Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel

In this paper, we study the large $N$ behavior of the moment-generating function (MGF) of the linear statistics of $N\times N$ Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite $N$ Fredholm determinant expression of the MGF of the linear statistics, we find the large $N$ asymptotics of the MGF associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then we obtain the mean and variance of the suitably scaled linear statistics. We show that there is an equivalence between the large $N$ behavior of the MGF of the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to the statistical equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In the end, we use the Coulomb fluid method to obtain the mean and variance of another type of linear statistics in GUE, which reproduces the result of Basor and Widom.


Introduction
In random matrix theory, the joint probability density function for the eigenvalues {x j } N j=1 of N ×N Hermitian matrices from an unitary ensemble (β = 2), symplectic ensemble (β = 4) or orthogonal ensemble (β = 1) is given by [15] w(x j ).
The moment-generating function (MGF) of the linear statistics N j=1 F (x j ) is, We write it in the following form where f (x) = e −λF (x) − 1. We assume f (x) lies in the Schwartz space [19] and 1 + f (x) = 0 over [a, b].
Min and Chen [16] expressed G (4) N (f ) and G (1) N (f ) as Fredholm determinants based on the work [12] and [22]. For the β = 1 case, we take N to be even for simplicity. We state the results as the following two lemmas [16]. (4) where K (1) N is an integral operator with kernel ν jk εψ j (x)ψ k (y).
We introduce here some notations, which will be used in the following sections of this paper.
Let K(x, y) be the Airy kernel The equality (1.4) is obtained by taking the limit y → x from (1.3) and using the property Ai ′′ (x) = x Ai(x).
We then define Finally, we mention that χ J (x) is the indicator function defined on the interval J, namely, The paper [16] studied the large N behavior of the MGF of the linear statistics in Gaussian ensembles associated with the sine kernel and Laguerre ensembles associated with the Bessel kernel.
This paper continues to study the large N behavior of the MGF in these Gaussian and Laguerre ensembles associated with the Airy kernel, from which we obtain the mean and variance of the scaled linear statistics. The unitary case is the simplest one among them. We established the relation between the mean and variance of the scaled linear statistics in symplectic, orthogonal and unitary ensembles. We also show that as N → ∞, the MGF of a suitably scaled linear statistics in the Gaussian ensembles and Laguerre ensembles are the same, which leads to the same mean and variance of the linear statistics between the Gaussian ensembles and Laguerre ensembles. For the problems on the mean and variance of linear statistics in unitary ensembles, see [4,5,11,17] for reference. Finally, we point out that the variance of linear statistics play an important role in the random matrix theory of quantum transport [8,9].
The rest of this paper is organized as follows. In Sec. 2, we study the large N behavior of the MGF of the scaled linear statistics in Gaussian unitary, symplectic and orthogonal ensembles, respectively. From this we obtain the mean and variance of the scaled linear statistics in the three Gaussian ensembles. In Sec. 3, we repeat the development of Sec. 2, but for the Laguerre ensembles.
In Sec. 4, we use two different methods to consider the large N behavior of another type of linear statistics in GUE. The mean and variance of the linear statistics are obtained. The conclusion is given in Sec. 5.

Gaussian Unitary Ensemble
In the Gaussian case, w(x) = e −x 2 , x ∈ R. From Lemma 1.1 we have where H j (x) are the Hermite polynomials of degree j.
We consider the large N asymptotics of G N (f ) in this subsection. It is well known that log det I + K (2) We state a theorem before our discussion.
where K(x, y) is the Airy kernel defined by (1.3).
Remark. The above result was obtained by [10,13,18], but they did not show the order term. See also [21] on the study of this Airy kernel.
We now use Theorem 2.1 to compute (2.2) term by term as N → ∞. We replace f (x) by in the following computations. The first term reads,

TrK
(2) The second term, It follows from (2.2) that log det I + K (2) We proceed to study the mean and variance of the scaled linear statistics N j=1 F 2 so we need to obtain the coefficients of λ and λ 2 from (2.4). From the relation of f (x) and F (x) we (2.5) Substituting (2.5) into (2.4), we have log det I + K (2) x j − √ 2N , and note that log G N f . Then we have the following theorem.
where K(x, y) is the Airy kernel defined by (1.3).

Gaussian Symplectic Ensemble
In this case, where ϕ j (x) is given by (2.1). It follows that M (4) is the direct sum of the N copies of [12,22]). From Lemma 1.2, we obtain the following result [16]. where and K 2N +1 is an operator on L 2 (R) with kernel We also have the following expansion formula, Similarly as Theorem 2.1, we have the following theorem.
Proof. From the definition (2.1) and the asymptotic formula (2.3), we readily obtain (2.9) and (2.10). It follows from the definition of ε that where use has been made of (2.9).
Similarly, we find 11 12 ). Now we use Theorem 2.4 and Theorem 2.5 to compute (2.8) as N → ∞. We will change f (x) in the following calculations. In this case, f ′ (x) becomes x − √ 4N . We consider Tr T GSE firstly, The first term reads, The second term, Then where L(x, y) is given by (1.5).
The third term, The fourth term, So we obtain We proceed to compute Tr T 2 GSE , Similarly, we compute the ten traces one by one. We write down the result here without the detailed calculations: Proceeding as in the previous subsection, we replace . Substituting these into (2.11) and (2.12), we finally find Denoting by µ x j − √ 4N , and noting that log G are given by (2 .6) and (2.7), respectively.

Gaussian Orthogonal Ensemble
It is convenient in this case to choose w(x) to be the square root of the Gaussian weight, and keep in mind that N is even. Define where ϕ j (x) is given by (2.1). It follows that M (1) is the direct sum of the N 2 copies of [12,22]). From Lemma 1.3, we obtain the following result [16].
Theorem 2.7. For the Gaussian orthogonal ensemble, we have and K N is an operator on L 2 (R) with kernel We also have log det(I + T GOE ) = Tr log(I + T GOE ) = Tr Similarly as the previous subsection, we have the following results. given by (1.2).
In the computations below, we replace f (x) by f 2 as N → ∞, we obtain the following results: where R contains the terms of integrals with integrands consisting of f , f , f ′ or f , f ′ , f ′ . These lead to at least power 3 of λ in the following discussions, and they will not affect the final results, so we need not write down the detailed results of R.

Laguerre Unitary Ensemble
In the Laguerre case, w(x) = x α e −x , α > −1, x ∈ R + . From Lemma 1.1 we have j (x) are the Laguerre polynomials of degree j. We state a theorem before our discussion.  Proof. From the asymptotic formula of Laguerre polynomials [20] (page 201), we have Using the Christoffel-Darboux formula, Replacing the variables x by 4N + 2α + 2 + 2 together with Stirling's formula, we obtain the desired result after some elaborate computations.
Remark. The above result was obtained by Forrester [13], but they also did not show the order term.
We now use Theorem 3.1 to compute (2.2) term by term as N → ∞. We replace f (x) by in the following computations. The first term reads,

TrK
(2) The second term, It follows from (2.2) that log det I + K (2) We proceed to study the mean and variance of the scaled linear statistics N j=1 . From the relation (2.5), we have log det I + K (2) be the mean and variance of the linear statistics N j=1 . Then we have the following theorem.
where K(x, y) is the Airy kernel defined by (1.3).
Remark. Comparing Sec. 2.1 and Sec. 3.1, we see that the large N behavior of the MGF of a suitably scaled linear statistics in GUE are the same with a suitably scaled linear statistics in LUE.
It follows that as N → ∞, the mean and variance of the corresponding linear statistics are also the same in GUE and LUE.

5)
and S (4) We also have the following expansion formula, Using the similar method in Theorem 3.1, we obtain the following theorem.  (1.3).
where B(x) is given by (1.2).
Proof. From the definition (3.4) and the asymptotics (3.1), we have where we have made use of the formula [1] (page 257) It follows that Similarly, we obtain and εϕ (α−1) The proof is complete. Now we use Theorem 3.4 and 3.5 to compute Tr T LSE and Tr T 2 LSE as N → ∞. We change f (x) 8N − 2α)) in the following computations. Firstly we have The first term reads, The second term, Let x = 8N + 2α + 2 The third term, Hence, Similarly we obtain the result for Tr T 2 LSE after some tedious computations, (3.7) From the above, we find that as N → ∞, It follows that as N → ∞, We have the following theorem. Theorem 3.6. As N → ∞, are given by (3.2) and (3.3), respectively.
Remark. We find that the large N behavior of the MGF of a suitably scaled linear statistics in LSE are the same with a suitably scaled linear statistics in GSE. It follows that as N → ∞, the mean and variance of the corresponding linear statistics are also the same in LSE and GSE.

Laguerre Orthogonal Ensemble
In this subsection, w(x) is taken to be the square root of the Laguerre weight, namely, and N is even. Following [16], we let Note that the definition of ϕ N is an integral operator with kernel We also have the following expansion formula, Similarly as previous subsection, we have the following theorems. Theorem 3.9. As N → ∞, we have . 4N − 2α − 4)) and use Theorem 3.8 and 3.9 to compute Tr T LOE and Tr T 2 LOE . We find that as N → ∞, It follows that as N → ∞, log det(I + T LOE ) = log det(I + T GOE ).
Denoting by µ x j − √ 2N , we have the following theorem. Remark. We find that the large N behavior of the MGF of a suitably scaled linear statistics in LOE are the same with a suitably scaled linear statistics in GOE. It follows that as N → ∞, the mean and variance of the corresponding linear statistics are also the same in LOE and GOE.

Gaussian Unitary Ensemble Continued
For the Gaussian unitary ensemble, if we change f (x) to f x − √ 2N , we can gain a better insight into the mean and variance of the corresponding linear statistics by using the result of Basor and Widom [7]. We see that as N → ∞, . This is because, as N → ∞, We now introduce the result of Basor and Widom as the following lemma [7]. where Hence we have the following theorem. respectively.
At the end of this section, we use another method, the coulomb fluid approach, to prove the above theorem. We state an important lemma [6]. For the Gaussian unitary ensemble, it is known that [6] σ(x) = √ b 2 − x 2 π , b = −a = √ 2N.

Conclusion
This paper studies the large N behavior of the MGF of the scaled linear statistics in Gaussian ensembles and Laguerre ensembles, from which we obtain the mean and variance of the corresponding linear statistics. We find that there is an equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In addition, we use the results of [7] and [6] to consider another type of linear statistics in GUE and also obtain the mean and variance of the corresponding linear statistics. For the GSE and GOE, we will deal with the corresponding type of linear statistics in the future.