The upper bound is obtained by splitting the r.h.s. of 3.14 into R
N
+ D
N
:= P
T
3 N
2 r=1
K
N
t + r · u
N
T
3 N
− r u
N
N − u
· P
∞ j=1
K
N
t + j +
P
T
3 N
2 r=1
K
N
t + T
3 N
− r · u
N
r u
N
N − u
· P
∞ j=1
K
N
t + j .
3.18 The term R
N
can be bounded from above by a constant by simply applying the upper bound in 2.21 to u
N
T
3 N
− r for all r ∈ {1, . . . , T
3 N
2} and the lower bound to u
N
N − u. To bound D
N
from above, we use the upper bound in 2.15, which, together with the fact that gT
N
+ φδ, T
N
∼ m
δ
T
3 N
, shows that there exists c
0 such that for N large enough and r ∈ {1, . . . , T
3 N
2} we have K
N
t + T
3 N
− r ≤ c
T
3 N
e
−gT
N
+φδ,T
N
t
. 3.19
Notice also that by 2.16 we can assert that
∞
X
j=1
K
N
t + j ≥ c
3
e
−gT
N
+φδ,T
N
t
. 3.20
Finally, 3.19, 3.20 and the fact that u
N
N − u ≥ c
1
T
3 N
for all u ∈ {0, . . . , V
N
} by 2.21 allow to write
D
N
≤ c
P
T
3 N
2 r=1
u
N
r c
1
c
3
. 3.21
By applying the upper bound in 2.21, we can check easily that P
T
3 N
2 r=1
u
N
r is bounded from above uniformly in N
≥ 1 by a constant. This completes the proof of the step.
3.4 Step 4
In this step we complete the proof of Theorem 1.1 1, by proving equation 1.5, that we rewrite for convenience: there exist 0
c
1
c
2
∞ such that for all a b ∈ R and for large N ∈ 2N for simplicity
c
1
Pa
Z ≤ b ≤ P
T
N
N , δ
a
Y
T
N
L
N
p v
δ
k
N
≤ b
≤ c
2
Pa Z ≤ b .
3.22 We recall 2.4 and we start summing over the location
µ
N
:= τ
T
N
L
N ,TN
of the last point in τ
T
N
before N :
P
T
N
N , δ
a
Y
T
N
L
N ,TN
p v
δ
k
N
≤ b
=
N
X
ℓ=0
P
T
N
N , δ
a
Y
T
N
L
N ,TN
p v
δ
k
N
≤ b µ
N
= N − ℓ
· P
T
N
N , δ
µ
N
= N − ℓ . 3.23
Of course, only the terms with ℓ even are non-zero. We want to show that the sum in the r.h.s. of
3.23 can be restricted to ℓ ∈ {0, . . . , V
N
}. To that aim, we need to prove that P
N ℓ=V
N
P
T
N
N , δ
µ
N
= N
− ℓ tends to 0 as N
→ ∞. We start by displaying a lower bound on the partition function Z
T
N
N , δ
.
2052
Lemma 3.1. There exists a constant c 0 such that for N large enough
Z
T
N
N , δ
≥ c
T
N
e
φδ,T
N
N
. 3.24
Proof. Summing over the location of µ
N
and using the Markov property, together with 2.11, we have
Z
T
N
N , δ
= E h
e
H
TN N ,
δ
S
i =
N
X
r=0
E h
e
H
TN N ,
δ
S
1
{µ
N
=r}
i
=
N
X
r=0
E h
e
H
TN r,
δ
S
1
{r∈τ
TN
}
i P
τ
T
N
1
N − r =
N
X
r=0
e
φδ,T
N
r
P
δ,T
N
r ∈ τ Pτ
T
N
1
N − r . 3.25
From 3.25 and the lower bounds in 2.14 and 2.21, we obtain for N large enough Z
T
N
N , δ
≥ const. e
φδ,T
N
N N
X
r=0
e
−[φδ,T
N
+gT
N
]N −r
min {
p N
− r + 1, T
N
} min{r + 1
3 2
, T
3 N
} .
3.26 At this stage, we recall that
φδ, T + gT = m
δ
T
3
+ o1T
3
as T → ∞, with m
δ
0, by 2.18. Since T
3 N
≪ N, we can restrict the sum in 3.26 to r ∈ {N − T
3 N
, . . . , N − T
2 N
}, for large N, obtaining Z
T
N
N , δ
≥ const. e
φδ,T
N
N
T
4 N
N −T
2 N
X
r=N −T
3 N
e
−
mδ T 3
N
+o
1 T 3
N
N −r
≥ const.
′
e
φδ,T
N
N
T
N
, 3.27
because the geometric sum gives a contribution of order T
3 N
. We can now bound from above using the Markov property and 2.11
N −V
N
X
l=0
P
T
N
N , δ
µ
N
= ℓ =
N −V
N
X
ℓ=0
E exp H
T
N
ℓ,δ
S 1
{ℓ∈τ}
· P τ
1
N − ℓ Z
T
N
N , δ
=
N −V
N
X
ℓ=0
P
δ,T
N
ℓ ∈ τ e
φδ,T
N
ℓ
P τ
1
N − ℓ Z
T
N
N , δ
≤ const.
N −V
N
X
ℓ=0
T
N
min {ℓ + 1
3 2
, T
3 N
} ·
e
−[φδ,T
N
+gT
N
]N −ℓ
min {
p N
− ℓ, T
N
} ,
3.28 where we have used Lemma 3.1 and the upper bounds in 2.14 and 2.21. For notational conve-
nience we set dT
N
= φδ, T
N
+ gT
N
. Then, the estimate 2.18 and the fact that V
N
≫ T
3 N
imply that
N −V
N
X
ℓ=0
P
T
N
n, δ
µ
N
= ℓ ≤ const. e
−dT
N
V
N
N −V
N
X
ℓ=0
e
−dT
N
N −V
N
−ℓ
min {ℓ + 1
3 2
, T
3 N
} ≤ const.
′
e
−dT
N
V
N
∞
X
ℓ=0
1 l + 1
3 2
+
∞
X
ℓ=0
e
−dT
N
ℓ
T
3 N
.
3.29
2053
Since dT
N
∼ m
δ
T
3 N
, with m
δ
0, and V
N
≫ T
3 N
we obtain that the l.h.s. of 3.29 tends to 0 as N
→ ∞. Thus, we can write
P
T
N
N , δ
a
Y
T
N
L
N ,TN
p v
δ
k
N
≤ b
=
V
N
X
ℓ=0
P
T
N
N , δ
a
Y
T
N
L
N ,TN
p v
δ
k
N
≤ b µ
N
= N − ℓ
P
T
N
N , δ
µ
N
= N − ℓ + ǫ
N
a, b , 3.30
where ǫ
N
a, b tends to 0 as N → ∞, uniformly over a, b ∈ R. At this stage, by using the Markov property and 2.10 we may write
P
T
N
N , δ
a
Y
T
N
L
N ,TN
p v
δ
k
N
≤ b µ
N
= N − ℓ
= P
T
N
N , δ
a
Y
T
N
L
N −ℓ,TN
p v
δ
k
N
≤ b N
− ℓ ∈ τ
T
= P
δ,T
N
a
Y
L
N −ℓ
p v
δ
k
N
≤ b N
− ℓ ∈ τ
. Plugging this into 3.30, recalling 3.10 and the fact that
P
V
N
ℓ=0
P
T
N
N , δ
µ
N
= N − ℓ → 1 by 3.29, it follows that equation 3.22 is proven, and the proof is complete.
4 Proof of Theorem 1.1: part 2
We assume that T
N
∼ const.N
1 3
and we start proving the first relation in 1.6, that we rewrite as follows: for every
ǫ 0 we can find M 0 such that for large N
P
T
N
N , δ
|S
N
| M · T
N
≤ ǫ . Recalling that L
T N
is the number of times the polymer has touched an interface up to epoch N , see 2.9, we have
|S
N
| ≤ T
N
· L
N ,T
N
+ 1, hence it suffices to show that
P
T
N
N , δ
L
N ,T
N
M ≤ ǫ .
4.1 By using 2.10 we have
P
T
N
N , δ
L
N ,T
M =
1 Z
T
N
N , δ
E h
e
H
TN N ,
δ
S
1
{L
N ,TN
M }
i
= 1
Z
T
N
N , δ
N
X
r=0
E h
e
H
TN r,
δ
S
1
{L
r,TN
M }
1
{r∈τ
TN
}
i P
τ
T
N
1
N − r =
1 Z
T
N
N , δ
N
X
r=0
e
φδ,T
N
r
P
δ,T
N
L
r,T
N
M , r ∈ τ
T
N
P τ
T
N
1
N − r .
2054
By 2.14 and 2.12 it follows easily that Z
T
N
N , δ
≥ Pτ
T
N
1
N ≥ const.
T
N
e
−
π2 2T 2
N
N
4.2 note that this bound holds true whenever we have const.N
1 4
≤ T
N
≤ const.
′
p N for large N .
Using this lower bound on Z
T
N
N , δ
, together with the upper bound in 2.14, the asymptotic expansions in 2.18 and 2.12, we obtain
P
T
N
N , δ
L
N ,T
M ≤ const. T
N N
X
r=0
P
δ,T
N
L
r,T
N
M , r ∈ τ
T
N
1 min
{ p
N − r + 1, T
N
} .
The contribution of the terms with r N − T
2 N
is bounded with the upper bound 2.21: T
N N
X
r=N −T
2 N
1 T
3 N
1 p
N − r + 1
≤ const.
T
N
−→ 0 N → ∞ ,
while for the terms with r ≤ N − T
2 N
we get T
N N
X
r=0
P
δ,T
N
L
r,T
N
M , r ∈ τ
T
N
1 T
N
= E
δ,T
N
L
N ,T
N
− M1
{L
N ,TN
M }
. Finally, we simply observe that
{L
N ,T
N
= k} ⊆ T
k i=1
{τ
i
− τ
i −1
≤ N}, hence P
δ,T
N
L
N ,T
N
= k ≤ P
δ,T
N
τ
1
≤ N
k
≤ c
k
, with 0
c 1, as it follows from 2.16 and 2.18 recalling that N = OT
3 N
. Putting together the preceding estimates, we have
P
T
N
N , δ
L
N ,T
N
M ≤ const. E
δ,T
N
L
N ,T
N
− M1
{L
N ,TN
M }
= const.
∞
X
k=M +1
k − M P
δ,T
N
L
N ,T
N
= k ≤ const.
∞
X
k=M +1
k − M c
k
≤ const.
′
c
M
, and 4.1 is proven by choosing M sufficiently large.
Finally, we prove at the same time the second relations in 1.6 and 1.7, by showing that for every ǫ 0 there exists η 0 such that for large N
P
T
N
N , δ
|S
N
| ≤ η T
N
≤ ǫ , 4.3
whenever T
N
satisfies const.N
1 3
≤ T
N
≤ const.
′
p N for large N . Letting P
k
denote the law of the simple random walk starting at k
∈ N and τ
∞ 1
its first return to zero, it follows by Donsker’s
2055
invariance principle that there exists c 0 such that inf
≤k≤ηT
N
P
k
τ
∞ 1
≤ η
2
T
2 N
, S
i
T
N
∀i ≤ τ
∞ 1
≥ c for large N . Therefore we may write
c P
T
N
N , δ
|S
N
| ≤ η T
N
= c
Z
T
N
N , δ
ηT
N
X
k=0
E h
e
H
TN N ,
δ
S
1
{|S
N
|=k}
i
≤ 1
Z
T
N
N , δ
ηT
N
X
k=0 η
2
T
2 N
X
u=0
E h
e
H
TN N ,
δ
S
1
{|S
N
|=k}
i P
k
τ
∞ 1
= u , S
i
T
N
∀i ≤ u
= 1
Z
T
N
N , δ
ηT
N
X
k=0 η
2
T
2 N
X
u=0
E h
e
H
TN N +u,
δ
S
1
{|S
N
|=k}
1
{|S
N +i
|T
N
∀i≤u}
1
{S
N +u
=0}
i .
Performing the sum over k, dropping the second indicator function and using equations 2.11, 2.21 and 2.3, we obtain the estimate
P
T
N
N , δ
|S
N
| ≤ η T
N
≤ 1
c Z
T
N
N , δ
η
2
T
2 N
X
u=0
E h
e
H
TN N +u,
δ
S
1
{N+u∈τ
TN
}
i
≤ 1
c Z
T
N
N , δ
η
2
T
2 N
X
u=0
e
φδ,T
N
N +u
P
δ,T
N
N + u ∈ τ ≤ const. η
2
T
2 N
Z
T
N
N , δ
T
3 N
e
−
π2 2T 2
N
N
. Then 4.2 shows that equation 4.3 holds true for
η small, and we are done.
5 Proof of Theorem 1.1: part 3
We now give the proof of part 3 of Theorem 1.1. More precisely, we prove the first relation in 1.7, because the second one has been proven at the end of Section 4 see 4.3 and the following
lines. We recall that we are in the regime when N
1 3
≪ T
N
≤ const. p
N , so that in particular C := inf
N ∈N
N T
2 N
0 . 5.1
We start stating an immediate corollary of Proposition 2.3.
Corollary 5.1. For every ǫ 0 there exist T
0, M
ǫ
∈ 2N, d
ǫ
0 such that for T T
d
ǫ
T
3
X
k=M
ǫ
P
δ,T
k ∈ τ
≤ ǫ . Note that we can restate the first relation in 1.7 as
P
T
N
N , δ
τ
T
N
L
N ,TN
≤ L ≥ 1 − ǫ. Let us define three
intermediate quantities, by setting for l ∈ N
B
1
l, N = P
