Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$, where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues within $\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.


I. INTRODUCTION AND MAIN RESULT
Denote by H n the set of n × n random Hermitian matrices endowed with the probability measure where Z n is the normalization constant and where for every M = (M ij ) 1≤i,j≤n in H n (R [z] being the real part of z ∈ C and I [z] its imaginary part). This set is known as the Gaussian Unitary Ensemble (GUE) and corresponds to the case where a n × n hermitian matrix M has independent, complex, zero mean, Gaussian distributed entries with variance E|M ij | 2 = 1 n above the diagonal while the diagonal entries are independent real Gaussian with the same variance. Much is known about the spectrum of M. Denote by λ (n) 1 , λ (n) 2 , · · · , λ (n) n the eigenvalues of M (all distinct with probability one), then : -The joint probability density function of the (unordered) eigenvalues (λ (n) 1 , · · · , λ (n) n ) is given by : where C n is the normalization constant.
- [9] The empirical distribution of the eigenvalues 1 n n i=1 δ λ (n) i (δ x stands for the Dirac measure at point x) converges toward Wigner's semi-circle law whose density is : - [1] The largest eigenvalue λ (n) max (resp. the smallest eigenvalue λ (n) min ) almost surely converges to 2 (resp. −2), the right-end (resp. left-end) point of the support of the semi-circle law as n → ∞. 2 - [6] The centered and rescaled quantity n 2 3 λ (n) max − 2 converges in distribution toward Tracy-Widom distribution function F + GU E which can be defined in the following way : where q solves the Painlevé II differential equation : and Ai(x) denotes the Airy function. In particular, F + GU E is continuous. Similarly, n 2 3 λ (n) . If ∆ is a Borel set in R, denote by : i.e. the number of eigenvalues in the set ∆. The following theorem is the main result of the article.

B. Application : Fluctuations of the condition number in the GUE
As a simple consequence of Corollary 1, we can easily describe the fluctuations of the condition number λmax λmin . Corollary 2: Let M be a n × n matrix from the GUE. Denote by λ min and λ max its smallest and largest eigenvalues, then : where D − → denotes convergence in distribution, λ − and λ + are independent random variable with respective distribution F − GU E and F + GU E . Proof: The proof is a mere application of Slutsky's lemma (see for instance [8,Lemma 2.8]). Write : Now, λmin+2 2λmin goes almost surely to zero 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 implies that λ min + 2 2λ min n Another application of Slutsky's lemma yields the convergence (in distribution) of the right-hand side of (6) to the limit of − 1 2 n 2 3 (λ max − 2) + n 2 3 (λ min + 2) , that is − 1 2 (X + Y ) with X and Y independent and distributed according to Denote by K n (x, y) the following Kernel on R 2 : = ψ hal-00556126, version 1 -15 Jan 2011 Equation (8) is obtained from (7) by the Christoffel-Darboux formula. We recall the two well-known asymptotic results Proposition 1: a) Bulk of the spectrum. Let µ ∈ (−2, 2).
where ρ(µ) = 2π . Furthermore, the convergence (9) is uniform on every compact set of R 2 . b) Edge of the spectrum.
where Ai(x) is the Airy function. Furthermore, the convergence (10) is uniform on every compact set of R 2 . We will need as well the following result on the asymptotic behavior of functions ψ (n) k . Proposition 2: Let µ ∈ (−2, 2), let k = 0 or k = 1 and denote by K a compact set of R. a) Bulk of the spectrum. There exists a constant C such that for large n, b) Edge of the spectrum. There exists a constant C such that for large n, The proof of these results can be found in [3,Chapter 7].
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 [2,Chapter 6] : Definition 1: Consider a linear operator S defined for any bounded integrable function f : R → R by where S(x, y) is a bounded integrable Kernel on R 2 → R with compact support. The Fredholm determinant D(z) associated with operator S is defined as follows : for each z ∈ C, i.e. it is an entire function. Its logarithmic derivative has the simple expression : where For details related to Fredholm determinants, see for instance [5], [7].
The following kernel will be of constant use in the sequel : where λ = (λ 1 , · · · , λ p ) ∈ R p or λ ∈ C p , depending on the need.

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Remark 1: Kernel K n (x, y) is unbounded and one cannot consider its Fredholm determinant without caution. Kernel S n (x, y) is bounded in x since the kernel is zero if x is outside the compact set ∪ p i=1 ∆ i , but a priori unbounded in y. In all the forthcoming computations, one may replace S n with the bounded kernelS n (x, y) = p i, =1 λ i 1 ∆i (x)1 ∆ (y)K 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 ; let = ( 1 , · · · , p ) ∈ N p and denote by ∆ = (∆ 1 , · · · , ∆ p ), where every ∆ i is a bounded Borel set. Assume that the ∆ i 's are pairwise disjoint. Then the following identity holds true : where S n (λ, ∆) is the operator associated to the kernel defined in (18).

Proof of Proposition 3 is postponed to Appendix A.
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.e. 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 : For 1 ≤ m, ≤ p, define : where S n (x, y; λ) is given by (21). Proposition 4: Let Λ ⊂ C p be a compact set. There exist two constants R = R(Λ) > 0 and C = C(Λ) > 0, independent from n, such that for n large enough, Proposition 4 is proved in Appendix B. Consider the iterated kernel |S n | (k) (x, y; λ) defined by : where S n (x, y; λ) is given by (21). The next estimates will be stated with λ ∈ C p fixed. Note that |S n | (k) is nonnegative and writes : As previously, define for 1 ≤ m, ≤ p : The following estimates hold true : Proposition 5: Consider the compact set Λ = {λ} and the associated constants R = R(λ) and C = C(λ) as given by Prop. 4. Let β > 0 be such that β > R −1 and consider ∈ (0, 1 3 ). There exists an integer N 0 = N 0 (β, ) such that for every n ≥ N 0 and for every k ≥ 1, Proposition 5 is proved in Appendix C.

B. End of proof
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. As previously, denote S n (x, y; λ) = S n (x, y; λ, ∆ n ) ; denote also S i,n (x, y; For every z ∈ C and λ ∈ C p , we use the following notations : The following controls will be of constant use in the sequel. Proposition 6: 1) Let λ ∈ C p be fixed. The sequences of functions : are uniformly bounded on every compact subset of C. 2) Let z = 1. The sequences of functions : Proof of Proposition 6 is provided in Appendix D.
We introduce the following functions : where denotes the derivative with respect to z ∈ C. We first prove that f n goes to zero as z → 0. 1) Asymptotic study of f n in a neighbourhood of z = 0: In this section, we mainly consider the dependence of f n in z ∈ C while λ ∈ C p is kept fixed. We therefore drop the dependence in λ to lighten the notations. Equality (16) yields : where denotes the derivative with respect to z ∈ C and T n (k) and T i,n (k) are as in (17), respectively defined by : Recall that D n and D i,n are entire functions (of z ∈ C). The functions D n D n and D i,n D i,n admit a power series expansion around zero given by (29). Therefore, the same holds true for f n (z), moreover : Lemma 1: Define R as in Proposition 4. For n large enough, f n (z) defined by (28) is holomorphic on B(0, R) := {z ∈ C, |z| < R}, and converges uniformly to zero as n → ∞ on each compact subset of B(0, R).
From (34), function j n,k is invariant up to any circular shift π q , so that j n,k (σ) coincides with j n (m, ,σ) for any σ = π q (m, ,σ) as above. Therefore, ×|K n (x 1 , x 2 ) · · · K n (x k , x 1 )|dx 1 · · · dx k hal-00556126, version 1 -15 Jan 2011 The latter writes It remains to notice that where (a) follows from (32), and (b) from the mere definition of the iterated kernel (24). Thus, for k ≥ 2, the following inequality holds true : For k = 1, let s n (1) = 0 so that equation T n (k) = i T i,n (k) + s n (k) holds for every k ≥ 1.
According to (28), f n (z) writes : Let us now prove that f n (z) is well-defined on the desired neighbourhood of zero and converges uniformly to zero as n → ∞. Let β > R −1 , then Propositions 4 and 5 yield : ×m,n |∆ m,n ||∆ ,n | ,

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where (a) follows from the fact that κ m + κ − 1 ≥ 1 3 . Clearly, the power series ∞ k=1 (k + 1)β k−1 z k converges for |z| < β −1 . As β −1 is arbitrarily lower than R, this implies that f n (z) is holomorphic in B(0, R). Moreover, for each compact subset K included in the open disk B(0, β −1 ) and for each z ∈ K, The right-hand side of the above inequality converges to zero as n → ∞. Thus, the uniform convergence of f n (z) to zero on K is proved ; in particular, as β −1 < R, f n (z) converges uniformly to zero on B(0, R). Lemma 1 is proved.
2) Convergence of d n to zero as n → ∞: In this section, λ ∈ C p is fixed. We therefore drop the dependence in λ in the notations. Consider function F n defined by : where log corresponds to the principal branch of the logarithm and D n and D i,n are defined in (29). As D n (0) = D i,n (0) = 1, there exists a neighbourhood of zero where F n is holomorphic. Moreover, using Proposition 6-3), one can prove that there exists a neighbourhood of zero, say B(0, ρ), where F n (z) is a normal family. Assume that this neighbourhood is included in B(0, R), where R is defined in Proposition 4 and notice that in this neighbourhood, F n (z) = f n (z) as defined in (28). Consider a compactly converging subsequence F φ(n) → F φ in B(0, ρ) (by compactly, we mean that the convergence is uniform over any compact set Λ ⊂ B(0, ρ)), then one has in particular We have proved that every converging subsequence of F n converges to zero in B(0, ρ). This yields the convergence (uniform on every compact of B(0, ρ)) of F n to zero in B(0, ρ). This yields the existence of a neighbourhood of zero, say B(0, ρ ) where : uniformly on every compact of B(0, ρ ). Recall that d n (z) = D n (z) − p i=1 D i,n (z). Combining (37) with Proposition 6-3) yields the convergence of d n (z) to zero in a small neighbourhood of zero. Now, Proposition 6-1) implies that d n (z) is a normal family in C. In particular, every subsequence d φ(n) compactly converges to a holomorphic function which coincides with 0 in a small neighbourhood of the origin, and thus is equal to 0 over C. We have proved that with λ ∈ C p fixed.
3) Convergence of the partial derivatives of λ → d n (1, λ) to zero: In order to establish Theorem 1, we shall rely on Proposition 3 where the probabilities of interest are expressed in terms of partial derivatives of Fredholm determinants. We therefore need to establish that the partial derivatives of d n (1, λ) with respect to λ converge to zero as well. This is the aim of this section.
In the previous section, we have proved that ∀(z, λ) ∈ C p+1 , d n (z, λ) → 0 as n → ∞. In particular, We now prove the following facts (with a slight abuse of notation, denote d n (λ) instead of d n (1, λ)) : 1) As a function of λ ∈ C p , d n (λ) is holomorphic.
2) The sequence (λ → d n (λ)) n≥1 is a normal family on C p .
3) The convergence d n (λ) → 0 is uniform over every compact set Λ ⊂ C p . Proof of Fact 1) is straightforward and is thus omitted. Proof of Fact 2) follows from Proposition 6-2). Let us now turn to the proof of Fact 3). As (d n ) is a normal family, one can extract from every subsequence a compactly converging one in C p (see for instance [4, Theorem 1.13]). But for every λ ∈ C p , d n (λ) → 0, therefore every hal-00556126, version 1 -15 Jan 2011 compactly converging subsequence converges toward 0. In particular, d n itself compactly converges toward zero, which proves Fact 3).
In order to conclude the proof, it remains to apply standard results related to the convergence of partial derivatives of compactly converging holomorphic functions of several complex variables, as for instance [4,Theorem 1.9]. As d n (λ) compactly converges to zero, the following convergence holds true : Let ( 1 , · · · , p ) ∈ N p , then This, together with Proposition 3, completes the proof of Theorem 1.

A. Proof of Proposition 3
Denote by E n ( , ∆) the probability that for every i ∈ {1, · · · , p}, the set ∆ i contains exactly i eigenvalues : Let P n (m) be the set of subsets of {1, · · · , n} with exactly m elements. If A ∈ P n (m), then A c is its complementary subset in {1, · · · , n}. The mere definition of E n ( , ∆) yields : Using the following formula : we obtain : Expanding the inner product and using the fact that the ∆ k 's are pairwise disjoint yields : Thus

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where (a) follows from the expansion of i (1 − k λ k 1 ∆k (x i )), (b) from the fact that the inner integral in the third line of the previous equation does not depend upon E due to the invariance of p n with respect to any permutation of the x i 's, and (c) follows from the determinantal representation (14). Therefore, Γ(λ, ∆) writes : where S n (x, y; λ, ∆) is the kernel defined in (18). As the operator S n (λ, ∆) has finite rank n, (39) coincides with the Fredholm determinant det(1 − S n (λ, ∆)) (see [7] for details). Proof of Proposition 3 is completed.

B. Proof of Proposition 4
In the sequel, C > 0 will be a constant independent from n, but whose value may change from line to line. First consider the case i = j. Denote by S µi (x, y) the following limiting kernel : Proposition 1 implies that n −κi K n (µ i + x/n κi , µ i + y/n κi ) converges uniformly to S µi (x, y) on every compact subset of R 2 , where κ i is defined by (20). Moreover, S µi (x, y) being bounded on every compact subset of R 2 , there exists a constant C i such that : It remains to take R as R −1 = max(C 1 , · · · , C p ) to get the pointwise or uniform estimate.
Consider now the case where i = j. Using notation κ i , inequalities (11) and (12) can be conveniently merged as follows : There exists a constant C such that for 1 ≤ i ≤ p, For n large enough, we obtain, using (8) : where (a) follows from (8) Let Λ = {λ} be fixed. We drop, in the rest of the proof, the dependence in λ in the notations. The mere definition of |S n | (k) yields : From the above inequality, the following is straightforward : Using Proposition 4, we obtain : where α := max(C|∆ 1 |, · · · , C|∆ p |). Now take β > R −1 and ∈ (0, where the last inequality follows from the fact that 2 + − 2κ m − 2κ i < 0, which implies that n 2+ −2κm−2κi → 0, which in turn implies that the term inside the parentheses is lower than one for n large enough. where the last inequality follows from the fact that the term inside the parentheses is lower than one for n large enough. Therefore, (25) holds for each k ≥ 1 and for n large enough.