acting on the Banach space of bounded continuous functions on Ω × Z
3
, equipped with the supre- mum norm. Let
S
t t
≥0
denote the semigroup generated by A .
Our aim is to make an absolutely continuous transformation of the measure P
η,x
with the help of an exponential martingale, in such a way that, under the new measure P
new η,x
, Z
t t
≥0
is a Markov process with generator
A
new
of the form A
new
f = e
−
1 κ
ψ
A e
1 κ
ψ
f −
e
−
1 κ
ψ
A e
1 κ
ψ
f . 2.9
This transformation leads to an interaction between the exclusion process part and the random walk part of Z
t t
≥0
, controlled by ψ: Ω × Z
3
→ R. As explained in [3], Section 4.2, it will be expedient to choose
ψ as ψ =
Z
T
ds S
s
φ 2.10
with T a large constant suppressed from the notation, implying that − A ψ = φ − S
T
φ. 2.11
It was shown in [3], Lemma 4.3.1, that N
t
= exp
1 κ
ψZ
t
− ψZ −
Z
t
ds e
−
1 κ
ψ
A e
1 κ
ψ
Z
s
2.12
is an exponential P
η,x
-martingale for all η, x ∈ Ω × Z
3
. Moreover, if we define P
new η,x
in such a way that
P
new η,x
A = E
η,x
N
t
1
1
A
2.13 for all events A in the
σ-algebra generated by Z
s s
∈[0,t]
, then under P
new η,x
indeed Z
s s
≥0
is a Markov process with generator
A
new
. Using 2.11–2.13 and E
new ν
ρ
,0
= R
Ω
ν
ρ
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
ds φZ
s
= E
new ν
ρ
,0
exp 1
κ ψZ
− ψZ
t
+ Z
t
ds e
−
1 κ
ψ
A e
1 κ
ψ
− A 1
κ ψ
Z
s
+ 1
κ Z
t
ds S
T
φ Z
s
. 2.14
The first term in the exponent in the r.h.s. of 2.14 stays bounded as t → ∞ and can therefore be
discarded when computing λ
∗
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 any κ 0,
λ
∗
κ ≤
≥ I
q 1
κ + I
r 2
κ 2.15
2097
with I
q 1
κ = 1
q lim
t →∞
1 t
log E
new ν
ρ
,0
exp q
Z
t
ds e
−
1 κ
ψ
A e
1 κ
ψ
− A 1
κ ψ
Z
s
, I
r 2
κ = 1
r lim
t →∞
1 t
log E
new ν
ρ
,0
exp r
κ Z
t
ds S
T
φ Z
s
, 2.16
where 1 q + 1r = 1, with q 0, r 1 in the first inequality and q 0, 0 r 1 in 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 choosing r 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 →∞
= lim
T →∞
lim
κ→∞
lim
t →∞
. 2.17
In the next proposition we write ψ
T
instead of ψ to indicate the dependence on the parameter T .
Proposition 2.2. For any α ∈ R,
lim sup
t, κ,T →∞
κ
2
t log E
new ν
ρ
,0
exp
α
Z
t
ds e
−
1 κ
ψ
T
A e
1 κ
ψ
T
− A 1
κ ψ
T
Z
s
≤
α 6
ρ1 − ρG. 2.18
Proposition 2.3. For any
α 0, lim
t, κ,T →∞
κ
2
t log E
new ν
ρ
,0
exp α
κ Z
t
ds S
T
φ Z
s
= [6α
2
ρ1 − ρ]
2
P
3
. 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
1 t
x, y and p
t
x, y = p
3 t
x, y be the transition kernels of simple random walk in d = 1 and d = 3, respectively, with step rate 1.
Lemma 2.4. There exists C 0 such that, for all t ≥ 0 and x, y, e ∈ Z
3
with kek = 1,
p
1 t
x, y ≤ C
1 + t
1 2
, p
t
x, y ≤ C
1 + t
3 2
, p
t
x + e, y − p
t
x, y ≤
C 1 + t
2
. 2.20
Proof. Standard. In the sequel we will frequently write p
t
x − y instead of p
t
x, y. 2098
From the graphical representation for SSE Liggett [7], Chapter VIII, Theorem 1.1 it is immediate that
E
η
ξ
t
x =
X
y ∈Z
d
p
t
x, y η y. 2.21
Recalling 2.4–2.5 and 2.10, we therefore have S
s
φη, x = E
η,x
φZ
s
= E
η
X
y ∈Z
3
p
6s
x, y ξ
s κ
y − ρ
= X
z ∈Z
3
p
6s1[ κ]
x, z ηz − ρ
2.22
and ψη, x =
Z
T
ds X
z ∈Z
3
p
6s1[ κ]
x, z ηz − ρ
, 2.23
where we abbreviate 1[
κ] = 1 + 1
6 κ
. 2.24
Lemma 2.5. For all
κ, T 0, η ∈ Ω, a, b ∈ Z
3
with ka − bk = 1 and x ∈ Z
3
, |ψη, b − ψη, a| ≤ 2C
p T
for T ≥ 1,
2.25 ψ η
a,b
, x − ψη, x
≤ 2G, 2.26
X
{a,b}
ψ η
a,b
, x − ψη, x
2
≤ 1
6 G,
2.27 where C
0 is the same constant as in Lemma 2.4, and G is the value at 0 of the Green function of simple random walk on Z
3
. 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 + e
1
with e
1
= 1, 0, 0. Then, by 2.23, we have |ψη, b − ψη, a| ≤
Z
T
ds X
z ∈Z
3
p
6s1[ κ]
z + e
1
− p
6s1[ κ]
z =
Z
T
ds X
z ∈Z
3
p
1 6s1[
κ]
z
1
+ e
1
− p
1 6s1[
κ]
z
1
p
1 6s1[
κ]
z
2
p
1 6s1[
κ]
z
3
= Z
T
ds X
z
1
∈Z
p
1 6s1[
κ]
z
1
+ e
1
− p
1 6s1[
κ]
z
1
= 2 Z
T
ds p
1 6s1[
κ]
0 ≤ 2C p
T . 2.28
In the last line we have used the first inequality in 2.20.
2099
Let G be the Green operator acting on functions V : Z
3
→ [0, ∞ as G V x =
X
y ∈Z
3
Gx − yV y,
x ∈ Z
3
, 2.29
with Gz = R
∞
d t p
t
z. Let k · k
∞
denote the supremum norm.
Lemma 2.6. For all V : Z
3
→ [0, ∞ and x ∈ Z
3
, E
X x
exp Z
∞
d t V X
t
≤ 1
− kG V k
∞ −1
≤ exp
kG V k
∞
1 − kG V k
∞
,
2.30 provided that
kG V k
∞
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 measure P
new ν
ρ
,0
are asymptotically the same as the expectations w.r.t. the old measure P
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
