by using 5.10. Recall that the eigenvalues are ordered, λ
1
λ
2
. . . λ
N
. Let L ≤ N − n n was
defined in 4.15 and define Π
L
λ := {λ
L+1
, λ
L+2
, . . . λ
L+n
} and
Π
c L
λ := {λ
1
, λ
2
, . . . λ
N
} \ Π
L
λ
its complement. For convenience, we will relabel the elements of Π
L
as x =
{x
1
, x
2
, . . . x
n
} in increasing order. The elements of Π
c L
will be denoted by Π
c L
λ := y = y
−L
, y
−L+1
, . . . y
−1
, y
1
, y
2
, . . . y
N −L−n
∈ Ξ
N −n
, again in increasing order Ξ was defined in 3.11. We set
J
L
:= {−L, −L + 1, . . . , −1, 1, 2, . . . N − L − n}
6.2 to be the index set of the y’s. We will refer to the y’s as
external points and to the x
j
’s as internal
points. Note that the indices are chosen such that for any j we have y
k
x
j
for k 0 and y
k
x
j
for k
0. In particular, for any fixed L, we can split any y ∈ Ξ
N −n
as y = y
−
, y
+
where
y
−
:= y
−L
, y
−L+1
, . . . y
−1
, y
+
:= y
1
, y
2
, . . . y
N −L−n
The set Ξ
N −n
with a splitting mark after the L-th coordinate will be denoted by Ξ
N −n L
and we use the
y
∈ Ξ
N −n
⇐⇒ y
−
, y
+
∈ Ξ
N −n L
one-to-one correspondance. For a fixed L we will often consider the expectation of functions O
y on Ξ
N −n
with respect to µ or
f µ; this will always mean the marginal probability:
E
µ
O := Z
O yuy
−
, x
1
, x
2
, . . . x
n
, y
+
dydx, y = y
−
, y
+
. 6.3
E
f
O := Z
O y f uy
−
, x
1
, x
2
, . . . x
n
, y
+
dydx. 6.4
For a fixed L ≤ N − n and y ∈ Ξ
N −n
let f
L
y
x = f
y
x = f
t
y, x
Z f
t
y, xµ
y
dx
−1
6.5 be the conditional density of
x given y with respect to the conditional equilibrium measure
µ
L
y
dx = µ
y
dx = u
y
xdx,
u
y
x := uy, x
Z u
y, xdx
−1
6.6 Here f
L
y
also depends on time t, but we will omit this dependence in the notation. Note that for any fixed
y
∈ Ξ
N −n
, any value x
j
lies in the interval I
y
:= [ y
−1
, y
1
], i.e. the functions u
y
x and f
y
x
are supported on the set Ξ
n y
:= n
x = x
1
, x
2
, . . . , x
n
: y
−1
x
1
x
2
. . . x
n
y
1
o ⊂ I
n
y
. 547
Now we localize the good set Ω introduced in Definition 4.1. For any fixed L and y = y
−
, y
+
∈ Ξ
N −n L
we define Ω
y
:= {Π
L
λ : λ ∈ Ω, Π
c L
λ = y} = {x = x
1
, x
2
, . . . , x
n
: y
−
, x, y
+
∈ Ω}. Set
Ω
1
= Ω
1
L := y
∈ Ξ
N −n L
: P
f
y
Ω
y
≥ 1 − C e
−n
γ12
. 6.7
Since P
Ω = P
f
P
f
y
Ω
y
, from 4.19 we have
P
f
Ω
1
≥ 1 − C e
−n
γ12
. 6.8
Here P
f
Ω
1
is a short-hand notation for the marginal expectation, i.e. P
f
Ω
1
:= P
f
Π
c L
−1
Ω
1
, but we will neglect this distinction.
Note that y
∈ Ω
1
also implies, for large N , that there exists an x
∈ I
n
y
such that y
−
, x, y
+
∈ Ω.
This ensures that those properties of λ ∈ Ω that are determined only by y’s, will be inherited to the
y’s. E.g. y ∈ Ω
1
will guarantee that the local density of y’s is close to the semicircle law on each
interval away from I
y
. More precisely, note that for any interval I = [E − η
∗ m
2, E + η
∗ m
2] of length η
∗ m
= 2
m
n
γ
N
−1
and center E, |E| ≤ 2 − κ, that is disjoint from I
y
, we have, by 4.16,
y ∈ Ω
1
, I
∩ I
y
= ; =⇒
N I N
|I| − ̺
sc
E ≤
1 N |I|
1 4
n
γ12
. 6.9
Moreover, for any interval I with |I| ≥ n
γ
N
−1
we have, by 4.18,
y ∈ Ω
1
, I
∩ I
y
= ; =⇒
N I ≤ KN|I|. 6.10
For any L with N κ
3 2
≤ L ≤ N1 − κ
3 2
, let E
L
= N
−1 sc
LN
−1
, i.e. N
Z
E
L
−2
̺
sc
λdλ = L. 6.11
Then we have − 2 + Cκ ≤ E
L
≤ 2 − Cκ, ̺
sc
E
L
≥ cκ
1 2
6.12 Using 4.21 and 4.22 from Lemma 4.3 on the set Ω see 4.18, we for any
y ∈ Ω
1
L we have | y
−1
− N
−1 sc
LN
−1
| ≤ C n
−γ6
, |I
y
| − n
N ̺
sc
E
L
≤ CN
−1
n
γ+34
6.13 in particular
| y
−1
|, | y
1
| ≤ 2 − κ2 and |I
y
| ≤ C n
N 6.14
with C = C κ.
Let Ω
2
= Ω
2
L = n
y
−
, y
+
∈ Ξ
N −n L
, : |I
y
| ≤ KnN
−1
o 6.15
with some large constant K. On the set Ω we have |I
y
| ≤ KnN see 6.14, thus Π
c L
Ω ⊂ Ω
2
L, i.e.
P
f
Ω
2
≥ 1 − C e
−n
γ6
. 6.16
548
6.2 Localization of the Dirichlet form
