6.2 Localization of the Dirichlet form
For any L ≤ N − n and any y ∈ Ξ
N −n L
, we define the Dirichlet form D
L, y
f := Z
1 2N
∇
x
f
2
d µ
L
y
x
for functions f = f x defined on Ξ
n y
. Hence from 6.1 we have the inequality 1
N 1 − 2κ
3 2
N 1 −κ
3 2
X
L=N κ
3 2
E
f
t
D
L, y
p f
y
x ≤ C nN
−1
D p
f
t
≤ C N
1+ α
n τ
−1
6.17 where the expectation E
f
t
is defined similarly to 6.4, with f replaced by f
t
. In the first inequality in 6.17, we used the fact that, by 6.5 and 6.6,
E
f
t
D
L, y
p f
y
x
= Z
d xdy f
t
y, xuy, x D
L, y
p f
y
x
= 1
8N Z
d xdy f
t
y, xuy, x
Z d
x
′
|∇
x
′
f
t
y, x
′
|
2
f
t
y, x
′
1 R
d x
′
f
t
y, x
′
uy, x
′
u y, x
′
= 1
8N
n
X
j=1
Z d
xdy
|∇
x
j
f
t
y, x|
2
f
t
y, x
u y, x
and therefore, when we sum over all L ∈ {Nκ
3 2
, . . . , N 1 − κ
3 2
} as on the l.h.s. of 6.17, every local Dirichlet form is summed over at most n times, so we get the total Dirichlet form with a
multiplicity at most n. We define the set
G
1
= n
N κ
3 2
≤ L ≤ N1 − κ
3 2
: E
f
t
D
L,
y
p f
y
x ≤ C N
1+ α
n
2
τ
−1
o ,
6.18 then the above inequality guarantees that for the cardinality of
G
1
, |G
1
| N 1
− 2κ
3 2
≥ 1 − C
n .
6.19 For L
∈ G
1
, we define Ω
3
= Ω
3
L := n
y
−
, y
+
∈ Ξ
N −n L
: D
L, y
p f
y
x ≤ C N
1+ α
n
4
τ
−1
o ,
6.20 then
P
f
Ω
c 3
≤ C n
−2
. 6.21
549
6.3 Local entropy bound
Suppose that L ∈ G
1
and fix it. For any y
∈ Ξ
N −n L
denote by H
y
x = N
n
X
i=1
1 2
x
2 i
− 2
N X
1 ≤i j≤n
log |x
j
− x
i
| − 2
N X
k ∈J
L
n
X
i=1
log |x
i
− y
k
|
6.22 Note that
Hess H
y
x ≥ inf
x ∈I
y
X
k ∈J
L
|x − y
k
|
−2
6.23 for any
x ∈ I
n
y
as a matrix inequality. On the set y
∈ Ω
2
L we have inf
x ∈I
y
X
k ∈J
L
|x − y
k
|
−2
≥ 1
| y
1
− y
−1
|
2
≥ cN
2
n
2
, y
∈ Ω
2
L. We can apply the logarithmic Sobolev inequality 5.3 to the local measure
µ
y
, taking into account Remark 5.1. Thus we have
S
µ
L
y
f
y
≤ c
−1
n
2
N
−1
D
L, y
p f
y
x ≤ C n
6
N
α
τ
−1
for any y
∈ Ω
2
L ∩ Ω
3
L, L ∈ G
1
. 6.24
Using the inequality p
S f ≥ C
Z | f − 1|dµ
6.25 for
µ = µ
y
and f = f
y
, we have also have Z
| f
y
− 1|dµ
y
2
≤ C n
6
N
α
τ
−1
for any y
∈ Ω
2
L ∩ Ω
3
L, L ∈ G
1
6.26 We will choose t = N
−1
τ with τ = N
β
such that C n
6
N
α
τ
−1
≤ n
−4
6.27 i.e.
β ≥ 10ǫ + α.
6.4 Good external configurations
Kategori