For the proof of 4.24, we note that it is sufficient to bound the event when N |λ
j
− λ
ℓ
| ≥ 1 after using 4.23. Inserting the partition 4.25, we get
1 N
E 1
Ω ∗
X
j ℓ
1N |λ
ℓ
− λ
j
| ≥ 1 N
λ
ℓ
− λ
j
= 1
N X
|r|,|s|≤r
E 1
Ω
X
j ℓ
1 λ
j
∈ I
r
, λ
ℓ
∈ I
s
1N |λ
ℓ
− λ
j
| ≥ 1 N
λ
ℓ
− λ
j
≤ C
N X
|r|,|s|≤r
E 1
Ω
N
I
r
N
I
s
n
γ
[|s − r| − 1]
+
+ 1 ≤
C n
2 γ
N X
|r|,|s|≤r
1 n
γ
[|s − r| − 1]
+
+ 1 ≤ C n
γ
log N . Recalling the choice of n completes the proof of Lemma 4.4.
5 Global entropy
5.1 Evolution of the entropy
Recall the definition of the entropy of f µ with respect to µ
S
µ
f := S f µ|µ = Z
f log f d µ
and let f
t
solve 3.9. Then the evolution of the entropy is given by the equation ∂
t
S f
t
= −D p
f
t
and thus using that S f
t
0 we have Z
t s
D p
f
u
du ≤ S f
s
. 5.1
For dynamics with energy H and the convexity condition
Hess H = ∇
2
H ≥ Λ 5.2
for some constant Λ, the following Bakry-Emery inequality [2] holds: ∂
t
D p
f
t
≤ − Λ
N D
p f
t
notice the additional N factor due to the N
−1
in front of the second order term in the generator L, see 3.7. This implies the logarithmic Sobolev inequality that for any probability density g, with
respect to µ,
D p
g = −
Z p
g L p
gd µ ≥
Λ N
Sg 5.3
542
In this case, the Dirichlet form is a decreasing function in time and we thus have for any t s that
D p
f
t
≤ S f
s
t − s
5.4 In our setting, we have
Hess H =
∂
2
H ∂ λ
i
∂ λ
j
= δ
i j
N + X
k 6= j
2 λ
j
− λ
k 2
− δ
i 6= j
2 λ
i
− λ
j 2
≥ N · Id 5.5
as a matrix inequality away from the singularities see remark below how to treat the singular set. Thus we have
∂
t
D p
f
t
≤ −D p
f
t
5.6 and by 5.3
∂
t
S f
t
≤ −S f
t
5.7 This tells us that S f
t
in 3.9 is exponential decaying as long as t ≫ 1. But for any time t ∼ 1 fixed, the entropy is still the same order as the initial one. Note that t
∼ 1 is the case considered in Johasson’s work [21].
Remark 5.1. The proof of 5.5 and the application of the Bakry-Emery condition in 5.6 requires further justification. Typically, Bakry-Emery condition is applied for Hamiltonians
H defined on spaces without boundary. Although the Hamiltonian
H 3.5 is defined on R
N
, it is however convex only away from any coalescence points
λ
i
= λ
j
for some i 6= j; the Hessian of the logarithmic
terms has a Dirac delta singularity with the wrong negative sign whenever two particles overlap. In accordance with the convention that we work on the space Ξ
N
throughout the paper, we have to consider
H restricted to Ξ
N
, where it is convex, i.e. 5.5 holds, but we have to verify that the Bakry-Emery result still applies. We review the proof of Bakry and Emery and prove that the
contribution of the boundary term is zero. Recall that the invariant measure exp
−H dλ and the dynamics L =
1 2N
[∆−∇H ∇] are restricted to Ξ = Ξ
N
. With h = p
f we have ∂
t
h
2
= Lh
2
= 2hLh + 1
N ∇h
2
, i.e.
∂
t
h = Lh + 1
2N h
−1
∇h
2
. Computing
∂
t
D p
f
t
, we have ∂
t
1 2N
Z
Ξ
∇h
2
e
−H
dλ =
1 N
Z
Ξ
∇h∇ Lh +
1 2N
h
−1
∇h
2
e
−H
dλ
= 1
N Z
Ξ
h ∇hL∇h −
1 2N
∇h∇
2
H ∇h + 1
2N ∇h∇[h
−1
∇h
2
] i
e
−H
dλ
= 1
2N
2
Z
Ξ
h − ∇h∇
2
H ∇h − X
i, j
∂
2 i j
h −
∂
i
h ∂
j
h h
2
i e
−H
dλ
≤ −D p
f
t
5.8 543
assuming that the boundary term Z
∂ Ξ
∂
i
h ∂
2 i j
h e
−H
= 0 5.9
in the integration by parts vanishes. To see 5.9, consider a segment
λ
i
= λ
i+1
of the boundary ∂ Ξ. From the explicit representation
5.11, 5.12 in the next section, we will see that f
t
≥ 0 is a meromorphic function in each variable in the domain Ξ for any t
0. It can be represented as by λ
i+1
− λ
i β
F λ with some β ∈ Z,
where F is analytic and 0 F ∞ near λ
i
= λ
i+1
. Since f
t
≥ 0, we obtain that the exponent β is non-negative and even. Therefore f
1 2
t
behaves as |λ
i+1
− λ
i
|
β2
with a non-negative integer exponent
β2 near λ
i
= λ
i+1
. It then follows that ∂
i
p f
∂
2 i j
p f e
−H
vanishes at the boundary due to the factor
λ
i+1
− λ
i 2
in e
−H
, i.e. the integral 5.9 indeed vanishes.
5.2 Bound on the entropy
