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A note on spiked Wishart matrices
Accepted Manuscript A note on spiked Wishart matrices Dang-Zheng Liu, Yanhui Wang PII: DOI: Reference:
S0167-7152(17)30119-0 http://dx.doi.org/10.1016/j.spl.2017.03.019 STAPRO 7897
To appear in:
Statistics and Probability Letters
Received date: 20 April 2016 Revised date: 2 January 2017 Accepted date: 19 March 2017 Please cite this article as: Liu, D.-Z., Wang, Y., A note on spiked Wishart matrices. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.03.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
A note on spiked Wishart matrices Dang-Zheng Liu∗ and Yanhui Wang† January 2, 2017
Abstract Consider the spiked complex Wishart matrices with sample size M and population size N . As N, M → ∞ such that N/M → γ ∈ (0, 1], Baik, Ben Arous and P´ech´e (Ann. Probab. 33: 1643–1697, 2005) established a phase transition of the largest eigenvalue. In this paper we show that some of their main results also hold true in the case where N/M → 0. More precisely, we prove that limiting distribution of the largest eigenvalue is the finite GUE distribution.
1
Introduction and main results
Consider M independent and identically distributed complex Gaussian vectors of dimension N , say Z1 , . . . , ZM , each having the common joint density function p(z) =
1 π N det(Σ)
∗ Σ−1 z
e−z
,
z ∈ CN
(1.1)
where Σ is a fixed N × N positive definite matrix. Set Z = (Z1 , . . . , ZM ) to 1 ZZ ∗ be the complex sample covariance be an N × M matrix, and let S = M matrix with covariance matrix Σ (also called sample covariance matrices of spiked population models). It appears as an important model in many fields such as mathematical statistics, mathematical finance, statistical physics or wireless communications; see [1, 2, 9] and references therein for the motivations, significance and related applications. −1 the eigenvalues of Σ, usually called popuWe denote by π1−1 ≥ · · · ≥ πN lation eigenvalues, and by λ1 , · · · , λN eigenvalues of S. Then we know from ∗ Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China. E-mail: [email protected] † Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China. E-mail: yh [email protected]
1
[2, Proposition 2.1] that the joint probability density function of eigenvalues of S reads off PM,N (λ) =
N j−1 1 det(e−M πj λk )1≤k,j≤N det(λk )1≤k,j≤N Y M −N λk , C det(πkj−1 )1≤k,j≤N
(1.2)
k=1
and can be further rewritten as a determinantal point process PM,N (λ) =
1 det(KM,N (λk , λj ))1≤k,j≤N , N!
where the correlation kernel KM,N (η, ζ) =
M (2πi)2
Z
dz
Γ
Z
(1.3)
dw e−ηM (z−q)+ζM (w−q) Σ
N 1 z M Y πk − w × . w−z w πk − z
(1.4)
k=1
Here 0 < q < min{π1 , . . . , πN }, Γ is a simple closed curve encircling points π1 , . . . , πN and lying in {t ∈ Γ : Re (t) > q}, Σ is a simple closed curve encircling the origin and lying in {t ∈ Σ : Re (t) < q}, both oriented counterclockwise. Moreover, for the largest eigenvalue denoted by λmax = max{λ1 , . . . , λN }, there is a Fredholm determinant expression P(λmax ≤ ξ) = det(I − KM,N |(ξ,∞) ),
(1.5)
where KM,N |(ξ,∞) is the operator acting on L2 ((ξ, ∞)) with kernel KM,N ; see [2, Section 2.2] for more discussion. As both M and N tend to infinity such that N/M → γ ∈ (0, 1], Baik, Ben Arous and P´ech´e [2] established the threshold for the size of population eigenvalues and proved the famous Baik-Ben Arous-P´ech´e phase transition of the largest eigenvalues λmax . In particular, in the supercritical regime of the spectrum, the scaled largest eigenvalue converges in distribution to that of the finite shifted mean GUE. Interestingly, in the case that the population size N is fixed but the sample size M → ∞, they also proved that the rescaled λmax converges in distribution to that of the finite shifted mean GUE; see [2, Proposition 1.1]. In the case both M and N tend to infinity but N/M → 0, El Karoui [6] showed that when Σ = I, i.e., the null case, the largest eigenvalue λmax converges to the Tracy-Widom distribution under certain centering and scaling. And latter, Chen and Pan [4] proved that the largest eigenvalue converges to 1 almost surely for i.i.d. case. We refer the reader to a recent survey [9] and references therein for relevant developments. In the present paper we will prove that the finite shifted mean GUE appears as the scaled limit of the largest eigenvalue as well, even when N/M → 0 as N → ∞ (we are partially inspired by the Deift-Menon-Trogdon 2
paper [5] where the scaled smallest eigenvalue of LUE is proved √ to also be the Tracy-Widom distribution in the special case of M − N = 4γN ). More precisely, we suppose that N = ⌊γM ι ⌋ = γM ι − κ
(1.6)
for some γ ∈ (0, ∞) and ι ∈ (0, 1). Here ⌊x⌋ denotes the largest integer less than or equal to x, which implies that κ ∈ [0, 1). To state our main result, we first introduce the shifted mean GUE kernel (see, e.g., [7, Chapter 5.8]) defined by KGUE (k; x, y) =
Z
Γ∞
dz 2πi
Z
−2ε+i∞ −2ε−i∞
1 2 2 k dw e 2 (w −z )+yw−xz Y θl − w , (1.7) 2πi w−z θl − z
l=1
where ε > − min {θ1 , . . . , θk , 0}, Γ∞ is a closed simple curve encircling the points θ1 , . . . , θk with min{Re z : z ∈ Γ∞ } > −c and oriented counterclockwise. For k ≥ 1, we define the distribution Z x Z x 1 det(KGUE (k; xi , xj ))1≤i,j≤k dx1 · · · dxk . ··· Gk (x) = (1.8) −∞ k! −∞ Theorem 1.1. With the correlation kernel KM (η, ζ) in (1.4) and the assumption (1.6), for fixed integers r ≥ k ≥ 1, real u ∈ (0, 1) and θ1 , . . . , θk , suppose that πr+1 = · · · = πN = 1, (1.9) θl , l = 1, · · · , k, (1.10) πl = u 1 + √ M and πk+1 , . . . , πr are in a compact subset of (u, ∞). Let η=
1 ρ x + 1−ι + √ , u M u M
where ρ=
(
ζ=
γ 1−u ,
ρ y 1 + 1−ι + √ , u M u M
ι ∈ ( 21 , 1); ι ∈ (0, 12 ],
0,
then we have
√ 1 1 √ e u M (u−q)(x−y) KM (η, ζ) = KGUE (k; x, y) M →∞ u M
lim
uniformly for x, y in any compact subset of R, and further √ ρ 1 M u λmax − + ≤ x = Gk (x). lim P M →∞ u M 1−ι
(1.11)
(1.12)
(1.13)
(1.14)
As a direct consequence of Theorem 1.1, use Slutsky’s theorem or follow almost the same proof of [2, Corollary 1.1] we have 3
Corollary 1.2. Under the same assumption of Theorem 1.1, as M → ∞ we have λmax → 1/u in probability. (1.15) Remark 1.3. We conjecture that for spiked real Wishart matrices, Theorem 1.1 still holds true for limiting distribution associated with shifted mean GOE but with the same scalings; cf. [3, 8, 10]. Also, for other deformed ensembles of random matrices without the Gaussian assumption Theorem 1.1 is expected to be true; cf. [9].
2
Proof of Theorem 1.1
Our task in this section is to complete the proof by taking similar steps as in that of [2, Theorem 1.1(b)]. However, the “phase functions” given in (2.1) below have three parts with different scalings because of introducing of a double scaling, which is different from [2, Theorem 1.1(b)]. The leading phase function f seems much simpler, and so it may be easier for us to choose proper “steepest descent” contours. Proof. First, we are devoted to the detailed proof of the statement (1.13) when ι ∈ ( 21 , 1). Similarly, we can complete the proof in the case of ι ∈ (0, 12 ]. Recalling (1.11), we rewrite (1.4) as √ Z Z √ √ M −M f (z)+M f (w) − Mr1 (z)+ M r2 (w) dw e e KM (x, y) : = dz (2πi)2 u Γ Σ (2.1) k 1 Y πl − w −M ι s(z)+M ι s(w) g(w) , ×e g(z) w − z πl − z l=1
where f (z) = u−1 (z − q) − ln z,
(2.2)
s(z) = ρ(z − q) + γ ln(1 − z),
r1 (z) = u
−1
r2 (z) = u
x(z − q),
and −r−κ
g(z) := (z − 1)
r Y
l=k+1
−1
(2.3) y(z − q)
1 . πl − z
(2.4)
(2.5)
Obviously, both f ′ (z) = 0 and s′ (z) = 0 have the same solution pc := u. Since 1 f ′′ (pc ) = 2 > 0, (2.6) pc the point pc is suitable for the steepest-descent analysis for KM (x, y). 4
Σ2 Σ1 0
Γ′
Γ′′2
Γ′′1
q pc
Γ′′3 1
π∗
Figure 1: Contours of double integrals Let ε be defined as (1.7), set εu q := pc − √ , M
(2.7)
and we choose two contours as depicted in Figure 1. Define 2εu Σ1 = pc − √ + iy : y ∈ [0, 1] , (2.8) M 1 2εu 2εu Σ2 = (pc − √ )2 + 1 2 eiθ : θ ∈ [arctan((pc − √ )−1 ), π] , (2.9) M M S S S and set Σ = Σ1 Σ2 Σ1 Σ2 (here ourSchoice is slightly different from that in [2]). For large R > 0, set Γ = Γ′ Γ′′ , where Γ′ is a small closed curve encircling θ1 , . . . , θk , pc and lying on the right of q and Γ′′ is a rectangle Γ′′ =
3 [
l=1
Γ′′l
!
∪
3 [
l=1
!
Γ′′l ,
(2.10)
with
p c + π∗ p c + π∗ + iy : y ∈ [0, ] , 2 a p c + π∗ p c + π∗ ′′ :x∈[ , R] , Γ2 = x + i a 2 p c + π∗ Γ′′3 = R + iy : y ∈ [0, ] . 2 Γ′′1 =
(2.11) (2.12) (2.13)
Here π∗ = min {πk+1 , . . . , πr , 1} and the positive parameter a will be chosen according to the condition (2.34) below. For any fixed δ > 0, denote Σ′ := Σ ∩ {z : |z − pc | < δ}, and Σ′′ := Σ\Σ′ . We choose δ < 1 small enough such that the Taylor series of f and g converge in the disk centered at pc with radius δ, and also such that Σ1 intersects 5
{z : |z − pc | = δ} at some point on the upper-half plane which is denoted by 2εu ˜ Note that δ˜ → δ as M → ∞. p∗ = pc − √ + iδ. M Next, we split KM into three parts Z Z Z Z Z Z dw · dz dw · + dw · + dz dz KM = (2.14) Σ′ Γ′′ Σ′′ Γ Σ′ Γ′ ′ ′′ ′′′ =: K + K + K , and will do asymptotic analysis by the steepest decent argument. And for this the following statement plays a key role. Claim: There exist constants M0 , C1 > 0 such that for every M ≥ M0 the following hold true. sup Re (f (z) − f (pc )) + M ι−1 (s(z) − s(pc )) ≤ −C1 , (2.15) z∈Σ′′
and
inf′′ Re (f (z) − f (pc )) + M ι−1 (s(z) − s(pc )) ≥ C1 .
z∈Γ
For simplicity, we define a function depending on ε > 0 as ( e−εx , x ≥ 0; Lε (x) = 1, x < 0.
(2.16)
(2.17)
Moreover, we consider the case that x, y ≥ W where W is a fixed real number. √ Notice the√inequalities M Re (r2 (w) − r2 (pc )) ≤ −2εy for y ≥ 0 and |w − z| ≥ εu/ M with w ∈ Σ, z ∈ Γ, by the Claim above, there are positive constants C2 and C3 independent of x, y such that ′′ K ≤
√ M M (r2 (pc )−r1 (pc )) e 2ε(2πi)2 u2
Z
dz e−M Re (f (z)−f (pc )) k √ Y 1 − M Re (r1 (z)−r1 (pc )) −M ι Re (s(z)−s(pc )) 1 e ×e g(z) πl − z l=1 Z √ ι dw eM Re (f (w)−f (pc )) e M Re (r2 (w)−r2 (pc )) eM Re (s(w)−s(pc )) × ′′ Σ k Y × g(w) (πl − w) Γ
l=1
≤ C3 M e−C2 M Lε (x)Lε (y).
(2.18)
Similay, we have ′′′ K ≤ C3 M e−C2 M Lε (x)Lε (y). 6
(2.19)
These show that both K′′ and K′′′ decay exponentially to zero. To estimate K′ , the Taylor expansions of f (z), s(z), r1 (z) and r2 (z) at z = pc give us 1 1 (z − pc )2 + O((z − pc )3 ), 2 p2c 1 γ s(z) = s(pc ) − (z − pc )2 + O((z − pc )3 ), 2 (1 − pc )2 r1 (z) = r1 (pc ) + u−1 x(z − pc ),
(2.21)
r2 (z) = r2 (pc ) + u−1 y(z − pc ).
(2.23)
f (z) = f (pc ) +
(2.20)
(2.22)
and Then, for sufficient large M , we have
1 K′ =O √ Lε′ (x)Lε′ (y)+ M 1 Z −2ε+i∞ Z 2 2 k dw e 2 (w −z )+yw−xz Y θl − w dz , e−ε(x−y) w−z θl − z Γ∞ 2πi −2ε−i∞ 2πi
(2.24)
l=1
where 0 < ε′ < ε. Combination of equations (2.18), (2.19) and (2.24) completes the proof of statement (1.13), since the convergence is obviously uniform for x, y in a compact subset of R. In order to prove the statement (1.14), we need to strengthen the statement (1.13) from uniform convergence into the trace norm convergence of the integral operators with respect to the correlation kernels. For this we can use the same argument as in the proof of Theorem 1.1 [5, Section 4.3], that is, it is sufficient to find a dominating function over (W, +∞) × (W, +∞) for KM (x, y). Actually, we easily see from (2.18), (2.19) and (2.24) that there exists a constant C4 > 0 such that |KM (x, y)| ≤ C4 Lε′ (x)Lε′ (y).
(2.25)
Finally, we need to verify (2.15) and (2.16). Note that both Σ and Γ are bounded curves, it suffices to verify the assertion only in the case of s(z) = 0. Actually, it is easy to see δ 2εu sup Re (f (z)) ≤ Re (f (p∗ )) ≤ Re f (pc − √ + i ) . 2 M z∈Σ′′
7
(2.26)
Thus, there exists a constant C1 > 0 such that sup {Re (f (z) − f (pc ))}
z∈Σ′′
2εu δ ≤ Re (f (pc − √ + i )) − Re (f (pc )) 2 M 1 2εu 2εu 2 δ2 1 = − √ − log (1 − √ ) + 2 u M 2 4pc M pc
(2.27)
≤ −C1 .
The assertion (2.15) is then proved. We turn to the assertion (2.16). When z ∈ Γ′′1 , since p + π 2 π∗ − p c 1 c ∗ 2 2 − log pc Re (f (z) − f (pc )) = − log y + 2 2u 2 (2.28) is a decreasing function of y, we have inf Re (f (z) − f (pc )) ≥ Re (f (p1 ) − f (pc ))
z∈Γ′′ 1
π∗ − p c 1 4 π∗ − p c − log 1 + 2 , − log 1 + u 2u 2 2 a pc +π∗ pc +π∗ where p1 = 2 + i a denotes the top-left corner of Γ′′ . When z ∈ Γ′′2 , simple calculation shows that =
h(x) : = Re (f (z) − f (pc )) p + π 2 1 1 c ∗ 2 2 − log pc . = (x − pc ) − log x + u 2 a
(2.29)
(2.30)
Further, we see from
h′ (x) =
x 1 1 1 − 2 ≥ − >0 p +π c ∗ 2 u x +( a ) u x
(2.31)
that inf Re (f (z) − f (pc )) ≥ Re (f (p1 ) − f (pc )).
z∈Γ′′ 2
For z ∈ Γ′′3 , we have inf′′ Re (f (z) − f (pc )) =
z∈Γ3
1 1 (R − pc ) − (log(R2 + y 2 ) − log p2c ) u 2
(2.32)
(2.33)
≥ Re (f (p2 ) − f (pc )) ≥ Re (f (p1 ) − f (pc )),
∗ where p2 = R + i pc +π denotes the top-right corner of Γ′′ . a Once we choose a such that −pc ) 2(1 + π∗2u , a > u−1 (π −p ) π ∗ c − (1 + ∗ −pc )2 )1/2 (e 2u
(2.34)
combination of (2.29), (2.32) and (2.33) shows that there exists a positive constant, say C1 , such that the inequality (2.16) holds true. 8
Acknowledgment We are grateful to Prof. Zhidong Bai for helpful discussions. The work of D.-Z. L. was supported by the National Natural Science Foundation of China (Grant # 11301499), and by the Fundamental Research Funds for the Central Universities (Grant # WK3470000008).
References [1] Z.D. Bai and J.-F. Yao. Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincar´e, 44(3):447–474, 2008. [2] J. Baik, G. Ben Arous and S. P´ech´e. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (5):1643–1697, 2005. [3] A. Bloemendal and B. Vir´ag. Limits of spiked random matrices II. Ann. Probab. 44(4): 2726–2769, 2016. [4] B. B. Chen and G. M. Pan. Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero. Bernoulli. 18, no. 4, 1405–1420, 2012. [5] P. Deift, G. Menon, and T. Trogdon. On the condition number of the critically-scaled laguerre unitary ensemble. arXiv:1507.00750, 2015. [6] N. El Karoui. On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity. arXiv:0309355v1, 2003. [7] P.J. Forrester. Log-gases and random matrices (LMS-34). Princeton University Press, Princeton, NJ, 2010. [8] M.Y. Mo. Rank 1 real Wishart spiked model. Commun. Pure and Applied Math. LXV, 1528–1638, 2012. [9] S. P´ech´e. Deformed ensembles of random matrices. Proceedings of ICM vol. III, 1159-1174, Seoul, 2014. [10] D. Wang. The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I. J. Stat. Phys. 146, no. 4, 719–761, 2012.
9
S0167-7152(17)30119-0 http://dx.doi.org/10.1016/j.spl.2017.03.019 STAPRO 7897
To appear in:
Statistics and Probability Letters
Received date: 20 April 2016 Revised date: 2 January 2017 Accepted date: 19 March 2017 Please cite this article as: Liu, D.-Z., Wang, Y., A note on spiked Wishart matrices. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.03.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
A note on spiked Wishart matrices Dang-Zheng Liu∗ and Yanhui Wang† January 2, 2017
Abstract Consider the spiked complex Wishart matrices with sample size M and population size N . As N, M → ∞ such that N/M → γ ∈ (0, 1], Baik, Ben Arous and P´ech´e (Ann. Probab. 33: 1643–1697, 2005) established a phase transition of the largest eigenvalue. In this paper we show that some of their main results also hold true in the case where N/M → 0. More precisely, we prove that limiting distribution of the largest eigenvalue is the finite GUE distribution.
1
Introduction and main results
Consider M independent and identically distributed complex Gaussian vectors of dimension N , say Z1 , . . . , ZM , each having the common joint density function p(z) =
1 π N det(Σ)
∗ Σ−1 z
e−z
,
z ∈ CN
(1.1)
where Σ is a fixed N × N positive definite matrix. Set Z = (Z1 , . . . , ZM ) to 1 ZZ ∗ be the complex sample covariance be an N × M matrix, and let S = M matrix with covariance matrix Σ (also called sample covariance matrices of spiked population models). It appears as an important model in many fields such as mathematical statistics, mathematical finance, statistical physics or wireless communications; see [1, 2, 9] and references therein for the motivations, significance and related applications. −1 the eigenvalues of Σ, usually called popuWe denote by π1−1 ≥ · · · ≥ πN lation eigenvalues, and by λ1 , · · · , λN eigenvalues of S. Then we know from ∗ Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China. E-mail: [email protected] † Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China. E-mail: yh [email protected]
1
[2, Proposition 2.1] that the joint probability density function of eigenvalues of S reads off PM,N (λ) =
N j−1 1 det(e−M πj λk )1≤k,j≤N det(λk )1≤k,j≤N Y M −N λk , C det(πkj−1 )1≤k,j≤N
(1.2)
k=1
and can be further rewritten as a determinantal point process PM,N (λ) =
1 det(KM,N (λk , λj ))1≤k,j≤N , N!
where the correlation kernel KM,N (η, ζ) =
M (2πi)2
Z
dz
Γ
Z
(1.3)
dw e−ηM (z−q)+ζM (w−q) Σ
N 1 z M Y πk − w × . w−z w πk − z
(1.4)
k=1
Here 0 < q < min{π1 , . . . , πN }, Γ is a simple closed curve encircling points π1 , . . . , πN and lying in {t ∈ Γ : Re (t) > q}, Σ is a simple closed curve encircling the origin and lying in {t ∈ Σ : Re (t) < q}, both oriented counterclockwise. Moreover, for the largest eigenvalue denoted by λmax = max{λ1 , . . . , λN }, there is a Fredholm determinant expression P(λmax ≤ ξ) = det(I − KM,N |(ξ,∞) ),
(1.5)
where KM,N |(ξ,∞) is the operator acting on L2 ((ξ, ∞)) with kernel KM,N ; see [2, Section 2.2] for more discussion. As both M and N tend to infinity such that N/M → γ ∈ (0, 1], Baik, Ben Arous and P´ech´e [2] established the threshold for the size of population eigenvalues and proved the famous Baik-Ben Arous-P´ech´e phase transition of the largest eigenvalues λmax . In particular, in the supercritical regime of the spectrum, the scaled largest eigenvalue converges in distribution to that of the finite shifted mean GUE. Interestingly, in the case that the population size N is fixed but the sample size M → ∞, they also proved that the rescaled λmax converges in distribution to that of the finite shifted mean GUE; see [2, Proposition 1.1]. In the case both M and N tend to infinity but N/M → 0, El Karoui [6] showed that when Σ = I, i.e., the null case, the largest eigenvalue λmax converges to the Tracy-Widom distribution under certain centering and scaling. And latter, Chen and Pan [4] proved that the largest eigenvalue converges to 1 almost surely for i.i.d. case. We refer the reader to a recent survey [9] and references therein for relevant developments. In the present paper we will prove that the finite shifted mean GUE appears as the scaled limit of the largest eigenvalue as well, even when N/M → 0 as N → ∞ (we are partially inspired by the Deift-Menon-Trogdon 2
paper [5] where the scaled smallest eigenvalue of LUE is proved √ to also be the Tracy-Widom distribution in the special case of M − N = 4γN ). More precisely, we suppose that N = ⌊γM ι ⌋ = γM ι − κ
(1.6)
for some γ ∈ (0, ∞) and ι ∈ (0, 1). Here ⌊x⌋ denotes the largest integer less than or equal to x, which implies that κ ∈ [0, 1). To state our main result, we first introduce the shifted mean GUE kernel (see, e.g., [7, Chapter 5.8]) defined by KGUE (k; x, y) =
Z
Γ∞
dz 2πi
Z
−2ε+i∞ −2ε−i∞
1 2 2 k dw e 2 (w −z )+yw−xz Y θl − w , (1.7) 2πi w−z θl − z
l=1
where ε > − min {θ1 , . . . , θk , 0}, Γ∞ is a closed simple curve encircling the points θ1 , . . . , θk with min{Re z : z ∈ Γ∞ } > −c and oriented counterclockwise. For k ≥ 1, we define the distribution Z x Z x 1 det(KGUE (k; xi , xj ))1≤i,j≤k dx1 · · · dxk . ··· Gk (x) = (1.8) −∞ k! −∞ Theorem 1.1. With the correlation kernel KM (η, ζ) in (1.4) and the assumption (1.6), for fixed integers r ≥ k ≥ 1, real u ∈ (0, 1) and θ1 , . . . , θk , suppose that πr+1 = · · · = πN = 1, (1.9) θl , l = 1, · · · , k, (1.10) πl = u 1 + √ M and πk+1 , . . . , πr are in a compact subset of (u, ∞). Let η=
1 ρ x + 1−ι + √ , u M u M
where ρ=
(
ζ=
γ 1−u ,
ρ y 1 + 1−ι + √ , u M u M
ι ∈ ( 21 , 1); ι ∈ (0, 12 ],
0,
then we have
√ 1 1 √ e u M (u−q)(x−y) KM (η, ζ) = KGUE (k; x, y) M →∞ u M
lim
uniformly for x, y in any compact subset of R, and further √ ρ 1 M u λmax − + ≤ x = Gk (x). lim P M →∞ u M 1−ι
(1.11)
(1.12)
(1.13)
(1.14)
As a direct consequence of Theorem 1.1, use Slutsky’s theorem or follow almost the same proof of [2, Corollary 1.1] we have 3
Corollary 1.2. Under the same assumption of Theorem 1.1, as M → ∞ we have λmax → 1/u in probability. (1.15) Remark 1.3. We conjecture that for spiked real Wishart matrices, Theorem 1.1 still holds true for limiting distribution associated with shifted mean GOE but with the same scalings; cf. [3, 8, 10]. Also, for other deformed ensembles of random matrices without the Gaussian assumption Theorem 1.1 is expected to be true; cf. [9].
2
Proof of Theorem 1.1
Our task in this section is to complete the proof by taking similar steps as in that of [2, Theorem 1.1(b)]. However, the “phase functions” given in (2.1) below have three parts with different scalings because of introducing of a double scaling, which is different from [2, Theorem 1.1(b)]. The leading phase function f seems much simpler, and so it may be easier for us to choose proper “steepest descent” contours. Proof. First, we are devoted to the detailed proof of the statement (1.13) when ι ∈ ( 21 , 1). Similarly, we can complete the proof in the case of ι ∈ (0, 12 ]. Recalling (1.11), we rewrite (1.4) as √ Z Z √ √ M −M f (z)+M f (w) − Mr1 (z)+ M r2 (w) dw e e KM (x, y) : = dz (2πi)2 u Γ Σ (2.1) k 1 Y πl − w −M ι s(z)+M ι s(w) g(w) , ×e g(z) w − z πl − z l=1
where f (z) = u−1 (z − q) − ln z,
(2.2)
s(z) = ρ(z − q) + γ ln(1 − z),
r1 (z) = u
−1
r2 (z) = u
x(z − q),
and −r−κ
g(z) := (z − 1)
r Y
l=k+1
−1
(2.3) y(z − q)
1 . πl − z
(2.4)
(2.5)
Obviously, both f ′ (z) = 0 and s′ (z) = 0 have the same solution pc := u. Since 1 f ′′ (pc ) = 2 > 0, (2.6) pc the point pc is suitable for the steepest-descent analysis for KM (x, y). 4
Σ2 Σ1 0
Γ′
Γ′′2
Γ′′1
q pc
Γ′′3 1
π∗
Figure 1: Contours of double integrals Let ε be defined as (1.7), set εu q := pc − √ , M
(2.7)
and we choose two contours as depicted in Figure 1. Define 2εu Σ1 = pc − √ + iy : y ∈ [0, 1] , (2.8) M 1 2εu 2εu Σ2 = (pc − √ )2 + 1 2 eiθ : θ ∈ [arctan((pc − √ )−1 ), π] , (2.9) M M S S S and set Σ = Σ1 Σ2 Σ1 Σ2 (here ourSchoice is slightly different from that in [2]). For large R > 0, set Γ = Γ′ Γ′′ , where Γ′ is a small closed curve encircling θ1 , . . . , θk , pc and lying on the right of q and Γ′′ is a rectangle Γ′′ =
3 [
l=1
Γ′′l
!
∪
3 [
l=1
!
Γ′′l ,
(2.10)
with
p c + π∗ p c + π∗ + iy : y ∈ [0, ] , 2 a p c + π∗ p c + π∗ ′′ :x∈[ , R] , Γ2 = x + i a 2 p c + π∗ Γ′′3 = R + iy : y ∈ [0, ] . 2 Γ′′1 =
(2.11) (2.12) (2.13)
Here π∗ = min {πk+1 , . . . , πr , 1} and the positive parameter a will be chosen according to the condition (2.34) below. For any fixed δ > 0, denote Σ′ := Σ ∩ {z : |z − pc | < δ}, and Σ′′ := Σ\Σ′ . We choose δ < 1 small enough such that the Taylor series of f and g converge in the disk centered at pc with radius δ, and also such that Σ1 intersects 5
{z : |z − pc | = δ} at some point on the upper-half plane which is denoted by 2εu ˜ Note that δ˜ → δ as M → ∞. p∗ = pc − √ + iδ. M Next, we split KM into three parts Z Z Z Z Z Z dw · dz dw · + dw · + dz dz KM = (2.14) Σ′ Γ′′ Σ′′ Γ Σ′ Γ′ ′ ′′ ′′′ =: K + K + K , and will do asymptotic analysis by the steepest decent argument. And for this the following statement plays a key role. Claim: There exist constants M0 , C1 > 0 such that for every M ≥ M0 the following hold true. sup Re (f (z) − f (pc )) + M ι−1 (s(z) − s(pc )) ≤ −C1 , (2.15) z∈Σ′′
and
inf′′ Re (f (z) − f (pc )) + M ι−1 (s(z) − s(pc )) ≥ C1 .
z∈Γ
For simplicity, we define a function depending on ε > 0 as ( e−εx , x ≥ 0; Lε (x) = 1, x < 0.
(2.16)
(2.17)
Moreover, we consider the case that x, y ≥ W where W is a fixed real number. √ Notice the√inequalities M Re (r2 (w) − r2 (pc )) ≤ −2εy for y ≥ 0 and |w − z| ≥ εu/ M with w ∈ Σ, z ∈ Γ, by the Claim above, there are positive constants C2 and C3 independent of x, y such that ′′ K ≤
√ M M (r2 (pc )−r1 (pc )) e 2ε(2πi)2 u2
Z
dz e−M Re (f (z)−f (pc )) k √ Y 1 − M Re (r1 (z)−r1 (pc )) −M ι Re (s(z)−s(pc )) 1 e ×e g(z) πl − z l=1 Z √ ι dw eM Re (f (w)−f (pc )) e M Re (r2 (w)−r2 (pc )) eM Re (s(w)−s(pc )) × ′′ Σ k Y × g(w) (πl − w) Γ
l=1
≤ C3 M e−C2 M Lε (x)Lε (y).
(2.18)
Similay, we have ′′′ K ≤ C3 M e−C2 M Lε (x)Lε (y). 6
(2.19)
These show that both K′′ and K′′′ decay exponentially to zero. To estimate K′ , the Taylor expansions of f (z), s(z), r1 (z) and r2 (z) at z = pc give us 1 1 (z − pc )2 + O((z − pc )3 ), 2 p2c 1 γ s(z) = s(pc ) − (z − pc )2 + O((z − pc )3 ), 2 (1 − pc )2 r1 (z) = r1 (pc ) + u−1 x(z − pc ),
(2.21)
r2 (z) = r2 (pc ) + u−1 y(z − pc ).
(2.23)
f (z) = f (pc ) +
(2.20)
(2.22)
and Then, for sufficient large M , we have
1 K′ =O √ Lε′ (x)Lε′ (y)+ M 1 Z −2ε+i∞ Z 2 2 k dw e 2 (w −z )+yw−xz Y θl − w dz , e−ε(x−y) w−z θl − z Γ∞ 2πi −2ε−i∞ 2πi
(2.24)
l=1
where 0 < ε′ < ε. Combination of equations (2.18), (2.19) and (2.24) completes the proof of statement (1.13), since the convergence is obviously uniform for x, y in a compact subset of R. In order to prove the statement (1.14), we need to strengthen the statement (1.13) from uniform convergence into the trace norm convergence of the integral operators with respect to the correlation kernels. For this we can use the same argument as in the proof of Theorem 1.1 [5, Section 4.3], that is, it is sufficient to find a dominating function over (W, +∞) × (W, +∞) for KM (x, y). Actually, we easily see from (2.18), (2.19) and (2.24) that there exists a constant C4 > 0 such that |KM (x, y)| ≤ C4 Lε′ (x)Lε′ (y).
(2.25)
Finally, we need to verify (2.15) and (2.16). Note that both Σ and Γ are bounded curves, it suffices to verify the assertion only in the case of s(z) = 0. Actually, it is easy to see δ 2εu sup Re (f (z)) ≤ Re (f (p∗ )) ≤ Re f (pc − √ + i ) . 2 M z∈Σ′′
7
(2.26)
Thus, there exists a constant C1 > 0 such that sup {Re (f (z) − f (pc ))}
z∈Σ′′
2εu δ ≤ Re (f (pc − √ + i )) − Re (f (pc )) 2 M 1 2εu 2εu 2 δ2 1 = − √ − log (1 − √ ) + 2 u M 2 4pc M pc
(2.27)
≤ −C1 .
The assertion (2.15) is then proved. We turn to the assertion (2.16). When z ∈ Γ′′1 , since p + π 2 π∗ − p c 1 c ∗ 2 2 − log pc Re (f (z) − f (pc )) = − log y + 2 2u 2 (2.28) is a decreasing function of y, we have inf Re (f (z) − f (pc )) ≥ Re (f (p1 ) − f (pc ))
z∈Γ′′ 1
π∗ − p c 1 4 π∗ − p c − log 1 + 2 , − log 1 + u 2u 2 2 a pc +π∗ pc +π∗ where p1 = 2 + i a denotes the top-left corner of Γ′′ . When z ∈ Γ′′2 , simple calculation shows that =
h(x) : = Re (f (z) − f (pc )) p + π 2 1 1 c ∗ 2 2 − log pc . = (x − pc ) − log x + u 2 a
(2.29)
(2.30)
Further, we see from
h′ (x) =
x 1 1 1 − 2 ≥ − >0 p +π c ∗ 2 u x +( a ) u x
(2.31)
that inf Re (f (z) − f (pc )) ≥ Re (f (p1 ) − f (pc )).
z∈Γ′′ 2
For z ∈ Γ′′3 , we have inf′′ Re (f (z) − f (pc )) =
z∈Γ3
1 1 (R − pc ) − (log(R2 + y 2 ) − log p2c ) u 2
(2.32)
(2.33)
≥ Re (f (p2 ) − f (pc )) ≥ Re (f (p1 ) − f (pc )),
∗ where p2 = R + i pc +π denotes the top-right corner of Γ′′ . a Once we choose a such that −pc ) 2(1 + π∗2u , a > u−1 (π −p ) π ∗ c − (1 + ∗ −pc )2 )1/2 (e 2u
(2.34)
combination of (2.29), (2.32) and (2.33) shows that there exists a positive constant, say C1 , such that the inequality (2.16) holds true. 8
Acknowledgment We are grateful to Prof. Zhidong Bai for helpful discussions. The work of D.-Z. L. was supported by the National Natural Science Foundation of China (Grant # 11301499), and by the Fundamental Research Funds for the Central Universities (Grant # WK3470000008).
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