Focusing on the case a = b = exp γ, we mention in particular equations 6.4–6.6 in [8], which
for a 2 read as
Z
n,w
a, a ≈ const.
n
3 2
f
phase
p n
w
, 1.13
where we have neglected a combinatorial factor 2
n
which just comes from a different choice of notation, and where the function f
phase
x is such that f
phase
x → 1 as x → 0 , f
phase
x ≈ x
3
e
−π
2
x
2
2
as x → ∞ .
1.14 The regime a
2 corresponds to γ log 2, hence, in view of the correspondence δ = γ − log 2 described above, we are exactly in the regime
δ 0 for our model P
T N ,
δ
. We recall 1.2 and, with the help of equation 2.11, we can express the partition function with boundary condition
S
N
∈ 2T Z as Z
T, {S
N
∈2T Z} N ,
δ
∼ O1 Z
T, {S
N
∈T Z} N ,
δ
∼ O1 e
φδ,T N
P
δ,T
N ∈ τ , where, with some abuse of notation, we denote by O1 a quantity which stays bounded away from
0 and ∞ as N → ∞. In this formula, φδ, T is the free energy of our model and {τ
n
}
n ∈Z
+
, P
δ,T
is a basic renewal process, introduced respectively in §2.1 and §2.2 below. In the case when T = T
N
→ ∞, we can use the asymptotic expansion 2.3 for φδ, T , which, combined with the bounds in 2.21, gives as N , T
→ ∞ Z
T, {S
N
∈2T Z} N ,
δ
= O1
N
3 2
max 1,
p N
T
3
exp
− π
2
2 N
T
2
+ 2
π
2
e
−δ
− 1 N
T
3
+ o N
T
3
.
Since Z
T, {S
N
∈2T Z} N ,
δ
= Z
n,w
a, a, we can rewrite this relation using the notation of [8]: Z
n,w
a, a ≈ const.
n
3 2
f
phase
p n
w
g n
1 3
w ,
where gx ≈ e
2 π2
e−δ −1
x
as x → ∞ .
We have therefore obtained a refinement of equations 1.13, 1.14. This is linked to the fact that we have gone beyond the first order in the asymptotic expansion of the free energy
φδ, T , making an additional term of the order N
T
3 N
appear. We stress that this new term gives a non-negligible in fact, exponentially diverging contribution as soon as T
N
≪ N
1 3
w ≪ n
1 3
in the notation of [8]. This corresponds to the fact that, by Theorem 1.1, the trajectories that touch the walls a number of
times of the order N T
3 N
are actually dominating the partition function when T
N
≪ N
1 3
. Of course, a higher order expansion of the free energy cf. Appendix A.1 may lead to further correction terms.
1.4 Outline of the paper
Proving Theorem 1.1 requires to settle some technical tools, partially taken from [3], that we present in Section 2. More precisely, in §2.1 we introduce the free energy
φδ, T of the polymer and we describe its asymptotic behavior as T
→ ∞ for fixed δ 0. In §2.2 we enlighten a basic corre- spondence between the polymer constrained to hit one of the interfaces at its right extremity and an
explicit renewal process. In §2.3 we investigate further this renewal process, providing estimates on
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the renewal function, which are of crucial importance for the proof of Theorem 1.1. Sections 3, 4, 5 and 6 are dedicated respectively to the proof of parts 1, 2, 3 and 4 of Theorem 1.1. Finally,
some technical results are proven in the appendices.
We stress that the value of δ 0 is kept fixed throughout the paper, so that the generic constants
appearing in the proofs may be δ-dependent.
2 A renewal theory viewpoint
In this section we recall some features of our model, including a basic renewal theory representation, originally proven in [3], and we derive some new estimates.
2.1 The free energy
Considering for a moment our model when T
N
≡ T ∈ 2N is fixed, i.e., it does not vary with N, we define the free energy
φδ, T as the rate of exponential growth of the partition function Z
T N ,
δ
as N
→ ∞: φδ, T := lim
N →∞
1 N
log Z
T N ,
δ
= lim
N →∞
1 N
log E e
H
T N ,
δ
. 2.1
Generally speaking, the reason for looking at this function is that the values of δ if any at which
δ 7→ φδ, T is not analytic correspond physically to the occurrence of a phase transition in the system. As a matter of fact, in our case
δ 7→ φδ, T is analytic on the whole real line, for every T
∈ 2N. Nevertheless, the free energy φδ, T turns out to be a very useful tool to obtain a path description of our model, even when T = T
N
varies with N , as we explain in detail in §2.2. For this reason, we now recall some basic facts on
φδ, T , that were proven in [3], and we derive its asymptotic behavior as T
→ ∞. We introduce
τ
T 1
:= inf {n 0 : S
n
∈ {−T, 0, +T }}, that is the first epoch at which the polymer visits an interface, and we denote by Q
T
λ := E e
−λτ
T 1
its Laplace transform under the law of the simple random walk. We point out that Q
T
λ is finite and analytic on the interval λ
T
, ∞, where
λ
T
0, and Q
T
λ → +∞ as λ ↓ λ
T
as a matter of fact, one can give a closed explicit expression for Q
T
λ, cf. equations A.4 and A.5 in [3]. A basic fact is that Q
T
· is sharply linked to the free energy: more precisely, we have
φδ, T = Q
T −1
e
−δ
, 2.2
for every δ ∈ R see Theorem 1 in [3]. From this, it is easy to obtain an asymptotic expansion of
φδ, T as T → ∞, for δ 0, which reads as φδ, T = −
π
2
2T
2
1 −
4 e
−δ
− 1 1
T + o
1 T
, 2.3
as we prove in Appendix A.1. We stress that this expansion is for fixed δ 0, in particular the term
o1 T in 2.3 does depend on δ.
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2.2 A renewal theory interpretation
