Application: Fluctuations of the ratio of the extreme eigenvalues in y; λ, ∆ and its iterations Consider µ, ∆ and ∆ y; λ the kernel y; λ := S y; λ, ∆ y; λ ,

380 Electronic Communications in Probability where by equations 1, 3 and 4 ε n α := −P n Π−α, x , n 2 3 λ max − 2 y o − P ¦ Π−α, x , ˜ Π y, α © − P n n 2 3 λ min + 2 x , ˜ Π y, α o + P ¦ N ∆ 1,n = 0 © P ¦ ˜ Π y, α © + P {Π−α, x} P ¦ N ∆ 2,n = 0 © − P {Π−α, x} P ¦ ˜ Π y, α © . Using the triangular inequality, we obtain: |ε n α| ≤ 6 max € P {Π−α, x} , P ¦ ˜ Π y, α ©Š . As { Π−α, x } ⊂ {n 2 3 λ min + 2 −α}, we have P {Π−α, x } ≤ P{n 2 3 λ min + 2 −α} −−→ n →∞ F − GU E −α −−→ α→∞ 0 . We can apply the same arguments to { ˜ Π y, α } ⊂ {n 2 3 λ max − 2 α}. We thus obtain: lim α→∞ lim sup n →∞ |ε n α| = 0 . 6 The difference P ¦ N ∆ 1,n = 0 , N ∆ 2,n = 0 © −P ¦ N ∆ 1,n = 0 © P ¦ N ∆ 2,n = 0 © in the right ­- hand side of 5 converges to zero as n → ∞ by Theorem 1 for every α large enough. We therefore obtain lim sup n →∞ |u n | = lim sup n →∞ |ε n α| . The lefthand side of the above equation is a constant w.r.t. α while the second term whose behaviour for small α is unknown converges to zero as α → ∞ by 6. Thus, lim n →∞ u n = 0. The mere definition of u n together with Tracy and Widom fluctuation results yields lim n →∞ P n n 2 3 λ min + 2 x , n 2 3 λ max − 2 y o = € 1 − F − GU E x Š F + GU E y . This completes the proof of Corollary 1.

2.2 Application: Fluctuations of the ratio of the extreme eigenvalues in

the GUE As a simple consequence of Corollary 1, we can easily describe the fluctuations of the ratio λ max λ min . The counterpart of such a result to Gaussian Wishart matrices is of interest in digital communication see [ 4 ] for an application in digital signal detection. Corollary 2. Let M be a nn matrix from the GUE. Denote by λ min and λ max its smallest and largest eigenvalues, then n 2 3 λ max λ min + 1 D −−→ n →∞ − 1 2 λ − + λ + , where D − → denotes convergence in distribution, λ − and λ + are independent random variable with respective distribution F − GU E and F + GU E . 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 Š 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 Œ . 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, ∆ :=

p X i=1 λ i 1 ∆ i xK n x, y, 19 where λ = λ 1 , · · · , λ p ∈ R p or λ ∈ C p , depending on the need, and ∆ = ∆ 1 , · · · , ∆ p is a collection of p bounded Borel sets in R. Asymptotic Independence in the Spectrum of the GUE 383 Remark 3. The kernel K n x, y is unbounded and one cannot consider its Fredholm determinant without caution. The kernel S n x, y is bounded in x since the kernel is zero if x is outside the compact closure of the set ∪ p i=1 ∆ i , but a priori unbounded in y. In all the forthcoming compu- tations, one may replace S n with the bounded kernel ˜ S n x, y = P p i,ℓ=1 λ i 1 ∆ i x1 ∆ ℓ yK n x, y and get exactly the same results. For notational convenience, we keep on working with S n . Proposition 3. Let p ≥ 1 be a fixed integer, ℓ = ℓ 1 , · · · , ℓ p ∈ N p and denote ∆ = ∆ 1 , · · · , ∆ p , where every ∆ i is a bounded Borel set. Assume that the ∆ i ’s are pairwise disjoint. Then the following identity holds true P ¦ N ∆ 1 = ℓ 1 , · · · , N ∆ p = ℓ p © = 1 ℓ 1 · · · ℓ p − ∂ ∂ λ 1 ℓ 1 · · · ‚ − ∂ ∂ λ p Œ ℓ p det 1 − S n λ, ∆ λ 1 =···=λ p =1 , 20 where S n λ, ∆ is the operator associated to the kernel defined in 19. Proof of Proposition 3 is postponed to Section 4.1.

3.1.3 Useful estimates for kernel S

n

x, y; λ, ∆ and its iterations Consider µ, ∆ and ∆

n as in Theorem 1. Assume moreover that n is large enough so that the Borel sets ∆ i,n ; 1 ≤ i ≤ p are pairwise disjoint. For i ∈ {1, · · · , p}, define κ i as κ i = ¨ 1 if − 2 µ i 2 2 3 if µ i = 2 . 21 Otherwise stated, κ 1 = κ p = 2 3 and κ i = 1 for 1 i p. Let λ ∈ C p . With a slight abuse of notation, denote by S n

x, y; λ the kernel

S n

x, y; λ := S

n

x, y; λ, ∆

n . 22 For 1 ≤ m, ℓ ≤ p and Λ ⊂ C p , define M mℓ,n Λ := sup λ ∈Λ sup x, y∈∆ m,n ∆ ℓ,n S n

x, y; λ ,

23 where S n

x, y; λ is given by 22. Proposition 4. Let Λ