)(x,y)_−∇R3f 2

L2_(R3₎

=6θ2α2ρ(1₋ρ)2_P3.

(5.120)

Combining (5.116) and (5.120), and lettingθ↓1, we arrive at the claim of Lemma 5.13.

This, after a long struggle by the authors and considerable patience on the side of the reader, com-pletes the proof of the upper bound in Proposition 3.4.

Remark. The reader might be surprised that the expression in the l.h.s. of (5.98) does not only vanish in the limit asε↓0 but vanishes forallε >0 sufficiently small. This fact is closely related to the observation that

P3 π3

=0 whereas _P3(_∞) =_P3>0 (5.121) with

P3(ε) = sup

f∈H1(R3)

kfk2=1 – Z

R3

d x f2(x) Z

R3

d y f2(y) Z ε

0

d r p(_rG)(x₋y)_−∇R3f 2

2

™

. (5.122)

Indeed, given a potential V _≥ 0 with _kGR3Vk_∞ < 1/2, where GR3 denotes the Green operator associated with∆R3, the method used in the proof of Lemma 5.12 leads to

lim R→∞

1

Rlog E

β

0

‚ exp

– 1

R

Z R

0 ds

Z R

0

des V(β_es−βs) ™Œ

=0. (5.123)

On the other hand, the large deviation principle for the occupation time measures ofβ shows that this limit coincides with

sup

f∈H1(R3)

kfk2=1 – Z

R3

d x f2(x) Z

R3

d y f2(y)V(x− y)_−∇R3f 2

2

™

. (5.124)

But, for 0< ε < π3the potential

Vε(x) =

Z ε

0

d r p(_rG)(x) (5.125)

6

Higher moments

In this last section we explain how to extend the proof of Theorem 1.1 to higher momentsp _≥2. Sections 6.1–6.3 parallel Sections 2.1, 3.2, 4 and 5.

6.1

Two key propositions

Our starting point is the Feynman-Kac representation for thep-th moment, E_ν

ρ

u(0,t)p=E(p)

ν_ρ;0

‚ exp

– Z t

0 ds

p X

j=1

ξ_s X_κj_s

™Œ

, (6.1)

where X1,· · ·,Xp are independent simple random walks on Z3 _{starting at 0 and} E(_νp)

ρ;x denotes

expectation w.r.t.P(_νp)

ρ;x =Pνρ⊗P

X1

x1⊗ · · · ⊗P Xp

xp, x = (x1,· · ·,xp)∈(

Z3₎p_.

The arguments in Sections 2 and 3 easily extend to this more general case by replacing Z, A, (_St)_t≥0, φ andψ by their p-dimensional analogues Z(p), _A(p), (_St(p))_t≥0, φ(p) and ψ(p). To be precise, consider the Markov process

Z_t(p)= ξt/κ,X1t,· · ·,X p t

onΩ_×(Z3₎p _(6.2)

with generator

A(p)= 1

κL+

p X

j=1

∆_j, (6.3)

where the lattice Laplacian∆_j acts on the j-th spatial variable. Denote by(_St(p))_t≥0 the associated semigroup. We define

φ(p)(η;x₁,_{· · ·},x_p) = p X

j=1

φ(η,x_j) = p X

j=1

(η(x_j)₋ρ) (6.4) and

ψ(p)= Z T

0

ds_Ss(p)φ(p). (6.5)

Then

ψ(p)(η;x₁,_{· · ·},x_p) = p X

j=1

ψ(η,x_j). (6.6)

In this way the proof of Theorem 1.1 for p_≥2 reduces to the proof of the following extension of Propositions 3.3 and 3.4.

Proposition 6.1. For all p_∈N_andα∈R_, lim sup

t,κ,T→∞

κ2

ptlogE (p)

ν_ρ;0

‚ exp

–

α

Z t

0

dse−κ1ψ

(p)

A(p)eκ1ψ

(p)

− A(p)1

κψ

(p) _Z(p)

s ™Œ ≤ α

6ρ(1−ρ)G.

Proposition 6.2. For all p_∈N_{andα >0,} lim

t,κ,T→∞

κ2 ptlogE

(p)

νρ;0

‚ exp

–

α κ

Z t

0

ds_ST(p)φ(p) Z_s(p)

™Œ

=6α2ρ(1₋ρ)p2_P3. (6.8)

Proposition 6.1 has already been proven for allp_∈N_{in[3], Proposition 4.4.2 and Section 4.8.}

6.2

Lower bound in Proposition 6.2

We use the following analogue of the variational representation (4.1) lim

t→∞ 1

t logE (p)

ν_ρ;0

‚ exp

–

α κ

Z t

0 ds

ST(p)φ

(p) _Z(p)

s ™Œ

= sup

F(p)∈D(A(p)) kF(p)k

L2(µ_ρp)=1

ZZ

Ω×(Z3)p

dµ_ρp

α κ

ST(p)φ

(p) _F(p)2

+F(p)_A(p)F(p)

. (6.9)

To obtain the appropriate lower bound, we use test functionsF(p)of the form

F(p)(η;x₁,_{· · ·},x_p) =F₁(η)F₂(x₁)_{· · ·}F₂(x_p), (6.10) whereF1,F2 andF =F(1)are the same as in (4.15), (4.6) and (4.2), respectively. One easily checks

that

κ2

p

ZZ

Ω×(Z3)p

dµ_ρp

_α

κ

ST(p)φ

(p) _F(p)2

+F(p)_A(p)F(p)

=(pκ)

2 p2

ZZ

Ω×Z3

dµρ

_p

α

pκ

STφF2+F

₁

pκL+ ∆

F

.

(6.11)

But this is 1/p2times the expression in Lemma 4.1 withαandκreplaced bypαandpκ, respectively. Hence, we may use the lower bounds for this expression in Section 4 to arrive at the lower bound in Proposition 6.2.

6.3

Upper bound in Proposition 6.2

1. Freezing and defreezing can be done in the same way as in Section 5.1, but withV(η,x)in (5.2) replaced by

V(p)(η,x) =α

κ

X y∈Z3

‚_Xp j=1

p_6T1[κ] x_j,y

Œ

η(y)₋ρ. (6.12)

This leads to the analogues of Lemmas 5.1 and 5.3 along the lines of Sections 5.1.2 and 5.1.4. 2. Considering

V_k(_,p_u)(η) = 1

Rκ

Z (k+1)R_κ kRκ

ds X

y∈Z3 ‚_Xp

j=1

p_6T1[κ]+s−u

κ X

j s,y

Œ

and

ER(4,,αp)(t) =Eν_ρ;0

‚ exp

–

α κ

⌊t_X/R_κ⌋ k=1

Z kRκ (k−1)R_κ

du V_k(_,p_u) ξu/κ

™Œ

(6.14) instead of (5.34–5.35), the proof reduces to the following analogue of Lemma 5.4.

Lemma 6.3. For eachα >0,

lim sup R→∞

lim sup t,κ,T→∞

κ2 ptlogE

(4,p)

R,α (t)≤

6α2ρ(1₋ρ)p2_P3. (6.15) 3. The proof of Lemma 6.3 follows the lines of Sections 5.2–5.3. The spectral bound is essentially the same as in Section 5.2.1. In Lemma 5.6 we have to replaceV_k,ubyV_k(_,p_u)andλk,uby

λ(_kp_,u)= lim t→∞

1

t logEνρ

‚ exp – α κ Z t 0

ds V_k(_,p_u) ξ_s/κ

™Œ

. (6.16)

Subsequently, we replaceV_k,ubyV_k(_,p_u)in (5.45), to obtain functionsφb(p),ψb(p)replacing (5.45–5.46), and

Ξ(_rp)(x) = 1

Rκ

Z (k+1)Rκ

kRκ ds

p X

j=1

p_6T1[κ]+s−u+r

κ X

j s,x

(6.17) replacing (5.50). Then, in the analogue of Lemma 5.7, instead of (5.58) we get the bound

_K(p)

k,u

1≤e

2Cα/T 2α

2

κ2_R2

κ

Z (k+1)R_κ kR_κ

ds

Z (k+1)R_κ kR_κ

d_es

Z M

0 d r

p X i,j=1

p_12T1[κ]+s+es−2u+2r

κ Xj

e

s −X i s

(6.18) along the line of Section 5.2.3. Similarly, the proof of the analogue of Lemma 5.8 follows the argument in Section 5.2.4, leading to a reduction of Lemma 6.3 to the analogue of Lemma 5.11, as in Section 5.2.5.

4. To make the small-time cut-off, instead of (5.95) we consider ER(8,,αp)(κ)

=EX₀ ‚

exp –

Θ_α,T,ρ

κ2_R2

κ

Z R_κ

0 ds

Z R_κ s

d_es

Z 0

−R_κ

du Z m 0 d r p X i,j=1

p_12T1[κ]+s+es−2u+2r

κ Xj

e

s −X i s

™Œ

. (6.19) Using the Chapman-Kolmogorov equation, we see that

Z R_κ

0 ds

Z R_κ

0 des

p X i,j=1

p_12T1[κ]+s+es−2u+2r

κ

X_esj−X_si

= X

z∈Z3 ‚_Xp

i=1

Z R_κ

0

ds p_6T1[κ]+s−u+r

κ X

i s,z

Œ2

≤pX

z∈Z3 p X i=1

‚ Z Rκ 0

ds p_6T1[κ]+s−u+r

κ X

i s,z

Œ2

=p

p X

i=1

Z Rκ 0

ds

Z Rκ 0

des p_12T1[κ]+s+es−2u+2r

κ

X_esi−X_si.

Substituting this into the r.h.s. of (6.19) and applying Hölder’s inequality for the p exponential factors, we reduce the problem to the consideration of a single random walk and can proceed as in Section 5.3, leading to the analogues of Lemmas 5.12–5.13.

5. The proof of the analogue of Lemma 5.12 runs along the line of Section 5.3.2. To prove the analogue of Lemma 5.13, instead of (5.96) we consider

ER(9,,αp)(κ)

=EX₀ ‚

exp –

Θα,T,ρ κ2_R2

κ

Z Rκ 0

ds

Z Rκ

s

des

Z 0

−R_κ

du

Z M m

d r

p X i,j=1

p_12T1[κ]+s+es−2u+2r

κ

X_esj−X_si

™Œ

. (6.21) As in Section 5.3.3, this leads to

lim sup

κ,T,R→∞ 1

pRlogE (9,p)

R,α (κ)

≤ 1

p νi∈Msup1(Q) 1≤i≤p

–

6θ2α2ρ(1₋ρ) p X i,j=1

Z Q

νi(d x) Z

Q

νj(d y) Z K

ε

d r p(_rG,Q)(x,y)₋ p X i=1

SQ(νi)

™ _(6.22)

instead of (5.116–5.117). Now we can proceed similarly as in[2], Lemma 7.3. Consider the Fourier transformsνb_i of the measuresν_i∈ M1(Q)defined by

b

νj(k) = Z

Q

e−i(2π/l(Q))k·xνj(d x), k∈Z3,j=1,· · ·,p. (6.23) The transition kernel p(G,Q)admits the Fourier representation

p(_rG,Q)(x) = 1

l(Q)3

X k∈Z3

e−(2π/l(Q))2|k|2re−i(2π/l(Q))k·x, (x,t)_∈Q×(0,_∞). (6.24) Therefore we may write

Z Q

νi(d x) Z

Q

νj(d y)pr(G,Q)(x,y) = 1

l(Q)3

X k∈Z3

e−(2π/l(Q))2|k|2rν_bi(k)νbj(k). (6.25) Using that

Re

b

νi(k)νbj(k)

≤1₂bνi(k) 2

+1 2 bνj(k)

2

, (6.26)

we obtain Z Q

ν_i(d x) Z

Q

ν_j(d y)p(_rG,Q)(x,y) ≤ 1

2 Z

Q

νi(d x) Z

Q

νi(d y)pr(G,Q)(x,y) + 1 2 Z

Q

νj(d x) Z

Q

νj(d y)pr(G,Q)(x,y).

(6.27)

Therefore the term inside the square brackets in the r.h.s. of (6.22) does not exceed p

X i=1

–

6θ2α2ρ(1₋ρ)p

Z Q

νi(d x) Z

Q

νi(d y) Z K

ε

d r p(_rG,Q)(x,y)₋SQ(νi) ™

and we arrive at

lim sup

κ,T,R→∞

p RlogE

(9,p)

R,α (κ)≤ P

(Q,p)

3 (θ;ε,K) (6.29)

with

P3(Q,p)(θ;ε,K) = sup

ν∈M1(Q) –

6θ2α2ρ(1₋ρ)p

Z Q

ν(d x) Z

Q

ν(d y) Z K

ε

d r p(_rG,Q)(x,y)₋SQ(ν) ™

, (6.30) which is the analogue of (5.116–5.117) forp_≥2. The rest of the proof can be easily obtained from the analogues of (5.119–5.120).

References

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