6 Sharp estimates on the mean hitting times
In this section we conclude the proof of Theorem 1.2. To do this we will use Equation 2.12 with A =
S [m
∗
] and B = S [M], where m
∗
is a local minimum of F
β,N
and M is the set of minima deeper than m
∗
. The denominator on the right-hand side of 2.12, the capacity, is controlled by Theorem 1.3. What we want to prove now is that the equilibrium potential, h
A,B
σ, is close to one in the neighborhood of the starting set A, and so small elsewhere that the contributions from
the sum over σ away from the valley containing the set A can be neglected. Note that this is not
generally true but depends on the choice of sets A and B: the condition that all minima m of F
β,N
such that F
β,N
m F
β,N
m
∗
belong to the target set B is crucial. In earlier work see, e.g., [5] the standard way to estimate the equilibrium potential h
A,B
σ was to use the renewal inequality h
A,B
σ ≤ cap
A,σ
cap
B,σ
and bounds on capacities. This bound cannot be used here, since the capacities of single points are too small. In other words,
hitting times of fixed microscopic configurations are separated on exponential scales from hitting times of the corresponding mesoscopic neighborhoods. We will therefore use another method to
cope with this problem.
6.1 Mean hitting time and equilibrium potential
Let us start by considering a local minimum m
∗
of the one-dimensional function F
β,N
, and denote by M the set of minima m such that F
β,N
m F
β,N
m
∗
. We then consider the disjoint subsets A
≡ S [m
∗
] and B ≡ S [M], and write Eq. 2.12 as X
σ∈A
ν
A,B
σE
σ
τ
B
= 1
capA, B X
m ∈[−1,1]
X
σ∈S [m]
µ
β,N
σh
A,B
σ. 6.1
We want to estimate the right-hand side of 6.1. This is expected to be of order Q
β,N
m
∗
, thus we can readily do away with all contributions where
Q
β,N
is much smaller. More precisely, we choose
δ 0 in such a way that, for all N large enough, there is no critical point z of F
β,N
with F
β,N
z ∈
F
β,N
m
∗
, F
β,N
m
∗
+ δ
, and define U
δ
≡ {m : F
β,N
m ≤ F
β,N
m
∗
+ δ}. 6.2
Denoting by U
c δ
the complement of U
δ
, we obviously have
Lemma 6.1.
X
m ∈U
c δ
X
σ∈S [m]
µ
β,N
σh
A,B
σ ≤ N e
−β Nδ
Q
β,N
m
∗
. 6.3
The main problem is to control the equilibrium potential h
A,B
σ for configurations σ ∈ S [U
δ
]. To do that, first notice that
U
δ
= U
δ
m
∗
[
m ∈M
U
δ
m, 6.4
where U
δ
m is the connected component of U
δ
containing m see Figure 4.. Note that it can happen that
U
δ
m = U
δ
m
′
for two different minima m, m
′
∈ M. With this notation we have the following lemma.
1587
m
∗
z m
∗
U
δ
m
∗
U
δ
m
∗
Figure 4: Decomposition of the magnetization space [ −1, 1]: The dotted lines and the continuous
lines correspond respectively to U
c δ
and U
δ
= U
δ
m
∗
S
m ∈M
U
δ
m.
Lemma 6.2. There exists a constant, c 0, such that
i for every m ∈ M,
X
σ∈S [U
δ
m]
µ
β,N
σh
A,B
σ ≤ e
−β N c
Q
β,N
m
∗
6.5 and
ii X
σ∈S [U
δ
m
∗
]
µ
β,N
σ
1 − h
A,B
σ
≤ e
−β N c
Q
β,N
m
∗
. 6.6
The treatment of points i and ii is completely similar, as both rely on a rough estimate of the probabilities to leave the starting well before visiting its minimum, and it will be discussed in the
next section.
Assuming Lemma 6.2, we can readily conclude the proof of Theorem 1.2. Indeed, using 6.5 together with 6.3, we obtain the upper bound
X
σ∈S
N
µ
β,N
σh
A,B
σ ≤ X
m ∈U
δ
m
∗
Q
β,N
m + O
Q
β,N
m
∗
e
−β N c
= Q
β,N
m
∗
È πN
2 β am
∗
1 + o1, 6.7
where am
∗
is given in 1.19. On the other hand, using 6.6, we get the corresponding lower
1588
bound X
σ∈S
N
µ
β,N
σh
A,B
σ ≥ X
m ∈U
δ
m
∗
X
σ∈S [m]
µ
β,N
σ
1 − 1 − h
A,B
σ
≥ X
m ∈U
δ
m
∗
Q
β,N
m − OQ
β,N
m
∗
e
−β N c
= Q
β,N
m
∗
È πN
2 β am
∗
1 + o1. 6.8
From Equation 1.12 for Q
β,N
m
∗
and Equation 1.32 for capA, B, we finally obtain E
ν
A,B
τ
B
= X
σ∈S
N
µ
β,N
σh
A,B
σ capA, B
= exp
β N
F
β,N
z
∗
− F
β,N
m
∗
× 2
πN β|ˆγ
1
| v
u u
t βE
h
1
− tanh
2
βz
∗
+ h
− 1 1
− βE
h
1
− tanh
2
βm
∗
+ h 1 + o1,
6.9 which proves Theorem 1.2.
6.2 Upper bounds on harmonic functions.
