Universality in the bulk of the spectrum for complex sample covariance matrices

www.imstat.org/aihp 2012, Vol. 48, No. 1, 80–106

DOI:10.1214/11-AIHP442

© Association des Publications de l’Institut Henri Poincaré, 2012

Universality in the bulk of the spectrum for complex sample covariance matrices

Sandrine Péché

Institut Fourier, 100 Rue des Maths, BP 74, F-38402 St Martin d’Heres, France. E-mail:sandrine.peche@ujf-grenoble.fr Received 22 December 2009; revised 23 November 2010; accepted 15 June 2011

Abstract. We consider complex sample covariance matricesM_N=_N¹Y Y^∗whereY is aN×prandom matrix with i.i.d. entries Y_ij,1≤i≤N,1≤j≤p, with distributionF. Under some regularity and decay assumptions onF, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit whereN→ ∞and lim_N→∞p/N=γ for any real number γ∈(0,∞).

Résumé. On considère des matrices de covariance empirique complexesM_N=_N¹Y Y^∗oùY est une matrice de tailleN×pdont les coefficientsY_ij,1≤i≤N,1≤j≤p, sont des variables aléatoires i.i.d. de loiF. Sous certaines hypothèses de régularité et de décroissance surF, on montre l’universalité de certaines statistiques locales de valeurs propres au milieu du spectre quandN→ ∞ et lim_N→∞p/N=γpour tout réelγ∈(0,∞).

MSC:60B20; 60B10; 60B12

Keywords:Random matrix; Bulk universality; Sample covariance matrices

1. Introduction

1.1. Model and result

This paper is concerned with universal properties of large complex sample covariance matrices in the bulk of the spectrum. We consider N ×p random matrices Y =(Y_ij)where the Y_ij’s are i.i.d. random variables with some probability distributionF. Let thenM_Nbe the sample covariance matrix:

M_N= 1 NY Y^∗.

In the whole paper we assume thatp≥Nand that

∃γ∈ [1,∞)such thatp/N→γ asN→ ∞.

The case wherep < N can be deduced from the above setting using the fact thatY Y^∗ andY^∗Y have the same non- zero eigenvalues. In the sequel we callλ1≥λ2≥ · · · ≥λN the ordered eigenvalues ofMNandπN:=_N¹ _N

i=1δλ_i its spectral measure. Assuming thatF has a finite varianceσ², Marˇcenko and Pastur ([18], see also [20]) have shown thatπ_N almost surely converges asN → ∞to the so-called Marˇcenko–Pastur distributionρ^MP

γ ,σ².This probability

distribution depends onσ²only and not on any other detail (higher moments e.g.) ofF: in this sense it isuniversal. It is defined by the density with respect to the Lebesgue measure:

dρ_{γ ,σ}^MP2(x)

dx =

(u₊−x)(x−u₋)

2πxσ² 1x∈[u₋,u₊], (1.1)

whereu₊=σ²(1+ √γ )²andu₋=σ²(1− √γ )².

It has been conjectured (see [19] for instance) that, in the large-N-limit, some finer properties of the spectrum are also universal. For instance, the spacing between nearest neighbor eigenvalues in the vicinity of a pointu∈(u₋, u₊) is expected to be universal, under the sole assumption that the variance ofF is finite. The spacing is actually believed to be “more universal” than the limiting Marˇcenko–Pastur distribution, in the sense that it is expected to be the same as for Hermitian Wigner matrices. To investigate such local properties of the spectrum, we introduce the so-called local eigenvalue statisticsin the bulk of the spectrum. Given a functionf ∈L^∞(R^m)(mfixed) with compact support, a pointu∈ [u₋, u₊],and a scaling factorρ_N, we define the local eigenvalue statisticS^(m)_N (f, u, ρ_N)by

S_N^(m)(f, u, ρN)=

i1,...,im

f

ρN(λi₁−u), . . . , ρN(λi_m−u)

, (1.2)

where the sum is over all distinct indices from{1, . . . , N}.Whenuis in the bulk of the spectrum, that isu∈(u₋, u₊), the natural choice for the scaling factor isρN=Nρ^MP

γ ,σ²(u), which should give the scale of the spacing between nearest neighbor eigenvalues in the vicinity ofu.

We here prove universality of some local linear statistics in the bulk of the spectrum for a wide class of complex sample covariance matrices. We follow the approach used by [14] (and a series of papers [11–13]) where universality in the bulk of Wigner matrices is proved. We now define the class of matrices under consideration in this paper. Let μbe the real Gaussian distribution with mean 0 and variance 1/2. LetF be a complex probability distribution whose real and imaginary parts are of the form

ν(dx)=e^{−V (x)}μ(dx), (1.3)

for some real functionV satisfying the following assumptions:

– V ∈C⁶and there exists an integerk≥1 such that 6

j=1

V^{(j )}(x)≤C_∗

1+x²k

, (1.4)

– there existδ₁, C>0 such that∀x∈R,

ν(x)≤Ce^−δ1|x|2. (1.5)

This assumption can actually be relaxed as explained in Remark1.2below to consider distributions ν with sub- exponential decay only.

– F is normalized so that

xdν(x)=0 and

|x²|dν(x)=1/2. (1.6)

In the sequel we consider sample covariance matricesM_N=_N¹Y Y^∗whereY =(Y_ij)is aN×prandom matrix such that:

Y_ij,1≤i≤N;1≤j≤p, are i.i.d. random variables with distributionF . (1.7) One shall remark that the condition (1.6) can always be achieved by rescaling and does not impact on the generality of our next results.

We now give our two main results. Let >0 small be given.

Theorem 1.1. Assume thatF satisfies(1.3)–(1.6).Let alsou∈(u₋+ , u₊− )be a point in the bulk of the spectrum andρ_N=Nρ_{γ ,1}^MP(u).Then

Nlim→∞ES_N⁽²⁾(f, u, ρ_N)=

R²f (x, y) 1− sin(x−y)π π(x−y)

2 dxdy.

Remark 1.1. It is also possible to prove universality of local eigenvalue statistics form=3using the approach devel- oped hereafter.Nevertheless to consider higher integersm,one needs to increase the regularity ofV (see Remark1.1 in[14]).

We can also prove that the spacing distribution close to a point u in the interior of the support of Marˇcenko–

Pastur’s law is universal. Lets≥0 and (t_N)be a sequence such that limN→∞t_N = +∞and limN→∞ tN

N^1−β =0 for someβ >0.Letu∈(u₋+ , u₊− )be given andρ_N=Nρ_{γ ,1}^MP(u).Define then the “spacing function” of the eigenvalues by

S_N(s, u):= 1 2tN

1≤j ≤N−1, λj+1−λ_j≤ s ρN

,|λ_j−u| ≤ t_N ρN

. (1.8)

Intuitively the expectation of the spacing function is the probability, knowing that there exists an eigenvalue in the interval[u−t_N, u+t_N], to find its nearest neighbor within a distance_ρ^s_N.Finally, we define

p(s)= d²

ds²det(I−K)_L2(0,s), whereK(x, y):=sinπ(x−y)

π(x−y) . (1.9)

Theorem 1.2. Assume thatF satisfies(1.3)–(1.6).Let alsou∈(u₋+ , u₊− )be a point in the bulk of the spectrum.

Then,

Nlim→∞ESN(s, u)= _s

0

p(w)dw. (1.10)

Remark 1.2. Using truncation and centralization techniques,it is possible to prove both Theorem1.1and Theorem1.2 when assumption(1.5)is replaced by the weaker assumption

∃C₁, C₂>0such thatν(x)≤C₁e^−C2|x|.

This extension is examined in full detail in Section5of[14]for Wigner random matrices and readily extends to sample covariance matrices.

The first proofs of universality in the bulk of the spectrum of large random matrices have been obtained for the so-called invariant ensembles [7–9]. Their proof relies on the fact that the joint eigenvalue density of such ensembles can be computed and the asymptotic local eigenvalue statistics can then be determined. A breakthrough in the proof of the conjecture was obtained in [17] (following the idea of [5] and [6]), proving universality in the bulk for the so-called Dyson Brownian motion model [10]. This then allowed to extend universality results to a wide class of non-invariant Hermitian random matrix ensembles – the so-called Gauss divisible ensembles (see Section 1.2for the definition). [4] have then obtained the same universality result for Gauss divisible complex sample covariance matrices. Very recently, such universality results have been greatly improved by [22] and [14] for Hermitian Wigner random matrices. Both the papers remove the Gauss divisible assumption and only assume sufficient decay of the entries of Wigner random matrices. The approach of [22] is to make a Taylor expansion of local eigenvalue statistics of Wigner matrices. The core of the proof is then to show that these statistics depend, in the large-N-limit, only on the first four moments of the entries of the Wigner matrix. Proving that any Wigner matrix can be matched to a Gauss divisible matrix with the same first 4 moments allows to prove a very general universality result. On the other hand,

the approach of [14] is to show that any Wigner matrix (under suitable decay of the entries) can be sufficiently well approximated by a Gauss divisible random matrix, so that the bulk universality follows. We also refer the reader to [15] where the two approaches are combined.

While writing this paper, a proof of universality in the bulk of the spectrum of large sample covariance matrices has been obtained in [23] in the sole case wherep−N=O(N^43/48), but with much milder assumptions on the decay of the distributionν. Thereinνsatifies

|x|^Codν(x) <∞for some sufficiently largeC_oin place of our assumptions (1.4) and (1.5). Their approach is based on the ideas developed in [22].

Our paper closely follows the ideas developed in [14]. We give an overview of the proof in the next subsection.

1.2. The idea of the proof

Following the pioneering work of [17] and [4,5] have shown universality of local eigenvalue statistics in the bulk of the spectrum for complex sample covariance matrices when the distribution of the sample isGauss divisible. We recall that a complex distributionμ_Gis Gauss divisible if there exist a complex probability distributionP and a non-trivial complex Gaussian distributionGsuch thatμ_G=P G. Equivalently [4] consider random matrices of the form

MN= 1

N(W +aX)(W+aX)^∗, (1.11)

whereW andXare independentN×pcomplex random matrices both with i.i.d. entries. TheW_ij’s areP distributed and theX_ij’s are complex standard Gaussian random variables. In the above contexta is real number independent ofN. The proof of [17] and [4] relies on three main steps:

– conditionnally onH=_N¹W W^∗, the eigenvalue process induced byM_Ndefines a so-called determinantal random point field;

– the corresponding correlation kernel can be expressed as a double integral in the complex plane depending onH through its sole spectral measureμ_N;

– under suitable assumptions onW and thanks to concentration results for the spectral measureμ_N established by [2,3] and [16], the asymptotic analysis of the correlation kernel (and local statistics) can be performed.

One may also point out that the correlation kernel is expressed in terms of Bessel functions with a large parameter and whose order may be bounded or not. The asymptotic analysis requires uniform asymptotic expansions of such Bessel functions. In [4], the authors consider the case whereγ =1 andp−N=O(N^43/48)only, for which uniform asymptotic expansions of Bessel functions are easier to handle. In this paper, we are able to consider the general case withγ≥1, using different uniform asymptotic expansions of Bessel functions.

The parameterais to be seen as the “order of the Gaussian regularization” ofP. In principle the above result would yield a full proof of the universality conjecture if one could letaapproach (and be smaller than) 1/√

N. Unfortunately this idea fails whatever sharp concentration results can be established forμN. The asymptotic analysis is not stable in this scale.

A breakthrough to overcome this difficulty is obtained in [14]. In [11–13] concentration results are deeply sharp- ened so as to be able to consider a Gaussian regularization of ordera^√1_N. Given an arbitrary (non-Gauss divisible) distributionF, the main point is however to show that one can find a Gauss divisible distribution approximatingF sufficiently well so that one can deduce universality in the bulk for a sample covariance matrix with i.i.d.F distributed entries. This is the main step achieved in [14], which we now briefly expose.

Consider the Ornstein–Uhlenbeck (OU) process L=1

4

∂²

∂x²−x 2

∂

∂x, ∂_tu=Lu.

Given a complex probability distributionμ, let thenXt be theN×pmatrix process with initial distributionμ^⊗Np and whose entries (real and imaginary parts) evolve independently according to the OU process. ThenXt evolves as

t→e^{−t /2}Hˆ+

e^t−11/2

X ,

whereHˆ has i.i.d. entries with distributionμ andX has standard complex Gaussian entries. Let e^tL:=(e^{t L})^⊗Np denote the dynamics of the OU process for all the matrix elements. Here, for smallt, one should think thatta².

Erd˝os et al. [14] prove that there exists a Gauss divisible distributionμ^t_GapproximatingF in the total variation norm in a sufficiently good way. Roughly speaking, let

μ^t_G=e^{t L}

1−t L+t L²/2

F^c, μ=

1−t L+t²L²/2 F^c,

whereF^cis obtained fromF after truncation and centralization. Intuitively, it is reasonable to expect thatμ^t_Gis a good approximation ofF in the scalet1. For sake of completeness we here recall their result (Proposition 2.1).

Letθbe a smooth cutoff function satisfyingθ (x)=1 if|x| ≤1 andθ (x)=0 for|x| ≥2.

Proposition 1.1. LetV satisfy(1.4), (1.5)for somekand (1.6).Letλ >0be sufficiently small andt=N^λ−1.Let c_N, d_N be real numbers so thatvdμdefines a centered probability density if the functionvis given by:

v(x):=e^−Vc(x), V^c(x):=V (x)θ

(x−c_N)N^−λ/4k +d_N.

Let thenL:=L^⊗Np, fN,p:=(e^−V)^⊗Np,andfc,N,p:=v^⊗Np.

1. There exist constantsC >0,c >0depending onkandλsuch that

|fc,N,p−fN,p|dμ^⊗Np≤Ce^−c(Np)c/2. 2. gt:=(1−t L+t²L²/2)vis a probability measure with respect todμ.SettingGt:= [gt]^⊗Np,there exists a constant

Cdepending onλand the constantsC_∗andδ1defined in(1.4)and(1.5)such that |e^tLGt−fc,N,p|²

e^tLGt

dμ^⊗Np≤CNpt^6−λ≤CN^−4+8λp N.

The idea is then to prove Theorem1.1and Theorem1.2for the Gauss divisible ensembles with small parameter a∼N^(λ−1)/2for someλ >0. Then, using Proposition1.1, and following the idea of [14], Section 4, one can extend universality of local eigenvalue statistics S^(m)_N (f, u, ρ_N)with m=2,3 and that of the spacing function to sample covariance matrices satisfying (1.7).

The paper is organized as follows. In Section2we study eigenvalue statitics in the bulk of the spectrum for Gauss divisible sample covariance matrices. To this aim, we first recall some properties of theDeformed Wishart ensemble.

This is the conditional distribution ofMNknowingW. Such an ensemble is in particular known to be determinantal, as we recall. We then establish some improved convergence rates for the spectral measure of sample covariance matrices whose entries have a sub-Gaussian tail. These concentration results then allow to compute the asymptotic correlation functions asN→ ∞in the regime wherea→0, a^√1_N. We then prove Theorem1.2and Theorem1.1in Section3.

In the whole paper, we useCandcto denote constants whose value may vary from line to line.

2. The Gauss divisible ensemble

In this section we establish some universality results for the following Gauss divisible ensemble: let MN= 1

N(W+aX)(W+aX)^∗, (2.1)

be a HermitianN×Nrandom matrix where

• (H₀) X and W are independent N ×p random matrices where p ≥ N and there exists γ ≥1 so that limN→∞p/N=γ;

• (H1) a=aN=

N^λ

N whereλ >0 is a (small) real number;

• (H₂) Xis aN×prandom matrix with complex standard Gaussian entries:

ReXij,ImXij∼N(0,1/2) ∀1≤i≤N,1≤j≤p;

• (H₃) W=(W_ij),1≤i≤N,1≤j≤p,is a complex random matrix such that ReWij,ImWij,1≤i≤N,1≤j≤ p, are i.i.d. and satisfy:

(A1) There exists a constantδo>0 such thatEe^δo|Wij|2<∞,∀i, j.

Without loss of generality, we also make the assumption that

E|W_ij|²=1/4. (2.2)

Note that (2.2) can always be achieved by rescalingMN. From now on, we denote byy1≥y2≥ · · · ≥yNthe ordered eigenvalues ofH=H_N=W W^∗/N and letμ_N=_N¹_N

i=1δ_yi be its spectral measure. Under assumption (2.2), it is known thatμ_N almost surely converges to the Marˇcenko–Pastur distribution with parametersγ and 1/4,ρ_{γ ,1/4}^MP = ρ^MP_γ , whose density function is given by (1.1) withσ²=1/4.Whenγ=1, we simply denoteρ₁^MPbyρ^MPfor short.

2.1. Correlation functions

The sample covariance Gauss divisible ensemble has a nice mathematical structure that we are going to make use of in the sequel: the conditional distribution ofM_N with respect toW is the so-called complex deformed Wishart ensemble. Such an ensemble has been widely studied in random matrix theory as it induces a determinantal random point field. We recall some results used in [4] (see references therein) that will be needed for the sequel. Let then P_N^H(λ1, λ2, . . . , λN)be the joint eigenvalue distribution induced by the conditional distribution ofMNw.r.t.H. Then (see Section 3 in [4]),P_N^H is absolutely continuous with respect to Lebesgue measure onR^N₊. Its densityf_N^H is given by:

f_N^H(x1, x2, . . . , x_N)=V (x) V (y)det 1

Se^{{−(yi+xj)/S}}I_ν 2√y_ix_j S

x_j y_i

ν/2N

i,j=1

, (2.3)

whereV (x):=

1≤i<j≤N(x_i−x_j),ν=p−N,I_νis the modified Bessel function of the first kind, andS=a²/N. The main tools to study local eigenvalue statistics are the so-called eigenvalue correlation functions (see Section3). They are defined for any integer 1≤m≤N by R_N^(m)(x₁, x₂, . . . , x_m;H )=_(N^N₋^!_m)!

R^N−m₊ f_N^H(x₁, x₂, . . . , x_m, λ_m+1, . . . , λ_N)_N

i=m+1dλi. Then, for any integer 1≤m≤N, one has that R^(m)_N (x₁, x₂, . . . , x_m;H )=det

K_N(x_i, x_j;H )m i,j=1,

for some correlation kernel K_N(u, v;H ). This gives the determinantal random point field structure. Furthermore K_N(u, v;H )is given by

K_N(u, v;H ):= 1 2i²π²S²

iR+A

dw

Γ

dze^(−z2+w2)/SK_ν 2w√ v S

I_ν 2z√ u S

× N i=1

w²−y_i z²−y_i

w z

ν w+z

w−z−w−z w+z

, (2.4)

whereK_ν is the modified Bessel function of the second kind.Γ is a contour oriented counterclockwise enclosing the

±y^1/2_i , i=1, . . . , N, andAis large enough so that the two contoursΓ andΥ =iR+Ado not cross each other (see Fig.1).

Due to the determinantal structure, correlation functions are not impacted by conjugation of the correlation kernel.

In particular, for anyb∈Rand any integerm≥1, det

K_N(u_i, u_j;H )m

i,j=1=det

K_N(u_i, u_j;H )e^{2b(√uj−√ui)/S}m i,j=1.

In the sequel we consider the conjugation for somebthat will be defined in the asymptotic analysis. We set

K_N^b(ui, uj;H ):=KN(ui, uj;H )e^{2b(√uj−√ui)/S}. (2.5)

Fig. 1. The two contours defining the correlation kernelK_N(u, v;H ).

The correlation kernelK_N^b(u, v;H )only depends on H through its spectral measureμ_N. As in the case where a is fixed (independent of N), the idea is to use the convergence of μ_N to the Marˇcenko–Pastur distribution. In particular, one would like to make the replacement (outside a suitably negligible set)N

i=1(w²−yi)=exp{N (1+ o(1))

ln(w²−y)dρ_γ^MP(y)}. In Section2.2, we prove that this replacement can be made in some sense provided Im(w²)is not too small.

Also, the kernelK_N^b(u, v;H )depends on Bessel functions with large argument and whose order can be bounded or grow unboundedly. For some technical reasons (namely uniform asymptotic expansions of Bessel functions with bounded/unbounded order), we will separate the asymptotic analysis of the kernel depending on the case whereνis bounded (Section2.3) or not (Section2.4). This has no impact (up to technicalities) on the strategy of the proof.

2.2. Concentration results

This subsection is devoted to the proof of the following Proposition2.1, which precises the rate of convergence ofμ_N toρ^MP. Before exposing this proposition we need a few definitions and notations. Given a complex numberz=u+iη, u∈R, η >0, we define the Stieltjes transform ofμN by

mN(z):=

1

λ−zdμN(λ), (2.6)

and the Stieltjes transform of the limiting Marˇcenko–Pastur distribution by mMP(z):=

1

λ−zdρ_γ^MP(λ). (2.7)

Let >0 be given (small enough).

Proposition 2.1. Let z=u+iη for some u∈ [u₋+ , u₊− ]and η >0. Then,there exist a constant c₁ and C_o, C >0, c >0depending on only such that∀δ < c₁ ,

P sup

u∈[u₋+ ,u₊−]

m_N(z)−m_MP(z)≥ δ+C_o p

N −γ

≤Ce^{−cδ√N η},

for any(lnN )⁴/N≤η≤1.Furthermore,givenη≥(lnN )⁴/N,there exist constantsc >0, C >0andK_osuch that

∀κ≥Ko, P

sup

|x|>( /200)²,y≥η

m_N(x+iy)≥κ

≤Ce^{−c√κN η}.

Proof. To ease the reading, the proof is postponed to AppendixA. The proof is actually a modification of that of Theorems 3.1 and 4.1 in [13], where the Stieltjes transform of Hermitian Wigner matrices is considered. We thus

simply indicate the main changes.

2.3. Asymptotics of the correlation kernel whenν=p−N is bounded The aim of this subsection is here to prove the following proposition.

Let >0 be given (small) independent ofN.

Proposition 2.2. Letu_∗∈ [u₋+ , u₊− ] = [ ,1− ]andm≥1be given.Consider a sequenceu=u_Nsuch that N^1−λ|u−u_∗| =o(1).Then,there existb∈R,a setΩ_N,and positive constantsC, csuch that:

– the complement ofΩ_Nis negligible:P(Ω_N^c)≤Ce^−cNλ/4, – onΩ_None has that

Nlim→∞det 1

Nρ^MP(u)K_N^b u+ xi

Nρ^MP(u), u+ x_j Nρ^MP(u);H

m

i,j=1

=det sin(π(x_i−x_j) π(xi−xj)

m

i,j=1

.

Remark 2.1. In Remark2.3we explain how to extend Proposition2.2to the case where|u−u_∗| =o(N^−c)withc >0 arbitrarily small.This extension is needed for the proof of Theorem1.2.

This subsection is devoted to the proof of Proposition2.2. The proof is divided into two parts: first we obtain a new expression for the correlation kernel, which then allows to derive its asymptotics by a saddle point argument.

2.3.1. Rewriting the kernel

Our strategy consists into two parts: we first replace the Bessel functions with their asymptotic expansion and then remove the singularity 1/(w−z)in the correlation kernel as it will be proved to prevent a direct saddle point analysis.

As this part is highly technical and needs a few notations, we summarize our main result in Lemma2.3stated at the end of this subsection. The reader might skip this part and is simply referred to the above cited lemma.

Our first task to replace in (2.4) Bessel functions with their asymptotic expansion given in AppendixB, formula (B.1). This replacement can be made, up to a negligible error, provided the contourΓ is cut in a small neighborhood of the imaginary axis: see Fig.2. In the sequel (see Fig.2) we callx^±_o (resp.x₁^±) the endpoints ofΓ_cwith Re(xo) <0 (resp. Re(x1) >0) with positive/negative imaginary part surrounding the imaginary axis:x^±_o andx₁^±will be chosen

Fig. 2. The contourΓchas been cut along the imaginary axis.

in Section2.3.2. We also callΓ₁(resp.Γ₂) the part ofΓ_clying to the right (resp. left) of iR. ThenΓ₁is the image of Γ₂byz→ −zandΓ_c=Γ₁∪Γ₂.One obtains the following lemma.

Lemma 2.1. For anyb∈R,one has that K_N^b(u, v;H )=

1+o(1)

K_N^1,b(u, v;H )+K_N^2,b(u, v;H )

, (2.8)

where

K_N^1,b(u, v;H )=e^{2b(√v−√u)/S} 4i²π²S

iR+A

dw

Γ₁

dze^(−z2+w2)/Se^−2w√v/Se^2z√u/S

× 1

√w√

z(uv)^1/4 N i=1

w²−yi

z²−y_i w

z ν

w+z w−z,

(2.9) K_N^2,b(u, v;H )=e^{2b(√v−√u)/S}

4i²π²S

iR+A

dw

Γ₁

dze^(−z2+w2)/Se^−2w√v/Se^−2z√u/S

× 1

√w√

z(uv)^1/4 N i=1

w²−y_i z²−y_i

w z

νw+z

w−ze^νiπ+iπ/2.

Proof. Here we will call on some arguments already used in [4], which we won’t develop entirely, yet trying to be the most self-contained as possible. Lemma2.1essentially follows from the two claims exposed in the sequel and whose proof relies on the saddle point analysis in the next subsection.

Claim 2.1. One has that

Nlim→∞

e^{2b(√v−√u)/S} 2i²π²S²

iR+A

dw

Γ\Γ_c

dze^(−z2+w2)/SK_ν 2w√ v S

I_ν 2z√ u S

,

(2.10) N

i=1

w²−y_i z²−yi

w z

ν w+z

w−z−w−z w+z

=0.

Claim2.1essentially follows from the fact that|Iν(z)| ≤e^zif Re(z) >0 (and a similar bound forKν) and the fact thatΓ \Γ_c lies far away from the critical points (a full justification can be derived from the saddle point analysis performed in Section2.3.2, see Remark2.2).

Claim 2.2. One has that

e^{2b(√v−√u)/S} 2i²π²S²

iR+A

dw

Γ_c

dze^(−z2+w2)/SK_ν 2w√ v S

I_ν 2z√ u S

× N i=1

w²−yi

z²−y_i w

z ν

w+z

w−z−w−z w+z

=e^{2b(√v−√u)/S} 4i²π²S

iR+A

dw

Γ₁

dze^(−z2+w2)/Se^{−2w√v/S+2z√u/S} 1 (uv)^1/4√

w√ z

× N i=1

w²−yi

z²−y_i w

z ν

w+z

w−z−w−z w+z

1+O

N^−λ/2

. (2.11)

By a straightforward change of variables, we can reduceΓ_ctoΓ₁. Claim2.2then follows by uniform asymptotic expansions of first the w-integral, using the asymptotics of the Bessel function K_ν given in (B.1), and then of the z-integral. Again this follows from the saddle point analysis of Section2.3.2. This saddle point analysis will prove that the correlation kernel does not vanish in the largeN-limit. Then splittingΓ₁into sufficiently many pieces lying to the left or to the right of the critical points and moving accordinglyΥ =iR+Ato pass close enough to the same critical points yields that the error term is of orderN^−λ/2. This yields Claim2.2and Lemma2.1. We skip the detail.

A saddle point analysis of the correlation kernel K_N^1,b cannot be performed at that point, due to the singularity 1/(w−z). Indeed, assuming that_N

i=1(w²−y_i)=e^{Nln(w2−y)dρMP(y)(1+o(1))},it is not difficult to see that thez- andw-integrands have the same critical points. Thus we first remove this singularity (see [4] for a more detailed explanation).

Define

f_u^N(z):=z²−2√ uz+S

N i=1

ln z²−yi

, fN(z)=f_u^N(z)+2b√ u,

f_v^N(z):=z²−2√ vz+S

N i=1

ln z²−yi

, fN(z)=f_v^N(z)+2b√ v,

g_N,u(w, z):=(w−b)(f_u^N)(w)−(f_u^N)(z)

w−z +

f_u^N

(z), (2.12)

g_N,u¹ (w, z)=νbS wz + Sb

2wz w−z

w+z, θ (w, b)= −e^{2(w−b)(√u−√v)/S}−1 2(w−b)(√

u−√ v),

θ¹(z, b)=e^{−2(z−b)(√u+√v)/S}−1 2(z−b)(√

u+√ v) .

Lemma 2.2. One has that

K_N^1,b(u, v;H )

= 1

(uv)^1/44i²π²S

iR+A

dw

Γ₁

dze^{(fN(w)−fN(z))/S}

× w+z

√w√ z

g_N,u(w, z)+g_N,u¹ (w, z) w z

ν

θ (w, b) (2.13)

− (x⁻₁ −b) 4i²π²(uv)^1/4

iR+A

dw w+x₁⁻

√w

x₁⁻ w x₁⁻

ν

e^{(fN(w)−fN(x1−))/S}θ (w, b)

+ (x⁺₁ −b) 4i²π²(uv)^1/4

iR+A

dw w+x₁⁺

√w

x₁⁺ w x₁⁺

ν

e^{(fN(w)−fN(x1+))/S}θ (w, b). (2.14)

SettingK_N^2,b(u, v;H ):=e^{−νiπ−iπ/2}K_N^2,b(u, v;H ),we have that

K_N^2,b(u, v;H )

=e^−4b√u/S 4i²π²S

iR+A

dw

Γ₁

dz

g_N,v(w, z)+g_N,v¹ (w, z)

e^{(fN(w)−fN(z))/S}

× w+z

√w√ z(uv)^1/4

w z

ν

θ¹(z, b)

−e^{−(4b√u)/S}(x₁⁻−b) 4i²π²(uv)^1/4

iR+A

dw w+x₁⁻

√w

x₁⁻ w x₁⁻

ν

e^{(fN(w)−fN(x−1))/S}θ¹(x₁⁻, b)

+e^−4b√u/S(x₁⁺−b) 4i²π²(uv)^1/4

iR+A

dw w+x₁⁺

√w

x₁⁺

e^{(fN(w)−fN(x+1))/S} w x₁⁺

ν

θ¹ x₁⁺, b

. (2.15)

Proof. We only considerK_N^1,b (the arguments are similar forK_N^2,b). To this aim we make the change of variables w=b+β(w−b),z=b+β(z−b)for someβreal close to 1 and get the following: set

E(w):=e^{(w2−2√vw)/S} N i=1

w²−y_i

; G(z)=e^{(z2−2√uz)/S} N i=1

z²−y_i

. (2.16)

Then one has that

K_N^1,b(u, v;H )= e^{2b(√v−√u)/S} (uv)^1/44i²π²S

iR+A

dw

Γ₁

dz 1

√wz E(w)

G(z) w

z ν

w+z w−z

= e^{2b(√v−√u)/S} 4i²π²S(uv)^1/4

iR+A

Γ₁

βdwdz

√(b+β(w−b))(b+β(z−b))

× b+β(w−b) b+β(z−b)

ν

2b+β(w+z−2b) w−z

E(b+β(w−b)) G(b+β(z−b))

− e^{2b(√v−√u)/S} 4i²π²S(uv)^1/4

iR+A

b+β(x₁⁺−b)

x₁⁺

dwdz

√wz w

z

νw+z w−z

E(w) G(z)

+ e^{2b(√v−√u)/S} 4i²π²S(uv)^1/4

iR+A

b+β(x₁⁻−b)

x₁⁻

dwdz

√wz w

z

νw+z w−z

E(w)

G(z). (2.17)

Differentiating with respect toβ(close to 1), we find that (see [17] Section 2 for the detail) (uv)^1/4K_N^1,b+√

v−√ u d

d√ v

(uv)^1/4K_N^1,b

=e^{2b(√v−√u)/S} 4i²π²S²

iR+A

dw

Γ1

dz 1

√wz w

z

νw+z w−z

E(w) G(z)

×

−2(w+z−b)+2√ u+S

N i=1

2yi(w+z)−b(2wz+y_i) (w²−y_i)(z²−y_i)

(2.18)

−νbS wz − Sb

2wz w−z w+z

−(x₁⁺−b)e^{2b(√v−√u)/S} 4i²π²S

iR+A

dw 1

wx₁⁺

w x₁⁺

νw+x₁⁺ w−x₁⁺

E(w) G(x₁⁺)

+(x₁⁻−b)e^{2b(√v−√u)/S} 4i²π²S

iR+A

dw 1

wx₁⁻

w x₁⁻

νw+x₁⁻ w−x₁⁻

E(w)

G(x₁⁻). (2.19)