whose eigenvalues are, for all γ 0 and for 0 ≤ k ≤ N − 1,
λ
k,N
= − 1
− 2γ sin
2
k π
N .
2.13 Set, for k
≥ 1, γ
N k
=
1 2 sin
2
kπN
. Then these eigenvalues can be written in the form
λ
k,N
= λ
N −k,N
= −1 +
γ γ
N k
, 1 ≤ k ≤ N − 1
λ
0,N
= λ = −1.
2.14
Note that γ
N k
⌊N2⌋ k=1
is a decreasing sequence, and so as γ increases, the number of non-positive
eigenvalues λ
k,N N
−1 k=0
decreases. When γ γ
N 1
, the only negative eigenvalue is −1. Thus
γ
N 1
= 1
2 sin
2
πN 2.15
is the threshold of the synchronization regime.
Lemma 2.1 Synchronization Regime. If
γ γ
N 1
, the only stationary points of F
γ,N
are I
±
and O. I
±
are minima, O is a 1-saddle. This lemma was proven in [2] by using a Lyapunov function. This configuration is called the syn-
chronization regime because the coupling between the particles is so strong that they all pass si- multaneously through their respective saddle points in a transition between the stable equilibria
I
±
. In this paper, we will focus on this regime.
2.3 Results for fixed N
Let ρ 0 and set B
±
≡ B
ρ
I
±
, where B
ρ
x denotes the ball of radius ρ centered at x. Equation 2.6 gives, with A = B
−
and D = B
+
, E
ν
B−,B+
[τ
B
+
] = R
B
c +
h
B
−
,B
+
ye
−F
γ,N
yǫ
d y capB
−
, B
+
. 2.16
First, we obtain a sharp estimate for this transition time for fixed N :
Theorem 2.2. Let N 2 be given. For γ γ
N 1
=
1 2 sin
2
πN
, let p
N ρ ≥ ε 0. Then
E
ν
B−,B+
[τ
B
+
] = 2πc
N
e
N 4
ε
1 + O p
ǫ| ln ǫ|
3
2.17 with
c
N
= 1
− 3
2 + 2 γ
eN 2
⌊
N −1
2
⌋
Y
k=1
1 −
3 2 +
γ γ
N k
2.18 where eN = 1 if N is even and 0 if N is odd.
327
Remark. The power 3 at ln ǫ is missing in [5] by mistake.
Remark. As mentioned above, for any fixed dimension, we can replace the probability measure ν
B
−
,B
+
by the Dirac measure on the single point I
−
, using Hölder and Harnack inequalities [5]. This gives the following corollary:
Corollary 2.3. Under the assumptions of Theorem 2.2, there exists α 0 such that
E
I
−
[τ
B
+
] = 2πc
N
e
N 4
ε
1 + O p
ǫ| ln ǫ|
3
. 2.19
Proof of the theorem. We apply Theorem 3.2 in [5]. For γ γ
N 1
=
1 2 sin
2
πN
, let us recall that there are only three stationary points: two minima I
±
and one saddle point O. One easily checks that F
γ,N
satisfies the following assumptions: • F
γ,N
is polynomial in the x
i i
∈Λ
and so clearly C
3
on R
N
. • F
γ,N
x ≥
1 4
P
i ∈Λ
x
4 i
so F
γ,N
−→
x →∞
+∞. • k∇F
γ,N
xk
2
∼ kxk
3 3
as kxk
2
→ ∞. • As ∆F
γ,N
x ∼ 3kxk
2 2
kxk
2
→ ∞, then k∇F
γ,N
xk − 2∆F
γ,N
x ∼ kxk
3 3
. The Hessian matrix at the minima I
±
has the form ∇
2
F
γ,N
I
±
= ∇
2
F
γ,N
O + 3Id, 2.20
whose eigenvalues are simply ν
k,N
= λ
k,N
+ 3. 2.21
Then Theorem 3.1 of [5] can be applied and yields, for p
N ρ ε 0, recall the the negative
eigenvalue of the Hessian at O is −1
E
ν
B−,B+
[τ
B
+
] = 2
πe
N 4
ε
p |det∇
2
F
γ,N
O| p
det ∇
2
F
γ,N
I
−
1 + O p
ε|ln ε|
3
. 2.22
Finally, 2.14 and 2.21 give: det
∇
2
F
γ,N
I
−
=
N −1
Y
k=0
ν
k,N
= 2ν
eN N
2,N ⌊
N −1
2
⌋
Y
k=1
ν
2 k,N
= 2
N
1 + γ
eN ⌊
N −1
2
⌋
Y
k=1
1 + γ
2 γ
N k
2
2.23
|det∇
2
F
γ,N
O| =
N −1
Y
k=0
λ
k,N
= λ
eN N
2,N ⌊
N −1
2
⌋
Y
k=1
λ
2 k,N
= 2γ − 1
eN ⌊
N −1
2
⌋
Y
k=1
1 −
γ γ
N k
2
. 2.24
Then, c
N
= p
det ∇
2
F
γ,N
I
−
p |det∇
2
F
γ,N
O| =
1 −
3 2 + 2
γ
eN 2
⌊
N −1
2
⌋
Y
k=1
1 −
3 2 +
γ γ
N k
2.25 and Theorem 2.2 is proved.
328
Let us point out that the use of these estimates is a major obstacle to obtain a mean transition time starting from a single stable point with uniform error terms. That is the reason why we have
introduced the equilibrium probability. However, there are still several difficulties to be overcome if we want to pass to the limit N
↑ ∞. i We must show that the prefactor c
N
has a limit as N ↑ ∞.
ii The exponential term in the hitting time tends to infinity with N . This suggests that one needs to rescale the potential F
γ,N
by a factor 1 N , or equivalently, to increase the noise strength by
a factor N . iii One will need uniform control of error estimates in N to be able to infer the metastable
behavior of the infinite dimensional system. This will be the most subtle of the problems involved.
3 Large N limit
As mentioned above, in order to prove a limiting result as N tends to infinity, we need to rescale the potential to eliminate the N -dependence in the exponential. Thus henceforth we replace F
γ,N
x by G
γ,N
x = N
−1
F
γ,N
x. 3.1
This choice actually has a very nice side effect. Namely, as we always want to be in the regime where γ ∼ γ
N 1
∼ N
2
, it is natural to parametrize the coupling constant with a fixed µ 1 as
γ
N
= µγ
N 1
= µ
2 sin
2 π
N
= µN
2
2 π
2
1 + o1. 3.2
Then, if we replace the lattice by a lattice of spacing 1 N , i.e. x
i i
∈Λ
is the discretization of a real function x on [0, 1] x
i
= xiN , the resulting potential converges formally to G
γ
N
,N
x →
N →∞
Z
1
1 4
[xs]
4
− 1
2 [xs]
2
ds + µ
4 π
2
Z
1
x
′
s
2
2 ds,
3.3 with x0 = x1.
In the Euclidean norm, we have kI
±
k
2
= p
N , which suggests to rescale the size of neighborhoods. We consider, for
ρ 0, the neighborhoods B
N ±
= B
ρ p
N
I
±
. The volume V B
N −
= V B
N +
goes to 0 if and only if
ρ 12πe, so given such a ρ, the balls B
N ±
are not as large as one might think. Let us also observe that
1 p
N kxk
2
−→
N →∞
kxk
L
2
[0,1]
= Z
1
|xs|
2
ds. 3.4
Therefore, if x ∈ B
N +
for all N , we get in the limit, kx − 1k
L
2
[0,1]
ρ. The main result of this paper is the following uniform version of Theorem 2.2 with a rescaled
potential G
γ,N
. 329
Theorem 3.1. Let µ ∈]1, ∞[, then there exists a constant, A, such that for all N ≥ 2 and all ǫ 0,
1 N
E
ν
BN −
,BN +
[τ
B
N +
] = 2πc
N
e
1 4ǫ
1 + Rǫ, N , 3.5
where c
N
is defined in Theorem 2.2 and |Rǫ, N| ≤ A
p ǫ| ln ǫ|
3
. In particular, lim
ǫ↓0
lim
N ↑∞
1 N
e
−14ǫ
E
ν
BN −
,BN +
[τ
B
N +
] = 2πV µ 3.6
where V
µ =
+∞
Y
k=1
h µk
2
− 1 µk
2
+ 2 i
∞. 3.7
Remark. The appearance of the factor 1 N may at first glance seem disturbing. It corresponds
however to the appropriate time rescaling when scaling the spatial coordinates i to i N in order to
recover the pde limit. The proof of this theorem will be decomposed in two parts:
• convergence of the sequence c
N
Proposition 3.2; • uniform control of the denominator Proposition 4.3 and the numerator Proposition 4.9 of
Formula 2.16.
Convergence of the prefactor c
N
. Our first step will be to control the behavior of c
N
as N ↑ ∞. We
prove the following:
Proposition 3.2. The sequence c
N
converges: for µ 1, we set γ = µγ
N 1
, then lim
N ↑∞
c
N
= V µ, 3.8
with V µ defined in 3.7.
Remark. This proposition immediately leads to Corollary 3.3. For
µ ∈]1, ∞[, we set γ = µγ
N 1
, then lim
N ↑∞
lim
ǫ↓0
e
−
1 4
ǫ
N E
ν
BN −
,BN +
[τ
B
N +
] = 2πV µ. 3.9
Of course such a result is unsatisfactory, since it does not tell us anything about a large system with specified fixed noise strength. To be able to interchange the limits regarding
ǫ and N , we need a uniform control on the error terms.
Proof of the proposition. The rescaling of the potential introduces a factor
1 N
for the eigenvalues, so that 2.22 becomes
E
ν
BN −
,BN +
[τ
B
N +
] =
2 πe
1 4
ε
N
−N2+1
p |det∇
2
F
γ,N
O| N
−N2
p det
∇
2
F
γ,N
I
−
1 + O p
ε|ln ε|
3
= 2
πN c
N
e
1 4
ε
1 + O p
ε|ln ε|
3
. 3.10
330
Then, with u
N k
=
3 2+
µ
γN 1
γN k
,
c
N
= 1
− 3
2 + 2 µγ
N 1
eN 2
⌊
N −1
2
⌋
Y
k=1
1 − u
N k
. 3.11
To prove the convergence, let us consider the γ
N k
N −1
k=1
. For all k ≥ 1, we have
γ
N 1
γ
N k
= sin
2 k
π N
sin
2 π
N
= k
2
+ 1 − k
2
π
2
3N
2
+ o 1
N
2
. 3.12
Hence, u
N k
−→
N →+∞
v
k
=
3 2+
µk
2
. Thus, we want to show that c
N
−→
N →+∞
+∞
Y
k=1
1 − v
k
= V µ. 3.13
Using that, for 0 ≤ t ≤
π 2
, t
2
1 − t
2
3 ≤ sin
2
t ≤ t
2
, 3.14
we get the following estimates for
γ
N 1
γ
N k
: set a = 1
−
π
2
12
, for 1 ≤ k ≤ N2,
ak
2
= 1
− π
2
12 k
2
≤ k
2
1 −
k
2
π
2
3N
2
≤ γ
N 1
γ
N k
= sin
2 k
π N
sin
2 π
N
≤ k
2
1 −
π
2
3N
2
. 3.15
Then, for N ≥ 2 and for all 1 ≤ k ≤ N2,
− k
4
π
2
3N
2
≤ γ
N 1
γ
N k
− k
2
≤ k
2
π
2
3N
2
1 −
π
2
3N
2
≤ k
2
π
2
N
2
. 3.16
Let us introduce V
m
=
⌊
m −1
2
⌋
Y
k=1
1 − v
k
, U
N ,m
=
⌊
m −1
2
⌋
Y
k=1
1 − u
N k
. 3.17
Then ln
U
N ,N
V
N
= ln
⌊
N −1
2
⌋
Y
k=1
1 − u
N k
1 − v
k
≤
⌊
N −1
2
⌋
X
k=1
ln 1
− u
N k
1 − v
k
. 3.18
Using 3.15 and 3.16, we obtain, for all 1 ≤ k ≤ N2,
v
k
− u
N k
1 − v
k
= 3
µ
γ
N 1
γ
N k
− k
2
−1 + µk
2
2 +
µ
γ
N 1
γ
N k
≤ µk
4
π
2
N
2
−1 + µk
2
2 + µak
2
≤ C
N
2
3.19
331
with C a constant independent of k. Therefore, for N N
, ln
1 − u
N k
1 − v
k
= ln
1 +
v
k
− u
N k
1 − v
k
≤
C
′
N
2
. 3.20
Hence ln
U
N ,N
V
N
≤ C
′
N −→
N →+∞
0. 3.21
As P
|v
k
| +∞, we get lim
N →+∞
V
N
= V µ 0, and thus 3.13 is proved.
4 Estimates on capacities
To prove Theorem 3.1, we prove uniform estimates of the denominator and numerator of 2.6, namely the capacity and the mass of the equilibrium potential.
4.1 Uniform control in large dimensions for capacities
