Asymptotic Independence in the Spectrum of the GUE 381
Proof. The proof is a mere application of Slutsky’s lemma see for instance [
18 , Lemma 2.8
]. Write
n
2 3
λ
max
λ
min
+ 1 =
1 λ
min
h n
2 3
λ
max
− 2 + n
2 3
λ
min
+ 2 i
. 7
Now, λ
min −1
goes almost surely to -2 as n → ∞, and n
2 3
λ
max
− 2 + n
2 3
λ
min
+ 2 converges in distribution to the convolution of F
− GU E
and F
+ GU E
by Corollary 1. Thus, Slutsky’s lemma yields the convergence in distribution of the right-hand side of 7 to
−
1 2
λ
−
+ λ
+
with λ
−
and λ
+
independent and distributed according to F
− GU E
and F
+ GU E
. Proof of Corollary 2 is completed.
3 Proof of Theorem 1
3.1 Useful results
3.1.1 Kernels
Let {H
k
x}
k ≥0
be the classical Hermite polynomials H
k
x := e
x
2
−
d d x
k
e
−x
2
and consider the function ψ
n k
x defined for 0 ≤ k ≤ n − 1 by: ψ
n k
x :=
n 2
1 4
e
−
nx2 4
2
k
k p
π
1 2
H
k
Ç n 2
x
. Denote by K
n
x, y the following kernel on R
2
K
n
x, y :=
n −1
X
k=0
ψ
n k
xψ
n k
y , 8
= ψ
n n
xψ
n n
−1
y − ψ
n n
yψ
n n
−1
x x
− y .
9 Equation 9 is obtained from 8 by the Christoffel-Darboux formula. We recall the two well-
known asymptotic results
Proposition 1. a Bulk of the spectrum. Let µ
∈ −2, 2. ∀x, y ∈ R
2
, lim
n →∞
1 n
K
n
µ +
x n
, µ + y
n
= sin πρµx
− y πx − y
, 10
where ρµ = p
4 −µ
2
2π
. Furthermore, the convergence 10 is uniform on every compact set of R
2
. b Edge of the spectrum.
∀x, y ∈ R
2
, lim
n →∞
1 n
23
K
n
2 +
x n
23
, 2 + y
n
23
=
AixAi
′
y − Ai yAi
′
x x
− y ,
11 where Aix is the Airy function. Furthermore, the convergence 11 is uniform on every
compact set of R
2
.
382 Electronic Communications in Probability
We will need as well the following result on the asymptotic behavior of functions ψ
n k
.
Proposition 2. Let µ ∈ −2, 2, k ∈ {0, 1} and denote by K a compact set of R.
a Bulk of the spectrum. There exists a constant C such that sup
x ∈K
ψ
n n
−k
µ +
x n
≤ C .
12 b Edge of the spectrum. There exists a constant C such that
sup
x ∈K
ψ
n n
−k
2
x n
23
≤ n
16
C . 13
The proof of these results can be found in [
11 , Chapter 7
], see also [ 1
, Chapter 3 ].
3.1.2 Determinantal representations, Fredholm determinants
There are determinantal representations using kernel K
n
x, y for the joint density p
n
of the eigenvalues λ
n i
; 1 ≤ i ≤ n, and for its marginals see for instance [
10 , Chapter 6
]: p
n
x
1
, · · · , x
n
= 1
n det
¦ K
n
x
i
, x
j
©
1 ≤i, j≤n
, 14
Z
R
n −m
p
n
x
1
, · · · , x
n
d x
m+1
· · · d x
n
= n − m
n det
¦ K
n
x
i
, x
j
©
1 ≤i, j≤m
m ≤ n . 15
Definition 1. Consider a linear operator S defined for any bounded integrable function f : R → R
by S f : x
7→ Z
R
Sx, y f yd y , where Sx, y is a bounded integrable Kernel on R
2
→ R with compact support. The Fredholm determinant Dz associated with operator S is defined as follows
∀z ∈ C, Dz := det1
− zS = 1 +
∞
X
k=1
−z
k
k Z
R
k
det ¦
Sx
i
, x
j
©
1 ≤i, j≤k
d x
1
· · · d x
k
. 16
It is in particular an entire function and its logarithmic derivative has a simple expression [ 17
, Section 2.5] given by
D
′
z Dz
= −
∞
X
k=0
T k + 1z
k
, 17
where T k =
Z
R
k
Sx
1
, x
2
Sx
2
, x
3
· · · Sx
k
, x
1
d x
1
· · · d x
k
. 18
For details related to Fredholm determinants, see for instance [
14 ,
17 ].
The following kernel will be of constant use in the sequel S
n
x, y; λ, ∆ :=
