544 Electronic Communications in Probability
If we choose c
N
:= C N η, from 31, 32 and 33 we obtain
ϕ
0,0 2N +1
0, 0 ≥ e
−M
γ
+V
1
2 λC
v0
2
kvk
2 ∞
η
3
1 N
= const.
N ,
which is the desired lower bound in 30. The upper bound is easier. By assumptions C1 and C2 both V
1
and V
2
are bounded from below, therefore we can replace V
1
∇ϕ
2N +1
, V
1
∇ϕ
2N
, V
2
∆ϕ
2N
and V
2
∆ϕ
2N −1
by the constant ec := inf
x ∈R
min {V
1
x, V
2
x} ∈ R getting the upper bound: ϕ
0,0 2N +1
0, 0 = 1
λ
2N +1
Z
R
2N −1
e
−H
[−1,2N+1]
ϕ 2N
−1
Y
i=1
d ϕ
i
≤ e
−4ec
λ
2N +1
Z
R
2N −1
e
−H
[−1,2N−1]
ϕ 2N
−1
Y
i=1
d ϕ
i
. Recalling Proposition 4 and Proposition 6, we obtain
ϕ
0,0 2N +1
0, 0 ≤ e
−4ec
λ
2
Z
R
2
v0 v
ϕ
2N −1
− ϕ
2N −2
P
0,0
W
2N −2
∈ dϕ
2N −2
, W
2N −1
∈ dϕ
2N −1
= v0
λ
2
e
−4ec
E
0,0
1 vY
2N −1
≤ v0
λ
2
e
−4ec
C = const. , which completes the proof of 30.
3 A lower bound on the partition function
We are going to give an explicit lower bound on the partition function in terms of a suitable renewal process. First of all, we rewrite equation 3 as
Z
ǫ,N
=
N −1
X
k=0
ǫ
k
X
A ⊆{1,...,N−1}
|A|=k
Z e
−H
[−1,N+1]
ϕ
Y
m ∈A
δ dϕ
m
Y
n ∈A
c
d ϕ
n
, 34
where we set A
c
:= {1, . . . , N − 1} \ A for convenience.
3.1 A renewal process lower bound
We restrict the summation over A in 34 to the class of subsets B
2k
consisting of 2k points organized in k consecutive couples:
B
2k
:= {t
1
− 1, t
1
, . . . , t
k
− 1, t
k
} | 0 = t t
1
. . . t
k
≤ N − 1 and t
i
− t
i −1
≥ 2 ∀i .
Localization for ∇ + ∆-pinning models
545 Plainly, B
2k
= ; for k N − 12. We then obtain from 34 Z
ǫ,N
≥
⌊N−12⌋
X
k=0
ǫ
2k
X
A ∈B
2k
Z e
−H
[−1,N+1]
ϕ
Y
m ∈A
δ dϕ
m
Y
n ∈A
c
d ϕ
n
=
⌊N−12⌋
X
k=0
ǫ
2k
X
0=t t
1
...t
k
t
k+1
=N +1 t
i
−t
i −1
≥2 ∀i≤k+1 k+1
Y
j=1
e Kt
j
− t
j −1
, 35
where we have set for n ∈ N
e Kn :=
if n = 1
e
−H
[−1,2]
0,0,0,0
= e
−2V
1
0−2V
2
if n = 2 Z
R
n −2
e
−H
[−1,n]
w
−1
,...,w
n
dw
1
· · · dw
n −2
with w
−1
= 0, w = 0, w
n −1
= 0, w
n
= 0
if n ≥ 3
. 36
We stress that a factorization of the form 35 is possible because the Hamiltonian H
[−1,N+1]
ϕ consists of two- and three-body terms and we have restricted the sum over subsets in B
2k
, that consist of consecutive couples of zeros. We also note that the condition t
i
− t
i −1
≥ 2 is immaterial, because by definition e
K1 = 0. We now give a probabilistic interpretation to the right hand side of 35 in terms of a renewal
process. To this purpose, for every ǫ 0 and for n ∈ N we define
K
ǫ
1 := 0 , K
ǫ
n := ǫ
2
λ
n
e Kn e
−µ
ǫ
n
= ǫ
2
ϕ
0,0 n
0, 0 e
−µ
ǫ
n
, ∀n ≥ 2 .
where the second equality follows recalling 36, Proposition 4 and the definition 14 of the density
ϕ
n
. The constant µ
ǫ
is chosen to make K
ǫ
a probability on N: X
n ∈N
K
ǫ
n = 1 , that is
∞
X
n=2
ϕ
0,0 n
0, 0 e
−µ
ǫ
n
= 1
ǫ
2
. 37
It follows from Proposition 8 that 0 µ
ǫ
∞ for every ǫ 0. We can therefore define a renewal process
{η
n
}
n ≥0
, P
ǫ
on N with inter-arrival law K
ǫ
·. More explicitly, η := 0 and
the increments {η
k+1
− η
k
}
k ≥0
are independent, identically distributed random variables with marginal law
P
ǫ
η
k+1
− η
k
= n = K
ǫ
n. Coming back to 35, we can write Z
ǫ,N
≥ λ
N +1
e
N +1µ
ǫ
ǫ
2 ⌊N−12⌋
X
k=0
X
0=t t
1
...t
k
t
k+1
=N +1 k+1
Y
j=1
K
ǫ
t
j
− t
j −1
= λ
N +1
e
N +1µ
ǫ
ǫ
2 ⌊N−12⌋
X
k=0
X
0=t t
1
...t
k
t
k+1
=N +1
P
ǫ
η
1
= t
1
, . . . , η
k+1
= t
k+1
= λ
N +1
e
N +1µ
ǫ
ǫ
2 ⌊N−12⌋
X
k=0
P
ǫ
η
k+1
= N + 1 =
λ
N +1
e
N +1µ
ǫ
ǫ
2
P
ǫ
N + 1 ∈ η
, 38
where in the last equality we look at η = {η
k
}
k ≥0
as a random subset of N , so that
{N + 1 ∈ η} = S
∞ m=1
{η
m
= N + 1} note that P
ǫ
η
k+1
= N + 1 = 0 for k ⌊N − 12⌋. We have thus obtained a lower bound on the partition function
Z
ǫ,N
of our model in terms of the renewal mass function or Green function of the renewal process
{η
n
}
n ≥0
}, P
ǫ
.
546 Electronic Communications in Probability
3.2 Proof of Theorem 1
