where I
fin
Q := HQ | q
⊗N ρ,ν
+ α − 1 m
Q
HΨ
Q
| ν
⊗N
. 2.14
The rate function I
que
is lower semi-continuous, has compact level sets, has a unique zero at Q = q
⊗N ρ,ν
, and is affine. Moreover, it is equal to the lower semi-continuous extension of I
fin
from P
inv,fin
e E
N
to P
inv
e E
N
. b In particular, if 2.1 holds with
α = 1, then for ν
⊗N
–a.s. all X , the family PR
N
∈ · | X satisfies the LDP with rate function I
ann
given by 2.7. Note that the quenched rate function 2.14 equals the annealed rate function 2.7 plus an addi-
tional term that quantifies the deviation of Ψ
Q
from the reference law ν
⊗N
on the letter sequence. This term is explicit when m
Q
∞, but requires a truncation approximation when m
Q
= ∞. We close this section with the following observation. Let
R
ν
:= Q
∈ P
inv
e E
N
: w−lim
L →∞
1 L
L −1
X
k=0
δ
θ
k
κY
= ν
⊗N
Q − a.s.
. 2.15
be the set of Q’s for which the concatenation of words has the same statistical properties as the letter sequence X . Then, for Q
∈ P
inv,fin
e E
N
, we have see [6], Equation 1.22 Ψ
Q
= ν
⊗N
⇐⇒ I
que
Q = I
ann
Q ⇐⇒
Q ∈ R
ν
. 2.16
3 Proof of Theorems 1.1, 1.3 and 1.6
3.1 Proof of Theorem 1.1
The idea is to put the problem into the framework of 2.1–2.5 and then apply Theorem 2.2. To that end, we pick
E := Z
d
, e
E = Ý Z
d
:= ∪
n ∈N
Z
d n
, 3.1
and choose νu := pu,
u ∈ Z
d
, ρn :=
p
2 ⌊n2⌋
2 ¯ G0
− 1 ,
n ∈ N,
3.2 where
pu = p0, u, u ∈ Z
d
, p
n
v − u = p
n
u, v, u, v ∈ Z
d
, ¯
G0 =
∞
X
n=0
p
2n
0, 3.3
the latter being the Green function of S − S
′
at the origin. Recalling 1.2, and writing
z
V
= z − 1 + 1
V
= 1 +
V
X
N =1
z − 1
N
V N
3.4 with
V N
= X
j
1
··· j
N
∞
1
{S
j1
=S
′ j1
,...,S
jN
=S
′ jN
}
, 3.5
562
we have E
z
V
| S
= 1 +
∞
X
N =1
z − 1
N
F
1 N
X , E
z
V
= 1 +
∞
X
N =1
z − 1
N
F
2 N
, 3.6
with F
1 N
X := X
j
1
··· j
N
∞
P S
j
1
= S
′ j
1
, . . . , S
j
N
= S
′ j
N
| X ,
F
2 N
:= E F
1 N
X ,
3.7 where X = X
k k
∈N
denotes the sequence of increments of S. The upper indices 1 and 2 indicate the number of random walks being averaged over.
The notation in 3.1–3.2 allows us to rewrite the first formula in 3.7 as F
1 N
X = X
j
1
··· j
N
∞ N
Y
i=1
p
j
i
− j
i −1
j
i
− j
i −1
X
k=1
X
j
i −1
+k
= X
j
1
··· j
N
∞ N
Y
i=1
ρ j
i
− j
i −1
exp
N
X
i=1
log
p
j
i
− j
i −1
P
j
i
− j
i −1
k=1
X
j
i −1
+k
ρ j
i
− j
i −1
.
3.8
Let Y
i
= X
j
i −1
+1
, · · · , X
j
i
. Recall the definition of f : Ý Z
d
→ [0, ∞ in 1.5, f x
1
, . . . , x
n
= p
n
x
1
+ · · · + x
n
p
2 ⌊n2⌋
[2 ¯ G0
− 1], n
∈ N, x
1
, . . . , x
n
∈ Z
d
. 3.9
Note that, since Ý Z
d
carries the discrete topology, f is trivially continuous. Let R
N
∈ P
inv
Ý Z
d N
be the empirical process of words defined in 2.5, and π
1
R
N
∈ P Ý Z
d
the projection of R
N
onto the first coordinate. Then we have F
1 N
X = E
exp
N
X
i=1
log f Y
i
X
= E
exp
N Z
Ý Z
d
π
1
R
N
d y log f y
X
, 3.10
where P is the joint law of X and τ recall 2.2–2.3. By averaging 3.10 over X we obtain recall
the definition of F
2 N
from 3.7 F
2 N
= E
exp
N Z
Ý Z
d
π
1
R
N
d y log f y
. 3.11
Without conditioning on X , the sequence Y
i i
∈N
is i.i.d. with law recall 2.4 q
⊗N ρ,ν
with q
ρ,ν
x
1
, . . . , x
n
= p
2 ⌊n2⌋
2 ¯ G0
− 1
n
Y
k=1
px
k
, n
∈ N, x
1
, . . . , x
n
∈ Z
d
. 3.12
Next we note that f in 3.9 is bounded from above.
3.13 563
Indeed, the Fourier representation of p
n
x, y reads p
n
x = 1
2π
d
Z
[−π,π
d
d k e
−ik·x
bpk
n
3.14 with
bpk = P
x ∈Z
d
e
ik ·x
p0, x. Because p ·, · is symmetric, we have bpk ∈ [−1, 1], and it follows
that max
x ∈Z
d
p
2n
x = p
2n
0, max
x ∈Z
d
p
2n+1
x ≤ p
2n
0, ∀ n ∈ N.
3.15 Consequently, f x
1
, . . . , x
n
≤ [2 ¯ G0
− 1] is bounded from above. Therefore, by applying the annealed LDP in Theorem 2.1 to 3.11, in combination with Varadhan’s lemma see Dembo and
Zeitouni [14], Lemma 4.3.6, we get z
2
= 1 + exp[−r
2
] with r
2
:= lim
N →∞
1 N
log F
2 N
≤ sup
Q ∈P
inv
Ý Z
d N
¨Z
Ý Z
d
π
1
Qd y log f y − I
ann
Q «
= sup
q ∈P Ý
Z
d
¨Z
Ý Z
d
qd y log f y − hq | q
ρ,ν
« 3.16
recall 1.3–1.4 and 3.6. The second equality in 3.16 stems from the fact that, on the set of Q’s with a given marginal
π
1
Q = q, the function Q 7→ I
ann
Q = HQ | q
⊗N ρ,ν
has a unique minimiser Q = q
⊗N
due to convexity of relative entropy. We will see in a moment that the inequality in 3.16 actually is an equality.
In order to carry out the second supremum in 3.16, we use the following.
Lemma 3.1. Let Z :=
P
y ∈Ý
Z
d
f yq
ρ,ν
y. Then Z
Ý Z
d
qd y log f y − hq | q
ρ,ν
= log Z − hq | q
∗
∀ q ∈ P Ý Z
d
, 3.17
where q
∗
y := f yq
ρ,ν
yZ, y ∈ Ý Z
d
. Proof. This follows from a straightforward computation.
Inserting 3.17 into 3.16, we see that the suprema are uniquely attained at q = q
∗
and Q = Q
∗
= q
∗ ⊗N
, and that r
2
≤ log Z. From 3.9 and 3.12, we have Z =
X
n ∈N
X
x
1
,...,x
n
∈Z
d
p
n
x
1
+ · · · + x
n n
Y
k=1
px
k
= X
n ∈N
p
2n
0 = ¯ G0
− 1, 3.18
where we use that P
v ∈Z
d
p
m
u + vpv = p
m+1
u, u ∈ Z
d
, m ∈ N, and recall that ¯
G0 is the Green function at the origin associated with p
2
·, ·. Hence q
∗
is given by q
∗
x
1
, . . . , x
n
= p
n
x
1
+ · · · + x
n
¯ G0
− 1
n
Y
k=1
px
k
, n
∈ N, x
1
, . . . , x
n
∈ Z
d
. 3.19
564
Moreover, since z
2
= ¯ G0
[ ¯ G0
− 1], as noted in 1.15, we see that z
2
= 1 + exp[− log Z], i.e., r
2
= log Z, and so indeed equality holds in 3.16. The quenched LDP in Theorem 2.2, together with Varadhan’s lemma applied to 3.8, gives z
1
= 1 + exp[
−r
1
] with r
1
:= lim
N →∞
1 N
log F
1 N
X ≤ sup
Q ∈P
inv
Ý Z
d N
¨Z
Ý Z
d
π
1
Qd y log f y − I
que
Q «
X − a.s.,
3.20 where I
que
Q is given by 2.13–2.14. Without further assumptions, we are not able to reverse the inequality in 3.20. This point will be addressed in Section 4 and will require assumptions
1.10–1.12.
3.2 Proof of Theorem 1.3
