Walid Hachem
email: walid.hachem@telecom-paristech.fr

Adrien Hardy
email: ahardy@kth.se

Jamal Najim
email: najim@univ-mlv.fr

Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges

Keywords: AMS 2000 subject classification: Primary 15A52, Secondary 15A18, 60F15 Large random matrices, Wishart matrix, Tracy-Widom fluctuations, Asymptotic independence, Bessel kernel

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the Tracy-Widom law at the scale N 2/3 . Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the scale N 2 . As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided.

Introduction

Correlated Wishart matrices and more generally empirical covariance matrices are ubiquitous models in applied mathematics. After Marčenko and Pastur's seminal contribution [START_REF] Marcenko | Distribution of eigenvalues in certain sets of random matrices[END_REF], a systematic study of their large dimension properties has been undertaken (see for instance [START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF][START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF] and the many references therein), which found many applications in multivariate statistics , e.g. [START_REF] Bai | Random matrix theory and its applications[END_REF], electrical engineering, e.g. [START_REF] Couillet | Random matrix methods for wireless communications[END_REF], mathematical finance [START_REF] Laloux | Potters: Noise dressing of financial correlation matrices[END_REF][START_REF] Münnix | A random matrix approach to credit risk[END_REF], etc. Now that many global properties of their spectrum are well-understood (cf. [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF][START_REF] Bai | Exact separation of eigenvalues of large-dimensional sample covariance matrices[END_REF][START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF][START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF][START_REF] Silverstein | Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices[END_REF]), attention has shifted to local properties (cf. [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF][START_REF] Bloemendal | Limits of spiked random matrices I[END_REF], etc.) and their underlying universal phenomenas (cf. [START_REF] Kuijlaars | Universality[END_REF] and references therein).

The main contribution of this article is to provide a local analysis of the spectrum of large complex correlated Wishart matrices near the edges of the limiting support: It is wellknown that such random Hermitian matrices have a real spectrum whose limiting support may display several disjoint intervals. Beside the behavior of the largest and smallest random eigenvalues, we investigate here the fluctuations of the eigenvalues that converge to any endpoint of the limiting support. These eigenvalues are referred to as extremal eigenvalues, for which we shall provide a precise definition later.

The model. Let X N be an N × n matrix with independent and identically distributed (i.i.d.) standard complex Gaussian entries N C (0, 1), and let Σ N be a n × n deterministic positive definite Hermitian matrix. The random matrix of interest here is the N × N matrix

M N = 1 N X N Σ N X * N . (1.1)
It has N non-negative eigenvalues 0 x 1 . . . x N , but which may be of different nature: min(n, N ) of them are non-negative random (i.e. non-deterministic) eigenvalues, whilst the other Nmin(n, N ) eigenvalues are deterministic and equal to zero. A companion matrix of interest is the n × n sample covariance matrix

M N = 1 N Σ 1/2 N X * N X N Σ 1/2 N , (1.2)
which models the empirical covariance of a sample of N independent observations

Σ 1/2 N [X * N ] k , 1 k N
where [X * N ] k stands for the k-th column of X * N , with population covariance matrix Σ N . Indeed, matrices M N and M N share the same non-null eigenvalues with the associated multiplicities.

We shall consider the asymptotic regime where n = n(N ), N → ∞ and lim

N →∞ n N = γ ∈ (0, ∞). (1.3)
This regime will be simply referred to as N → ∞ in the sequel.

The random matrix M N can also be interpreted as a multiplicative deformation of the Laguerre Unitary Ensemble (LUE) and is related to multiple Laguerre polynomials. A close matrix model is the additive deformation of the Gaussian Unitary Ensemble (GUE), also known as GUE with an external source; it involves multiple Hermite polynomials instead. For further information, see [START_REF] Bleher | Integral representations for multiple Hermite and multiple Laguerre polynomials[END_REF] and references therein. Capitaine and Péché [START_REF] Capitaine | Fluctuations at the edge of the spectrum of the full rank deformed gue[END_REF] recently studied the fluctuations of extremal eigenvalues for this model.

We now briefly review the literature and present our contribution.

Global regime. Denote by µ N the empirical distribution of the eigenvalues of M N , also called spectral measure in the sequel. Namely,

µ N = 1 N N i=1 δ x i
where δ x is the Dirac measure at point x; we shall also refer to µ N as the spectral distribution of M N . In the uncorrelated case where Σ N = I n , it is well-known [START_REF] Marcenko | Distribution of eigenvalues in certain sets of random matrices[END_REF] that µ N almost surely (a.s.) converges weakly towards the Marčenko-Pastur ( MP) distribution of parameter γ,

µ γ MP (dx) = (1 -γ) + δ 0 + 1 2πx (b -x) x -a) 1 [a,b] (x)dx, (1.4)
where x + = max(x, 0) and the endpoints of its support read a = (1 -√ γ) 2 and b = (1 + √ γ) 2 .

In the general case where Σ N is not the identity, say with eigenvalues 0 < λ 1 • • • λ n , a similar result holds true [START_REF] Silverstein | Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices[END_REF] under the additional assumption that the spectral measure

ν N = 1 n n j=1 δ λ j (1.5)
of Σ N converges weakly towards a limiting distribution ν. In the latter case, the limit µ of µ N only depends on the limiting parameters γ and ν but is no longer explicit; this dependence µ = µ(γ, ν) will be indicated when needed. However, its Cauchy-Stieltjes transform satisfies an explicit fixed-point equation from which many properties of µ can be inferred. For example, it is known that if ν({0}) = 0, then

µ(dx) = (1 -γ) + δ 0 + ρ(x)dx, (1.6)
where ρ(x) is a non-negative and continuous function on (0, +∞). Depending on the properties of γ and ν, the support of ρ(x)dx may have several connected components, see Section 2 for more precise informations. Alternatively, one can describe µ(γ, ν) in terms of the free multiplicative convolution of MP distribution (1.4) with ν, see [START_REF] Voiculescu | Limit laws for random matrices and free products[END_REF]. From now we shall refer to the support of ρ(x)dx as the bulk and to the endpoints of its connected components as the edges. Also, a positive edge is called soft edge and the terminology hard edge is here used when the edge is the origin.

Left and right edges. We say that an edge a is a left edge, resp. b is a right edge, if for every δ > 0 small enough, The leftmost edge can be a soft edge or an hard edge depending on the value of γ, as explained in Section 2. Of course, any other left edge and any right edge are soft edges.

Local regime: Behavior at the rightmost edge. If Σ N is the identity, Geman [START_REF] Geman | A limit theorem for the norm of random matrices[END_REF] proved the a.s. convergence of the largest eigenvalue x max of M N to the right edge of MP's bulk b = (1 + √ γ) 2 , for independent, not necessarily Gaussian, real entries of X N . Johansson [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] established Tracy-Widom fluctuations for x max at the scale N 2/3 for complex Gaussian entries; Johnstone [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] established a similar result for real Gaussian entries. Subsequent works [START_REF] Péché | Universality results for the largest eigenvalues of some sample covariance matrix ensembles[END_REF][START_REF] Pillai | Universality of covariance matrices[END_REF][START_REF] Soshnikov | A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices[END_REF][START_REF] Wang | Random covariance matrices: universality of local statistics of eigenvalues up to the edge[END_REF] then relaxed the Gaussian assumption, illustrating a phenomenon of universality.

If Σ N is a finite-rank perturbation of the identity, the limiting eigenvalue distribution is still given by MP distribution (1.4). Baik and Silverstein [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF] studied the limiting behaviour of x max for general entries. In the complex Gaussian case, Baik, Ben Arous and Péché [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] thoroughly described the fluctuations of the largest eigenvalues at the right edge and unveiled a remarkable phase transition phenomenon (referred to as BBP phase transition in the sequel). They established, for γ 1, that the convergence and fluctuations of x max are actually highly sensitive on the way ν N converges to δ 1 . More precisely, depending on the strength of the perturbation, they established that deformed Tracy-Widom fluctuations near the right edge b at the scale N 2/3 can arise, and that x max may also converge outside of the bulk with Gaussian-like1 fluctuations at the scale N 1/2 ; in the latter case x max is referred to as an outlier. Thus, depending on the way ν N converges towards its limit, the universality phenomenon may break down. Finally, Bloemendal and Virág [START_REF] Bloemendal | Limits of spiked random matrices I[END_REF][START_REF] Bloemendal | Limits of spiked random matrices II[END_REF] and Mo [START_REF] Mo | Rank 1 real Wishart spiked model[END_REF] extended the results in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] for real Gaussian entries, see also [START_REF] Bloemendal | On the principal components of sample covariance matrices[END_REF] for further extensions.

For general Σ N 's and complex Gaussian matrices, El Karoui [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF] (n/N 1) and then Onatski [START_REF] Onatski | The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices[END_REF] (n/N > 1) followed the approach developed in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] to establish Tracy-Widom fluctuations for x max , under mild conditions concerning Σ N 's spectral measure ν N provided that the rightmost edge satisfies some regularity condition. The Gaussian assumption has recently been relaxed by Bao et al. in [START_REF] Bao | Universality for the largest eigenvalue of sample covariance matrices with general population[END_REF].

Local regime: Behavior at the leftmost edge. If Σ N is the identity and γ 1, Bai and Yin [START_REF] Bai | Limit of the smallest eigenvalue of a largedimensional sample covariance matrix[END_REF] established the a.s. convergence of the smallest eigenvalue x min of M N to MP's left edge a = (1 -√ γ) 2 , see also [START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF]Chapter 5]. The nature of the fluctuations of x min dramatically changes whether γ = 1 (hard edge) or γ > 1 (soft edge). In the soft edge case, the fluctuations remain of a Tracy-Widom nature, see Borodin and Forrester [START_REF] Borodin | Increasing subsequences and the hard-to-soft edge transition in matrix ensembles[END_REF] and further extensions by Feldheim and Sodin [START_REF] Feldheim | A universality result for the smallest eigenvalues of certain sample covariance matrices[END_REF]. In the hard edge case, the fluctuations of x min are of order N 2 ; if N = n then the limiting distribution follows the exponential law as shown by Edelman [START_REF] Edelman | Eigenvalues and condition numbers of random matrices[END_REF] (cf. [START_REF] Tao | Random matrices: the distribution of the smallest singular values[END_REF] for further extensions), while if n = N + α with α independent of N , then the limiting distribution has been described by Forrester [START_REF] Forrester | The spectrum edge of random matrix ensembles[END_REF] with the help of Bessel kernels; the so-called hard-edge Tracy-Widom law, see Section 3 for a precise definition.

To the best knowledge of the authors, no result for the fluctuations at the leftmost edge in the general Σ N case is available in the literature.

Local regime: Asymptotic independence. When Σ N is the identity and γ > 1 (and also in the case of the GUE), Basor, Chen and Zhang [START_REF] Basor | PDEs satisfied by extreme eigenvalues distributions of GUE and LUE[END_REF] proved that x min and x max , properly rescaled, are asymptotically independent as N → ∞. Their approach heavily relies on orthogonal polynomials techniques, which are not available for complex correlated Wishart matrices. Using different techniques, the asymptotic independence for the GUE's smallest and largest eigenvalues was also obtained by Bianchi et al. [START_REF] Bianchi | Asymptotic independence in the spectrum of the Gaussian unitary ensemble[END_REF] and Bornemann [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF].

Again, it seems there is no result concerning the asymptotic independence for the extremal eigenvalues, even for the smallest and largest eigenvalues, in the general Σ N case.

Main results. Recall the asymptotic regime (1.3) of interest. We first state the main assumption related to matrix M N (cf. (1.1)) and then informally state the main results of the paper; pointers to the precise definitions and statements are provided in the next paragraph.

Assumption 1. The entries of X N are i.i.d. standard complex Gaussian random variables.

Assumption 2.

1. The spectral measure ν N of Σ N weakly converges towards a limiting probability distribution ν as N → ∞.

2. The eigenvalues 0 < λ 1 . . . λ n of Σ N stay in a compact subset of (0, +∞) which is independent of N , namely, lim inf

N →∞ λ 1 > 0, lim sup N →∞ λ n < +∞. (1.7)
In particular, ν({0}) = 0.

Another important assumption is the fact that the considered edges need to be regular. By this, we mean an edge which satisfies the regularity condition of Definition 2.5. This condition essentially rules out pathological behaviors at edges, e.g. when the limiting eigenvalue density does not vanish like a square root. It however enables the appearance of outliers. Close to our work is the recent paper by Capitaine and Péché [START_REF] Capitaine | Fluctuations at the edge of the spectrum of the full rank deformed gue[END_REF] where the fluctuations of the extremal eigenvalues for the additive deformation of the GUE are established, that is the counterpart of Part (b) of Theorem 1, and the Gaussian-like fluctuations for outliers as well. Besides the model, there are also other differences: While they also study the appearance of the Pearcey process when two bulks merge together, we investigate the asymptotic independence for the extremal eigenvalues and the appearance of the Bessel process at the hard-edge. As the involved techniques are extremely model-dependent, the technical difficulties are substantially different for the model under study. In particular, the assumptions on the convergence of ν N toward ν here seem less restrictive than its counterpart in [START_REF] Capitaine | Fluctuations at the edge of the spectrum of the full rank deformed gue[END_REF].

Let us now briefly comment on Theorem 1. In Part (a), we rely on results by Silverstein et al. [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF][START_REF] Bai | Exact separation of eigenvalues of large-dimensional sample covariance matrices[END_REF][START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] on the support of limiting spectral distributions and on fine asymptotic properties of the empirical spectrum to define regular edges and to properly express the convergence of extremal eigenvalues.

In Part (b), we first obtain an asymptotic Fredholm determinantal representation of the extremal eigenvalues' distribution and then perform an asymptotic analysis of the associated kernels to prove convergence toward the Airy kernel. The latter analysis is based on a steepest descent analysis involving contours deformations. Contrary to the analysis performed by Baik-Ben Arous-Péché [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF], El Karoui [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF] and Péché-Capitaine [START_REF] Capitaine | Fluctuations at the edge of the spectrum of the full rank deformed gue[END_REF] which work out explicit deformed contours, our analysis relies on a more abstract argument where the existence of appropriate contours is obtained by mean of the maximum principle for subharmonic functions. This argument has the advantage to work for every regular right or left edges up to minor modifications. Let us also stress that we do not follow the same strategy as in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF] concerning the involved operators convergence.

In Part (c), our proof of the asymptotic independence builds upon the operator-theoretic approach developed by Bornemann [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF] in the context of the GUE. We actually show that a weaker mode of convergence for the involved operators than the one required in [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF] is sufficient to establish the asymptotic independence; it has the advantage to be compatible with the previous asymptotic analysis.

Part (d) also relies on an asymptotic analysis of the rescaled kernel. It is based on an appropriate representation of the Bessel kernel as a double complex integral.

Organization of the paper. In Section 2, we provide a precise description for the bulk and the extremal eigenvalues and introduce the notion of regular edge. The precise statement of Part (a) of Theorem 1 is provided in Theorem 2 and proved.

In Section 3, we state our results concerning the fluctuations of the extremal eigenvalues and their asymptotic independence. Parts (b), (c) and (d) of Theorem 1 are respectively stated in Theorem 3, Theorem 4 and Theorem 5. We also recall there the definition of the Tracy-Widom and hard-edge Tracy-Widom distributions (Sections 3.1 & 3.3). We close the section with an asymptotic study of the condition number of large correlated Wishart matrices, a discussion on the non-regular edges, on spikes phenomena and provide some graphical illustrations.

Section 4 is devoted to the proof for Theorem 3 (Tracy-Widom fluctuations). Section 5 is devoted to the proof of Theorem 4 (asymptotic independence for extremal eigenvalues). Finally, Section 6 is devoted to the proof of Theorem 5 (hard-edge fluctuations).

Acknowledgements. AH and JN are pleased to thank the organizers of the 2011 France-China summer school in Changchun "Random Matrix Theory and High-dimensional Statistics" where this project began. The authors are indebted to Steven Delvaux for providing an important argument in the asymptotic analysis, see Section 4.4. Moreover, AH would like to thank Sandrine Péché for interesting discussions, and Manuela Girotti for generously sharing her computations on the double integral representation for the Bessel kernel, see the proof of Lemma 6.2. During this work, AH was supported by the KU Leuven research grant OT/12/73 and the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation. The work of WH and JN was partially supported by the program "modèles numériques" of the French Agence Nationale de la Recherche under the grant ANR-12-MONU-0003 (project DIONISOS).

Bulk description, regularity and extremal eigenvalues

In this section, we introduce the notion of regular soft edges (cf. Def. 2.5) and extremal eigenvalues (cf. Theorem 2). In particular, Theorem 2 provides a precise statement for Theorem 1-(a). We finally gather their main properties in Propositions 2.11 and 2.12. Before this, we provide a precise description of the bulk, mainly based on [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF].

Description of the limiting bulk

In [START_REF] Marcenko | Distribution of eigenvalues in certain sets of random matrices[END_REF], Marčenko and Pastur characterized the Cauchy-Stieltjes transform2 of the limiting

distribution µ = µ(γ, ν) of the eigenvalues of M N as N → ∞, m(z) = 1 z -λ µ(dλ), z ∈ C + = {z ∈ C : Im(z) > 0}, as the unique solution m ∈ C -= {z ∈ C : Im(z) < 0} of the fixed-point equation m = z -γ λ 1 -mλ ν(dλ) -1
for any z ∈ C + .

(2.1)

Recall that, by Assumption 1, γ = lim n/N ∈ (0, +∞), the probability measure ν is the limiting eigenvalue distribution of Σ N and its compact support is included in (0, +∞). In particular, ν({0}) = 0.

In [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF], Silverstein and Choi showed that

µ(dx) = (1 -γ) + δ 0 + ρ(x)dx, (2.2)
where ρ is a non-negative and continuous function on (0, ∞) which is analytic wherever it is positive. Moreover, following a procedure already described by Marčenko and Pastur, they showed rigorously how to extract from the fixed point equation above a characterization of the support of µ, and thus of ρ(x)dx. Specifically, the function m(z) has an explicit inverse given by

g(z) = 1 z + γ λ 1 -zλ ν(dλ) (2.3)
which analytically extends to the open subset of the real line

D = x ∈ R : x = 0, x -1 / ∈ Supp(ν) . (2.4)
Except in the proof of Proposition 2.7 below, we shall confine the notation g to the restriction of this function to D. On any interval I of R \ Supp(µ), the function m exists, is real and is decreasing (as a Cauchy-Stieltjes transform). Consequently, its inverse also exists and is decreasing on m(I). Silverstein and Choi showed that g is this inverse, and that R \ Supp(µ) coincides with the values of g(x) where this function is decreasing on D: Remark 2.2. This proposition has the following practical importance: In order to find Supp(µ), plot the function g on D; whenever g is decreasing (g ′ (x) < 0), remove the corresponding points g(x) from the vertical axis. What is left on this axis is precisely Supp(µ).

As an example, a plot of the function g is provided in Figure 1 along with Supp(µ) in the case where ν is the weighted sum of two Dirac measures and γ < 1.

The soft edges of the bulk are described more precisely by the next proposition.

Proposition 2.3 (Silverstein & Choi [60]). Any soft left edge a satisfies one of the two following properties:

(a) There exists a unique c ∈ D such that a = g(c), g ′ (c) = 0 and g ′′ (c) < 0.

(b) There exists a unique c ∈ ∂D such that (c, c + ε) ⊂ D for some ε > 0 small enough, the function g is decreasing on (c, c + ε), and a = lim x↓c g(x). In that case, we write a = g(c).

Conversely, for any point c satisfying one of these properties, a = g(c) is a soft left edge.

Similarly, any (soft) right edge b of the measure µ satisfies one of the two following properties: Hence, any soft edge of the bulk coincides with a unique extremum c of the function g and it reads g(c). These extrema may or may not be attained on D. In case they are, the second derivative of g is never equal to zero there, and it has been proved in [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] that the density vanishes like a square root at the associated edges. We shall see later that the Tracy-Widom fluctuations appear in this case. A right edge b = g(d) together with its preimage d are plotted in Figure 1.

The next proposition provides additional information on the bulk that will be useful in the sequel. Its proof is in Appendix A.

Proposition 2.4. Let Assumption 2 hold true. Let a be the leftmost edge of the bulk. The following facts hold true:

(a) If γ > 1, then a > 0.
Moreover, the function g(x) increases from zero to a then decreases from a to -∞ as x increases from -∞ to zero. In particular, if γ > 1, then a is the unique maximum for g on (-∞, 0).

(b) If γ 1, the function g is negative and decreasing on (-∞, 0).

(c) If γ < 1, then a > 0. Moreover, if we set η = inf Supp(ν) > 0, then a = g(c) is the supremum of g on (1/η, ∞). In addition, g increases to a on (1/η, c) whenever this interval is non empty, then decreases from a to zero on (c, ∞).

Let b = g(d) be a right edge of the bulk. Then the following facts hold true:

(d) [d, ∞) ⊂ D.
(e) Assume b is the rightmost edge of the bulk. For any γ ∈ (0, ∞), if we set ξ = sup Supp(ν) < ∞, then g decreases from infinity to b on (0, d) and increases on (d, 1/ξ) if this last interval is not empty. In particular, d is the unique extremum of g on (0, 1/ξ).

Fact (a) shows that when γ > 1, the study of g on (-∞, 0) allows to locate the leftmost edge a and this edge only. Facts (a) and (b) show that if γ 1 then it suffices to study g on D ∩ (0, ∞) to locate the edges of the bulk. In particular, if γ < 1, Fact (c) shows that the location of a is provided by the study of g on (1/η, ∞). This is illustrated by Figure 1, where a is the rightmost maximum of the function g. Fact (d) shows that when b = g(d) is a right edge of the bulk, then d cannot belong to the unbounded connected component of D in (0, ∞). Finally, the behavior of g described by (e) is illustrated on Figure 1 by the plot of this function on the interval (0, 1/3).

Regularity condition and its consequences

So far, we have thoroughly described the edges of the limiting eigenvalue distribution. Remember however that BBP phase transition [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] may occur regardless of the limiting spectral distribution (which is always MP distribution in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF]). As we shall see later, the notion of regular endpoint captures a joint condition on the limiting spectral distribution µ and on the convergence ν N → ν, which will guarantee Tracy-Widom fluctuations (cf. Theorem 3). (b) If a is an endpoint satisfying one of the options labelled (a) in Proposition 2.3, and if, furthermore, the distance dist(λ i , Supp(ν)) satisfies max

1 j n dist(λ j , Supp(ν)) ----→ N →∞ 0 ,
then a is a regular endpoint of Supp(µ). However, this last condition is not necessary. Further comments will be made in Section 3.1 below.

(c) If γ > 1, then the leftmost edge is regular (for a proof of this fact, simply write the leftmost edge as g(c), then Proposition 2.4-(a) shows that c < 0, which immediatly implies (2.5)).

Let γ N = n/N and consider now the probability measure µ(γ N , ν N ), which is the unique solution of the fixed point equation (2.1) associated with the data γ N , ν N . It is a finite-N deterministic equivalent of the spectral measure of M N . Associated to µ(γ N , ν N ) is the function

g N (z) = 1 z + γ N λ 1 -zλ ν N (dλ) = 1 z + 1 N n j=1 λ j 1 -zλ j , (2.6)
(cf. (2.3)). Similarly to µ(γ, ν), the measure µ(γ N , ν N ) has a density on (0, ∞) and its support can be also characterized with the help of Proposition 2.1 (simply replace g by g N ).

We furthermore have the following proposition:

Proposition 2.7. Let Assumption 2 hold true. Let g(c) be a regular soft edge. Then, for N large enough, (a) g N is analytic in a neighborhood of c which is independent of N .

(b) g N converges to g uniformly on the compact sets of this neighborhood, and so does its k-th order derivative g (k)

N to g (k) , for any k 1.

(c) There exists a sequence of positive numbers c N , unique up to a finite number of terms, such that c N → c, g ′ N (c N ) = 0, and g

(k) N (c N ) → g (k) (c) as N → ∞ for any k.
This proposition shows in particular that when a soft edge g(c) is regular, there is a sequence g N (c N ) of endpoints of Supp(µ(γ N , ν N )) that converge to g(c), and c N satisfies lim inf

N →∞ n min j=1 |c N -λ -1 j | > 0. (2.7)
Proof. Set η = min(|c|/2, lim inf N min j |λ -1 j -c|), and let B = B(c, η/2) be the open ball with center c and radius η/2. Since

λ i |1 -zλ j | = 1 |λ -1 j -z| 1 |λ -1 j -c| -|z -c| 3 η
for z ∈ B and for all N large, the functions g N are analytic and uniformly bounded on B for all N large. This establishes (a) in particular. Moreover, this yields that the family of analytic functions g N is uniformly bounded on B. Thus, by Montel's theorem, the family g N is normal. It follows from the convergences γ N → γ and ν N → ν provided by Assumption 1 that g N converges pointwise to g on B. Consequently, g N converges to g uniformly on the compact subsets of B, and the same is true for the convergence of the g

(k)
N to g (k) by [START_REF] Rudin | Real and complex analysis[END_REF]Th. 10.28]. Turning to (c), notice that c is a zero of g ′ by the regularity assumption, see Remark 2.6-(a). Since g ′ N converges to g ′ uniformly on the compact sets of B and g ′ is analytic there, Hurwitz's theorem shows that g ′ N has a zero c N that converges to the zero c of g ′ and that this zero is unique provided N is large enough. Write |g

(k) N (c N ) -g (k) (c)| |g (k) N (c N ) -g (k) (c N )| + |g (k) (c N ) -g (k) (c)|. Since for any k, g (k)
N converge uniformly to g k on the compact subsets of B, the first term at the right hand side vanishes as N → ∞. The second term vanishes as N → ∞ by the continuity of g (k) . This establishes (c). Let φ(N ) be the integer defined as

Extremal eigenvalues and their convergence

λ φ(N )+1 > m(v) -1 and λ φ(N ) < m(u) -1 .
(2.8)

Then P xφ(N)+1 > v, xφ(N) < u for all large N = 1 .
(2.9)

Remark 2.10. In [START_REF] Bai | Exact separation of eigenvalues of large-dimensional sample covariance matrices[END_REF], the result was established for matrices X N taken from a doubly infinite array of i.i.d. random variables with finite fourth moment. If the entries are Gaussian, one can relax the doubly infinite array assumption and establish Proposition 2.9 by using the completely different tools of [START_REF] Loubaton | Almost sure localization of the eigenvalues in a Gaussian information plus noise model-application to the spiked models[END_REF].

We are now in position to properly state and prove part (a) of Theorem 1.

Theorem 2 (Extremal eigenvalues). Let Assumptions 1 and 2 hold true Proof. We shall only prove the result for a right edge b. By Proposition 2.7, we can choose a compact neighborhood C of d such that g N and g ′ N converge uniformly to g and g ′ respectively on C. Let p, q, r, s be real numbers such that p < q < r < s < c, [p, s] ⊂ C, and g ′ (x) < 0 for x ∈ [p, s]. This last condition is made possible by the fact that b is a right edge of Supp(µ). Let u = g(r) and v = g(q). Since g N and g ′ N converge uniformly to g and g ′ respectively on [p, s], it holds that g ′ N (x) < 0 on [p, s], and [u, v] ⊂ [g N (s), g N (p)] for all N large. Proposition 2.1 applied to µ(γ N , ν N ) show then that [u, v] lies in an open set outside Supp(µ(γ N , ν N )) for all N sufficiently large. Now the integer φ(N ) defined in the statement is characterized by the inequalities λ -1 φ(N ) > d > λ -1 φ(N )+1 . Since no λ -1 j 's can belong to C for N large enough, we can equivalently write

λ -1 φ(N ) > r = m(u) > q = m(v) > λ -1 φ(N )+1
which is (2.8). By Proposition 2.9, we get (2.9). Since v > b, we have lim inf N (x φ(N )+1b) > 0 with probability one. Moreover, we know that a.s., the number of xi in [bε, b] is non zero for any ε > 0 and for all large N . Making r ↑ d, we get u = g(r) ↓ b. Since xφ(N) < u a.s. for all large N , we get that xφ(N) → b a.s. when N → ∞.

Summary of the properties of regular edges

For the reader's convenience and constant use in the sequel, we gather in the two following propositions some of the most important properties of regular edges introduced above. Recall the convention in Remark 2.8. -If γ > 1 then a = g(c) > 0 with c < 0 and a is a regular soft edge.

-If γ < 1 then a = g(c) > 0 with c > 0; a is a soft edge but its regularity is a priori not granted.

(b) Assume now that a is a regular left soft edge. Then a = g(c) with g ′ (c) = 0 g ′′ (c) < 0 and c < 0 if a is the leftmost edge and γ > 1, c > 0 otherwise. .

For N large enough, there exists a unique sequence c N such that g ′ N (c N ) = 0 and (by g For N large enough, there exists a unique sequence d N such that g ′ N (d N ) = 0 and

(0) N , g (0) we mean g N , g) c N ----→ N →∞ c , g (k) N (c N ) ----→ N →∞ g (k) (c) for any k 0 .
d N ----→ N →∞ d , g (k) N (d N ) ----→ N →∞ g (k) (d) for any k 0 .
Finally, there exists a deterministic sequence (φ(N )) such that almost surely,

lim N →∞ xφ(N) = b and lim inf N →∞ (x φ(N )+1 -b) > 0 .
3 Fluctuations around the edges

In this section, we state the main results of the paper, namely the fluctuations of the extremal eigenvalues and their asymptotic independence. Parts (b), (c) and (d) of Theorem 1 are respectively associated with Theorem 3 (Section 3.1), Theorem 4 (Section 3.2) and Theorem 5 (Section 3.3). We also provide a discussion on non-regular edges and spikes phenomena with graphical illustrations.

As an application, we obtain in Section 3.4 new results for the asymptotic behavior of the condition number of complex correlated Wishart matrices.

Tracy-Widom fluctuations at the regular soft edges

We first introduce the Tracy-Widom distribution. The Airy function Ai is the unique solution of the differential equation Ai ′′ (x) = xAi(x) which satisfies the asymptotic behavior

Ai(x) = 1 2 √ πx 1/4 e -2 3 x 3/2 1 + o(1) , x → +∞.
With a slight abuse of notation, denote by K Ai the integral operator associated with the Airy kernel

K Ai (x, y) = Ai(x)Ai ′ (y) -Ai(y)Ai ′ (x) x -y . ( 3.1)
A real-valued random variable X is said to have Tracy-Widom distribution if

P(X s) = det I -K Ai L 2 (s,∞) , s ∈ R,
where the right hand side stands for the Fredholm determinant of the restriction to L 2 (s, ∞) of the operator K Ai (see also Section 4.2). Tracy and Widom [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF] established the famous representation

det I -K Ai L 2 (s,∞) = exp - ∞ s (x -s)q(x) 2 dx ,
where q is the Hastings-McLeod solution of the Painlevé II equation, namely the unique solution of q ′′ (x) = 2q(x) 3 + xq(x) with boundary condition q(x) ∼ Ai(x) as x → ∞.

We are now in position to state our result concerning the Tracy-Widom fluctuations. Recall that g N has been introduced in (2.6).

Theorem 3. Let Assumptions 1 and 2 hold true.

(a) Let a be a left regular soft edge, and xϕ(N) and (c N ) N be as in Proposition 2.11. Set

a N = g N (c N ), σ N = 2 -g ′′ N (c N ) 1/3
.

Then, for every s ∈ R, lim

N →∞ P N 2/3 σ N a N -xϕ(N) s = det I -K Ai L 2 (s,∞) . (3.2)
(b) Let b be a right regular soft edge, and xφ(N) and (d N ) N be as in Proposition 2.12. Set

b N = g N (d N ), δ N = 2 g ′′ N (d N ) 1/3 .
Then, for every s ∈ R, lim

N →∞ P N 2/3 δ N xφ(N) -b N s = det I -K Ai L 2 (s,∞) . (3.3)
The proof is deferred to Section 4; see Section 4.1 for an outline.

Connexion with El Karoui's result.

Let us first comment the last theorem in the light of El Karoui's result [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF], see also Onatski's work [START_REF] Onatski | The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices[END_REF]. If we assume that lim inf

N →∞ d N λ n < 1, (3.4)
then, as a consequence of the analysis provided in Section 2, the sequence (d N ) N is associated with the rightmost edge b and the associated extremal eigenvalue has to be the largest eigenvalue of M N (or equivalently of M N ). Moreover, (3.4) implies that b is regular, so that Theorem 3 applies. This is the result of El Karoui announced in the introduction, which he actually proves in a more general setting. Indeed, in [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF] the weak convergence of ν N towards some limiting probability distribution and the convergence of n/N to some limit were not assumed; it is only assumed that n/N stays in a bounded set of (0, 1] (actually of (0, +∞) after [START_REF] Onatski | The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices[END_REF]) together with (3.4). Let us mention that under these only assumptions, by compactness one can always extract converging subsequences for ν N and n/N so that our result applies along a subsequence.

Notice also that the condition (3.4) is stronger than our regularity condition, since b can be regular with lim inf N d N λ n > 1. In this case, the extremal eigenvalue associated with the rightmost edge is no longer the largest eigenvalue of M N ; this entails the presence of outliers, as we shall explain in the next paragraph. Our result then states that the largest eigenvalue which actually converges to the rightmost edge b fluctuates for large N according to the Tracy-Widom law.

Non-regular edges and spikes phenomena.

In Remark 2.6-(b), we explained that when a soft edge reads b = g(d) with d ∈ ∂D, and when the Hausdorff distance between Supp(ν N ) and Supp(ν) converges to zero, then the endpoint b is regular. Still assuming that d ∈ ∂D, let us now assume instead that

ν N = k n δ ζ + νN
where k is a fixed positive integer, ζ > 0 is fixed and lies outside Supp(ν), and the Hausdorff distance between Supp(ν N ) and Supp(ν) converges to zero. The eigenvalue ζ of Σ N with multiplicity k is often called a spike. Assume without generality loss that b is a right edge and that that 1/ζ belongs to the same connected component of D as d. Three situations that we describe without formal proofs are of interest:

1. The spike ζ satisfies g ′ (1/ζ) < 0. This can only happen if 1/ζ < d as shown in Figure 2.

In that case, ζ produces k outliers, i.e., eigenvalues of M N which converge to a value outside the bulk, see [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF][START_REF] Benaych-Georges | The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices[END_REF]. In terms of the support of µ(γ N , ν N ), the location of these outliers corresponds to a small interval in Supp(µ(γ N , ν N )) (see Figure 2) which is absent from Supp(µ(γ, ν)). The width of this new interval is actually of order

N -1/2 . Since 1/ζ < d,
′ N , one can show that |1/ζ -d ′ N | is of order N -1/2 .
In particular, the regularity assumption (2.7) is not satisfied for d ′ N . In fact, it is known that when they are scaled by N 1/2 , the k outliers asymptotically fluctuate up to a multiplicative constant as the eigenvalues of a k × k matrix taken from the GUE ensemble, see [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF][START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF] among others. 3. The spike depends generally on N and satisfies 1/ζ → d as N → ∞. Here, we are at the crossing point of the phase transition discovered in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] between the "Tracy Widom regime" and the "GUE regime". More specifically, under an additional condition (see (B.2)) we shall briefly show in Appendix B that at the scale N 2/3 the asymptotic fluctuations are described by the so-called deformed Tracy-Widom law whose distribution function F k is defined in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF]Eq. 17].

All these arguments can be straightforwardly generalized to the case where a finite number of different spikes are present.

As explained in 3., we can tackle the situation where an edge satisfies a weak kind of non-regularity. Nevertheless, our approach breaks down when dealing with non-regular edges in the case where ν N is as above but with k → ∞ as N → ∞, or in the general case of a limiting measure ν for which Proposition 2.3-(b) occurs.

Asymptotic independence

Our next result states that the fluctuations of the extremal eigenvalues associated with any finite number of regular soft edges are asymptotically independent. Let xϕ i (N ) and c i,N be associated to a i as in Proposition 2.11, and denote by

a i,N = g N (c i,N ), σ i,N = 2 -g ′′ N (c i,N ) 1/3 , . b N d N -0.5 0 0.5 1 1.5 2
. Similarly, let xφ j (N ) and d j,N be associated to b j as in Proposition 2.12, and denote by

b j,N = g N (d j,N ), δ j,N = 2 g ′′ N (d j,N ) 1/3
, Then, for every real numbers (s i ) i∈J , (t j ) j∈J , we have lim

N →∞ P N 2/3 σ i,N a i,N -x ϕ i (N ) s i , N 2/3 δ j,N x φ j (N ) -b j,N t j , (i, j) ∈ I × J = i∈I det I -K Ai L 2 (s i ,∞) j∈J det I -K Ai L 2 (t j ,∞) .
We prove Theorem 4 in Section 5. Our strategy is to build on the operator-theoretic proof of Bornemann in the case of the smallest and largest eigenvalues of the GUE [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF]; it essentially amounts to prove that the off-diagonal entries of a two by two operator valued matrix decay to zero in the trace class norm. In our setting, the problem involves a larger operator valued matrix and we show that obtaining the decay to zero for the off-diagonal entries in the Hilbert-Schmidt norm is actually sufficient. We establish the latter by using the estimates established in Section 4.

A comment on universality. The results presented in this paper rely on the fact that the entries of X N are complex Gaussian random variables, a key assumption in order to take advantage of the determinantal structure of the eigenvalues of the model under study. A

. b N d N -0.5 0 0.5 1 1.5 2
. recent work [START_REF] Knowles | Anisotropic local laws for random matrices[END_REF] by Knowles and Yin enables to transfer the results presented here (except the hard edge fluctuations, see Theorem 5 below) to the case of complex, but not necessarily Gaussian, random variables. Indeed, by combining the local convergence to the limiting distribution established in [START_REF] Knowles | Anisotropic local laws for random matrices[END_REF] together with Theorems 3 and 4 , one obtains Tracy-Widom fluctuations and asymptotic independence in this more general setting, provided that the entries of matrix X N fulfill some moment condition. This also provides a similar generalization of our Proposition 3.2 describing the asymptotic behavior for the condition number of M N when γ > 1. Let us stress that the case of real Gaussian random variables, of important interest in statistical applications, remains open.

Fluctuations at the hard edge

Proposition 2.4 shows that when the leftmost edge is a hard edge, γ = 1 (actually, one can show that this is an equivalence). In order to study the smallest random eigenvalue fluctuations at the hard edge, we restrict ourselves to the case where n = N + α, where α ∈ Z is independent of N . Thus, the smallest random eigenvalue of M N is

x min = x 1 = xα+1 if α 0, x 1-α = x1 if α < 0.
We shall prove that the fluctuations of x min around the origin follows the hard-edge Tracy-Widom distribution of parameter α, that we introduce now.

The Bessel function of the first kind J α with parameter α ∈ Z is defined by

J α (x) = x 2 α ∞ n=0 (-1) n n! Γ(n + α + 1) x 2 2n , x > 0. (3.5)
Note that when α < 0, the first |α| terms in the series vanish since the Gamma function Γ has simple poles on the non-positive integers. Denote by K Be,α the Bessel kernel

K Be,α (x, y) = √ yJ α ( √ x)J ′ α ( √ y) - √ xJ ′ α ( √ x)J α ( √ y) 2(x -y) , (3.6)
and by extension K Be,α the associated integral operator. A non-negative real-valued random variable X is said to have hard-edge Tracy-Widom distribution if

P(X s) = det I -K Be,α L 2 (0,s) , s > 0,
where the right hand side stands for the Fredholm determinant of the restriction to L 2 (0, s) of the integral operator K Be,α . When α = 0 this is actually the distribution of an exponential law of parameter 1, namely det I -K Be,0 L 2 (0,s) = e -s . As for the name, it comes from the following alternative representation due to Tracy and Widom [START_REF] Tracy | Level spacing distributions and the Bessel kernel[END_REF],

det

I -K Be,α L 2 (0,s) = exp - 1 4 s 0 (log s -log x)q(x) 2 dx ,
where q is the solution of a differential equation which is reducible to a particular case of the Painlevé V equation (involving α in its parameters) and boundary condition q(x) ∼ J α ( √ x)

as x → 0.
Let us now state our result for the fluctuations around the hard edge.

Theorem 5. Let Assumptions 1 and 2 hold true; assume moreover that n = N + α, where α ∈ Z is independent of N . Set

σ N = 4 N n j=1 1 λ j . (3.7)
Then, for every s > 0, we have lim

N →∞ P N 2 σ N x min s = det I -K Be,α L 2 (0,s) . (3.8)
In particular, if N = n, then we have for every s > 0 lim

N →∞ P N 2 σ N x min s = e -s . (3.9) Remark 3.1.
The assumption that ν N converges weakly toward some limit ν is actually not used in the proof of Theorem 5. Namely, this result holds true under Assumption 1 and Assumption 2-2 only.

We provide a proof for Theorem 5 in Section 6. It is also based on a asymptotic analysis for the rescaled kernel; the key observation here is that when an edge is the hard edge, the associated critical point c is located at infinity (when embedding the complex plane into the Riemann sphere).

Application: Condition numbers

The condition number of the matrix M N with eigenvalues 0

x 1 • • • x N is defined by κ N = x N x 1 , provided it is finite, that is n/N 1.
If n/N < 1, one may instead consider the condition number associated to M N defined as κN = xn /x 1 . The study of condition numbers is important in numerical linear algebra [START_REF] Neumann | Numerical inverting of matrices of high order[END_REF][START_REF] Goldstine | Numerical inverting of matrices of high order[END_REF] and random matrix theory has already provided interesting theoretical [START_REF] Edelman | Eigenvalues and condition numbers of random matrices[END_REF][START_REF] Basor | PDEs satisfied by extreme eigenvalues distributions of GUE and LUE[END_REF] and applied [START_REF] Matthaiou | On the condition number distribution of complex wishart matrices[END_REF][START_REF] Bianchi | Performance of statistical tests for single-source detection using random matrix theory[END_REF] results. As a consequence of our former results, we provide an asymptotic study for κ N (one can easily derive similar results for κN ). Proposition 3.2. Let Assumptions 1 and 2 hold true and γ > 1. Let a be the leftmost edge, assume it is regular and let (c N ) N and c be as in Proposition 2.11. Let b be the rightmost edge, assume it is regular and let (d N ) N and d be as in Proposition 2.12. Denote by

a N = g N (c N ) , σ N = 2 -g ′′ N (c N ) 1/3 and b N = g N (d N ) , δ N = 2 g ′′ N (d N ) 1/3 . Assume moreover that x 1 → a and x N → b a.s. Then κ N a.s. ----→ N →∞ b a and N 2/3 κ N - b N a N D ----→ N →∞ X δa + bY σa 2
where D -→ denote convergence in distribution, X and Y are two independent Tracy-Widom distributed random variables, and where

σ = 2 -g ′′ (c) 1/3 = lim N →∞ σ N and δ = 2 g ′′ (d) 1/3 = lim N →∞ δ N .
Remark 3.3. The condition that x 1 → a and x N → b a.s. imposes that neither x N nor x 1 are outliers, otherwise their fluctuations (together with those of κ N ) would be of order N 1/2 and a different asymptotic analysis (somewhat easier) should be conducted. We do not pursue in this direction here.

Proof. Only the convergence in distribution requires an argument. Write

N 2/3 κ N - b N a N = N 2/3 x N x 1 - b N a N = N 2/3 a N x N -b N x 1 x 1 a N = N 2/3 x 1 a N {a N (x N -b N ) -b N (x 1 -a N )} = 1 x 1 δ N N 2/3 δ N (x N -b N ) + b N x 1 a N σ N N 2/3 σ N (a N -x 1 ) .
Using the asymptotically independent Tracy-Widom fluctuations of We now handle the case where γ = 1.

N 2/3 δ N (x N -b N ) and N 2/3 σ N (a N -x 1 ) (cf.
Proposition 3.4. Let Assumptions 1 and 2 hold true and let n = N + α where α ∈ N is independent of N . Let

σ N = 4 N n j=1 1 λ j and σ = 4 1 x dν(x) = lim N →∞ σ N .
Assume that a.s. x N → b for some b > 0. Then

1 N 2 κ N D ----→ N →∞ bσ X
where D -→ denote convergence in distribution and X is a hard-edge Tracy-Widom distributed random variable of parameter α.

Proof. Write κ N N 2 = σ N (x N -b) N 2 σ N x 1 + σ N b N 2 σ N x 1 .
Since by assumption x Nb → 0 a.s. and by Theorem 5 (N

2 σ N x 1 ) -1 → X -1 in distribution,
where X is a hard-edge Tracy-Widom distributed random variable of parameter α, we have

σ N (x N -b) N 2 σ N x 1 L ----→ N →∞ 0 .
By Slutsky's lemma, N -2 κ N then converges towards bσX -1 in distribution.

Remark 3.5. Interestingly, in the square case where γ = 1, the fluctuations of the largest eigenvalue x N (either of order N 1/2 if x N is an outlier, see the next paragraph, or of order N 2/3 in the Tracy-widom regime) have no influence on the fluctuations of κ N as these are imposed by the limiting distribution of x 1 .

Proof of Theorem 3: Tracy-Widom fluctuations

This section is devoted to the proof of Theorem 3.

Outline of the proof

Step 1 (preparation): As in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] and [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF], the starting point to establish Tracy-Widom fluctuations is that the random eigenvalues of M N or M N form a determinantal point process, so that the gap probabilities can be expressed as Fredholm determinants of an integral operator K N with kernel K N (x, y). We provide all the necessary material from operator theory in Section 4.2. In Section 4.3 we first recall the double contour integral formula for K N (x, y) obtained in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] Onatski | The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices[END_REF]. Next, we show using Theorem 2 that one can represent the cumulative distribution functions for the extremal eigenvalues as Fredholm determinants involving K N asymptotically. As a consequence, proving the Tracy-Widom fluctuations boils down to establish the appropriate convergence of rescaled versions K N (x, y) of the kernel K N (x, y) towards the Airy kernel. To this end, we split K N (x, y) into two parts, K

N (x, y) and K

N (x, y), each involving different integration contours.

Step 2 (contours deformations): Anticipating the forthcoming asymptotic analysis, we focus in Section 4.4 on right edges and prove the existence of appropriate integration contours coming with K (0) N (x, y) and K

(1) N (x, y); the case of a left edge is deferred to Section 4.6. To obtain appropriate explicit contours is usually the hard part in the asymptotic analysis, see in particular [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF]. Here, we instead provide a non-constructive proof for the existence of appropriate contours by mean of the maximum principle for subharmonic functions, and which has the advantage to work for every regular edge up to minor modifications.

Step 3 (asymptotic analysis): Still focusing on the right edge setting, we prove in Section 4.5.1 that K (0) N (x, y) does not contribute in the large N limit. Moreover, we prove the convergence of kernel K

(1) N (x, y) to the Airy kernel in an apropriate sense and then complete the proof of Theorem 3-(b). For this last step, we use a different approach than in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF]: Instead of relying on a factorization trick and the Hölder inequality to obtain the trace class convergence, we use an argument involving the regularized Fredholm determinant det 2 to show the convergence of the Fredholm determinants. Finally, in Section 4.6, we adapt the arguments to the left edge setting and complete the proof of Theorem 3.

Operators, Fredholm determinants and determinantal processes

Trace class operators and Fredholm determinants. We provide hereafter a few elements of operator theory; classical references are [START_REF] Davies | of Cambridge Studies in Advanced Mathematics[END_REF][START_REF] Gohberg | Traces and determinants of linear operators[END_REF][START_REF] Simon | Trace ideals and their applications[END_REF]. Consider a compact linear operator A acting on a separable Hilbert space H (we write A ∈ L(H)), and denote by (s n ) ∞ n=1 the singular values of A repeated according to their multiplicities, i.e. the eigenvalues of (AA * ) 1/2 . The set

J 1 = A ∈ L(H) , ∞ n=1 s n < ∞
is the (sub)algebra of trace class operators endowed with the norm A 1 = ∞ n=1 s n . which is complete. If A ∈ J 1 with eigenvalues (λ n ) ∞ n=1 (repeated according to their multiplicities), then the trace and the Fredholm determinant of A, respectively defined by Similarly, let

Tr(A) = ∞ n=1 λ n and det(I -A) = ∞ n=1 (1 -λ n ) ,
J 2 = A ∈ L(H) , ∞ n=1 s 2 n < ∞
be the (sub)algebra of Hilbert-Schmidt operators endowed with the norm

A 2 = ∞ n=1 s 2 n 1/2 . The set (J 2 , • 2 ) is complete. If A ∈ J 2 with eigenvalues (λ n ) ∞ n=1 (repeated according to their multiplicities), then the regularized 2-determinant of A, det 2 (I -A) = ∞ n=1 (1 -λ n )e λn , (4.2)
is well-defined and finite. The map A → det 2 (I -A) is continuous on (J 2 , • 2 ). The inclusion J 1 ⊂ J 2 is straightforward. Moreover, the Hölder inequality AB 1 A 2 B 2 yields that if A, B are Hilbert-Schmidt then both AB and BA are trace class. The following simple property will play a key role in the sequel:

Proposition 4.1. Let A ∈ J 1 then det 2 (1 -A) = det(1 -A)e -Tr(A) .

As a consequence, if the operators

A n , A ∈ J 1 are such that Tr(A n ) → Tr(A) and A n -A 2 → 0 as n → ∞, then det(I -A n ) ---→ n→∞ det(I -A) .
Integral operators. When working on H = L 2 (R), we identify a given kernel (x, y) → K(x, y) with its associated integral operator Kf = K(•, y)f (y) dy acting on L 2 (R), provided the latter makes sense. Let J ⊂ R be a Borel set and 1 J be the orthogonal projection of L 2 (R) onto L 2 (J). The restriction K| J of K to L 2 (J) is defined by

K| J f (x) = 1 J (x) J K(x, y)f (y) dy, f ∈ L 2 (J),
and is associated to the kernel (x, y) → 1 J (x)K(x, y)1 J (y), namely K| J = 1 J K1 J . In order to keep track of these projections when dealing with Fredholm determinants, we shall often write det(I -K) L 2 (J) for det(I -

1 J K1 J ). Given a measurable kernel K : R × R → R, the associated integral operator K on L 2 (R) is Hilbert-Schmidt if and only if R R K(x, y) 2 dxdy < ∞ ,
and in this case we have

K 2 = R R K(x, y) 2 dxdy 1/2 . (4.3)
We finally recall (cf. [START_REF] Gohberg | Traces and determinants of linear operators[END_REF]Th. 8

.1]) that if K : [a, b] × [a, b] → R is a continuous kernel whose associated operator 1 (a,b) K1 (a,b) is trace class 4 on L 2 (R), then Tr 1 (a,b) K1 (a,b) = b a K(x, x) dx . (4.4)
Convention: From now, the trace Tr and the Hilbert-Schmidt norm • 2 will always refer to the Hilbert space L 2 (R).

Determinantal point process. Real random variables x 1 , . . . , x m are said to form a determinantal point process with kernel K : R × R → R (and Lebesgue measure for reference measure) if its gap probabilities are expressed as Fredholm determinants, namely for any Borel set J ⊂ R we have

P ♯ 1 k m : x k ∈ J = 0 = det I -K L 2 (J) ,
provided that the right hand side makes sense; the latter stands for the Fredholm determinant of the restriction to L 2 (J) of the integral operator with kernel K(x, y).

The kernel of a correlated Wishart matrix and its properties

The next proposition will be of fundamental use in this paper.

Proposition 4.2. Let Assumption 1 holds true. Then, for every N , the min(n, N ) random eigenvalues of M N (and equivalently of M N ) form a determinantal point process associated with the kernel

K N (x, y) = N (2iπ) 2 Γ dz Θ dw e -N x(z-q)+N y(w-q) 1 w -z z w N n j=1 w -λ -1 j z -λ -1 j , (4.5)
where the real q ∈ (0, λ -1 n ) is a free parameter and we recall that the λ i 's are the eigenvalues of Σ N . Γ and Θ are disjoint closed contours, both oriented counterclockwise, such that Γ encloses the λ -1 j 's and lies in {z ∈ C : Re z > q}, whereas Θ encloses the origin and lies in {z ∈ C : Re z < q}.

By convention, all the contours we shall consider will be assumed to be simple and oriented counterclockwise. The integration contours are shown in Figure 4.

This proposition can be found in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] (n/N 1) where it is attributed to Johansson, and in [START_REF] Onatski | The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices[END_REF] (n/N > 1). Notice that since the pioneering work of Brézin and Hikami [START_REF] Brézin | Universal singularity at the closure of a gap in a random matrix theory[END_REF], many such double integral representations appeared for determinantal point processes.

. The assumption over q, i.e. q ∈ (0, λ -1 n ), ensures that K N with kernel (4.5) is trace class on L 2 (R). In the sequel, we shall only need K N to be locally trace class, that is trace class on L 2 (J) for every compact subset J ⊂ R. As an important consequence, we can choose q ∈ R with no further restriction. In fact, let q ∈ (0, λ -1 n ), q ′ ∈ R and J ⊂ R a compact set, then the multiplication operator E : f (x) → e (q ′ -q)N x f (x) and its inverse E -1 are trace class on L 2 (J). Write K N = K N E -1 E and use (4.1) to get

λ -1 n-1 λ -1 n λ -1 1 Γ Θ .
det(I -K N ) L 2 (J) = det(I -EK N E -1 ) L 2 (J) .
The kernel of EK N E -1 is simply obtained by (4.5) where q has been replaced by q ′ and our claim follows.

Asymptotic determinantal representation for the law of extremal eigenvalues.

Recall that to prove Theorem 3 for the maximal eigenvalue xn of M N , a classical way to proceed is to identify the events

{N 2/3 σ N (x n -b N ) s} = {no xi 's in (b N +s/(N 2/3 σ N ), ∞)} , to use the determinantal representation P N 2/3 σ N (x n -b N ) s = det(I -K N ) L 2 (b N +s/(N 2/3 σ N ),∞) ,
and to prove the convergence of operator K N to the Airy operator K Ai after the rescaling x → b N + x/(N 2/3 σ N ) for the trace class topology. This would yield the desired result since the Fredholm determinant is continuous for that topology.

Since the probabilities of interest P(N 2/3 σ N (x φ(N ) -b N ) s) and P(N 2/3 σ N (a N -x ϕ(N ) ) s) cannot be expressed as gap probabilities in general, we provide below an asymptotic Fredholm determinant representation as N → ∞ for these. (a) For every ε > 0 small enough and for every sequence (η N ) N of positive numbers satisfying lim N η N = +∞,

P η N a N -xϕ(N) s = det I -K N L 2 (a N -ε, a N -s/η N ) + o(1) (4.6)
as N → ∞.

(b) For every ε > 0 small enough and for every sequence (η N ) N of positive numbers satisfying lim N η N = +∞,

P η N xφ(N) -b N s = det I -K N L 2 (b N +s/η N , b N +ε) + o(1) (4.7)
as N → ∞.

Proof. We only prove (b), proof of (a) being similar. Observe that Theorem 2-(b) and the convergence b N → b yield together the existence of ε > 0 small enough such that

P η N xφ(N) -b N s = P η N xφ(N) -b N s , xφ(N)+1 b N + ε + o(1) (4.8)
as N → ∞. Now, ε being fixed, use the determinantal representation to write

det I -K N L 2 (b N +s/η N , b N +ε) = P ♯ ℓ k n : b N + s/η N xk b N + ε = 0 (4.9)
where ℓ = nmin(N, n) + 1. Recall the notational convention in Remark 2.8; we obtain by splitting along disjoint events

P ♯ ℓ k n : b N + s/η N xk b N + ε = 0 = P η N xφ(N) -b N s , xφ(N)+1 b N + ε + P xℓ b N + ε (4.10) + n k=ℓ, k = φ(N ) P xk b N + s/η N , xk+1 b N + ε .
Since we have the upper bounds

φ(N )-1 k=ℓ P xk b N + s/η N , xk+1 b N + ε P xφ(N) b N + ε , n k=φ(N )+1 P xk b N + s/η N , xk+1 b N + ε P xφ(N)+1 b N + s/η N ,
we obtain from (4.9), (4.10), Th. Rescaling and splitting the kernel K N . We introduce hereafter the rescaled kernel K N and provide an alternative integral representation with new contours. The aim is to prepare the forthcoming asymptotic analysis for right regular edges.

Let b be a soft regular right edge. By Proposition 2.12, there exist

d > 0 such that b = g(d), g ′ (d) = 0, g ′′ (d) > 0, (4.12)
and an associated sequence (d N ) such that g

(k) N (d N ) → g (k) (d). Denote by b N = g N (d N ), δ N = 2 g ′′ N (d N ) 1/3 , (4.13)
then in particular

g ′ N (d N ) = 0, lim N →∞ d N = d, lim N →∞ b N = b, lim N →∞ δ N = 2 g ′′ (d) 1/3 . (4.14)
In particular c N , g ′′ N (c N ) and σ N are positive numbers for every N large enough, and (σ N ) N is a bounded sequence.

It follows from the definition of the extremal eigenvalue xφ(N) , see Theorem 2, and Proposition 4.4 that for every ε > 0 small enough

P N 2/3 δ N xφ(N) -b N s = det I -K N L 2 (b N +s/(N 2/3 δ N ), b N +ε) + o(1) (4.15)
as N → ∞. By a change of variable, we can write

det I -K N L 2 (b N +s/(N 2/3 δ N ), b N +ε) = det I -1 (s, εN 2/3 δ N ) K N 1 (s, εN 2/3 δ N ) L 2 (s,∞) , (4.16)
where the scaled integral operator K N has kernel

K N (x, y) = 1 N 2/3 δ N K N b N + x N 2/3 δ N , b N + y N 2/3 δ N (4.17)
and where K N (x, y) was introduced in (4.5). Consider the map

f N (z) = -b N (z -d N ) + log(z) - 1 N n i=1 log(1 -λ i z). (4.18)
Remark 4.5. In order to fully define f N , one needs to specify the determination of the logarithm. This will be done when needed. Notice however that functions Re f N , exp(f N ) and the derivatives f

(k)
N are always well-defined.

By taking q = d N in (4.5), which is possible according to Remark 4.3, we have

K N (x, y) = N (2iπ) 2 Γ dz Θ dw e -N x(z-d N )+N y(w-d N ) 1 w -z z w N n j=1 1 -λ j w 1 -λ j z , (4.19)
where we recall that the contour Γ encloses the λ -1 j 's whereas the contour Θ encloses the origin and is disjoint from Γ. It then follows from the definition (4.17

) of K N that K N (x, y) = N 1/3 (2iπ) 2 δ N Γ dz Θ dw 1 w -z e -N 1/3 x(z-d N )/δ N +N 1/3 y(w-d N )/δ N +N f N (z)-N f N (w) .
(4.20) The key observation here is the identity

f ′ N (z) = g N (z) -g N (d N ), (4.21)
which follows from (2.6) and (4.13). As a byproduct, (4.14) yields that d N is a root of multiplicity two for f ′ N , and more precisely

f ′ N (d N ) = f ′′ (d N ) = 0, f (3)
N (d N ) = g ′′ N (d N ) > 0. (4.22)
The aim is to perform a saddle point analysis for f N around its critical point d N . To this end, we deform the contours Γ and Θ in a way that they pass near d N .

If d N is smaller than all the λ -1 j 's, as it is the case in [START_REF] El Karoui | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices[END_REF] when dealing with the maximal eigenvalue, then go directly to Section 4.4, set Γ (1) = Γ, K

N = K N and disregard every statement related to Γ (0) .

If not, then we proceed in two steps. First, we split Γ into two disjoint contours Γ (0) and Γ (1) as shown on Figure 5: the contour Γ (0) encloses the λ -1 j 's which are smaller than d N , while Γ (1) encloses the λ -1 j 's which are larger that d N . Notice that Proposition 2.

4-(d)

.

λ -1 1 λ -1 n d N Γ (0) Γ (1) Θ λ -1 φ(N ) λ -1 n-1 λ -1 φ(N )+1
.

Figure 5: The new contours Γ (0) and Γ (1) .

applied to the measure ν N shows that the set {j , 1 j n : λ -1 j > d N } is not empty. Therefore, the contour Γ (1) is always well-defined.

We now introduce for α ∈ {0, 1} the kernels

K (α) N (x, y) = N 1/3 (2iπ) 2 δ N Γ (α) dz Θ dw 1 w -z e -N 1/3 (z-d N )x/δ N +N 1/3 (w-d N )y/δ N +N f N (z)-N f N (w) , (4.23)
then it follows from the residue theorem that

K N (x, y) = K (0) N (x, y) + K (1) N (x, y) , (4.24)
and a similar identity for the associated operators.

In the second step, we modify the contour Θ in order for it to surround Γ (0) while remaining at the left of d N , cf. Figure 6. This can be done with no harm for the kernel K

(1) N . As for K (0) N , this modification for the contours yields a residue term, coming with the singularity (wz) -1 of the integrand. The latter residue term equals

- N 1/3 2iπσ N Γ (0) e N 1/3 (y-x)(z-c N )/σ N dz
and thus identically vanishes since the integrand is analytic.

.

λ -1 1 Θ Γ (1)
Γ (0) λ -1 n d N λ -1 n-1 λ -1 φ(N ) λ -1 φ(N )+1
.

Contours deformations and subharmonic functions

We now provide the existence of deformations for the contours Γ (0) , Γ (1) and Θ which are appropriate for the asymptotic analysis. These new contours will be referred to as Υ (0) , Υ (1) and Θ.

Proposition 4.6. For every ρ > 0 small enough, there exists a contour Υ (0) independent of N and two contours Υ (1) = Υ (1) (N ) and Θ = Θ(N ) which satisfy for every N large enough the following properties.

(1) (a) Υ (0) encircles the λ -1 j 's smaller than d N , (b) Υ (1) encircles all the λ -1 j 's larger than d N , (c) Θ encircles all the λ -1 j 's smaller than d N and the origin.

(

) (a) Υ (1) = Υ * ∪ Υ 2
res where

Υ * = {d N + te ±iπ/3 : t ∈ [0, ρ]}. (b) Θ = Θ * ∪ Θ res where Θ * = {d N -te ±iπ/3 : t ∈ [0, ρ]}.
(3) There exists K > 0 independent of N such that

(a) Re f N (z) -f N (d N ) -K for all z ∈ Υ (0) (b) Re f N (z) -f N (d N ) -K for all z ∈ Υ res (c) Re f N (w) -f N (d N ) K for all w ∈ Θ res (4) There exists d > 0 independent of N such that inf |z -w| : z ∈ Υ (0) , w ∈ Θ d , inf |z -w| : z ∈ Υ * , w ∈ Θ res d , inf |z -w| : z ∈ Υ res , w ∈ Θ * d , inf |z -w| : z ∈ Υ res , w ∈ Θ res d .
(5) (a) The contours Θ and Υ (0) are disjoint, so are Υ (0) and Υ (1) , but Θ ∩ Υ (1) = {d N }, (b) The contours Υ (1) and Θ lie in a bounded subset of C independent of N , (c) The lengths of Υ (1) and Θ are uniformly bounded in N .

Note that both the contours Υ (1) and Θ pass through the critical point d N .

In order to provide a proof for Proposition 4.6, we first establish a few lemmas. We recall that B(z, ρ) for z ∈ C and ρ > 0 stands for the open ball of C with center z and radius ρ.

Recall that 0

< inf N λ -1 n sup N λ -1 1 < ∞ by Assumption 2.
By the regularity assumption, namely lim inf N min n j=1 |d-λ -1 j | > 0, there exists ε > 0 such that λ -1 j ∈ (0, +∞)\B(d, ε) for every 1 j n and every N large enough. Denote by K the compact set

K = inf N 1 λ n , sup N 1 λ 1 \ B(d, ε) ∪ {0} . (4.25)
Notice that by construction {x ∈ R : x -1 ∈ Supp(ν N )} ⊂ K for every N large enough, and also that {x ∈ R :

x -1 ∈ Supp(ν)} ⊂ K because of the weak convergence ν N → ν.
Recall the definition (4.18) of f N and introduce its asymptotic counterpart:

f (z) = -b(z -d) + log(z) -γ log(1 -xz)ν(dx) . (4.26)
Notice that whereas f and f N are defined up to a determination of the complex logarithm,

Re f (z) = -b Re(z -d) + log |z| -γ log |1 -zx|ν(dx) (4.
f N (z) -f N (d N ) -g ′′ N (d N )(z -d N ) 3 /6 ∆|z -d N | 4 , and
Re(f N (z) -f N (d N )) -g ′′ N (d N ) Re[(z -d N ) 3 ]/6 ∆|z -d N | 4 .
(c) There exists ρ 0 > 0 and

∆ = ∆(ρ 0 ) > 0 such that B(d, ρ) ⊂ C \ K for every ρ ∈ (0, ρ 0 ] and Re(f (z) -f (d)) -g ′′ (d) Re[(z -d) 3 ]/6 ∆|z -d| 4 .
Proof

′ N (d N ) = f ′′ N (d N ) = 0, f (3)
N (d N ) = g ′′ N (d N ) > 0 and f (4) N = g (3)
N for all N large enough, we can perform a Taylor expansion for f N around d N in order to get

f N (z) -f N (d N ) -g ′′ N (d N )(z -d N ) 3 /6 |z -d N | 4 24 max w∈B(d N ,ρ) g (3)
N (w) , |z -d N | 4 24 max w∈B(d,2ρ 0 ) g (3)
N (w)

provided that z ∈ B(d N , ρ) and ρ ∈ (0, ρ 0 ]. Proposition 2.7 moreover provides that g

(3) N converges uniformly on B(d, 2ρ 0 ) to g (3) which is bounded there. The existence of ∆ = ∆(ρ 0 ) independent of N and 0 < ρ < ρ 0 satisfying the first inequality in Part (b) of the proposition follows. The inequality for the real part follows directly and Part (b) of the proposition is proved; so is Part (c) by using similar arguments.

We now provide a qualitative analysis for the map Re f . First, we study the behavior of Re f (z) as |z| → ∞. To do so, we introduce the sets

Ω -= z ∈ C : Re f (z) < Re f (d) , Ω + = z ∈ C : Re f (z) > Re f (d) , (4.29)
and prove the following.

Lemma 4.8. Both Ω + and Ω -have a unique unbounded connected component. Moreover, given any α ∈ (0, π/2), there exists R > 0 large enough such that

Ω R -= z ∈ C : |z| > R, - π 2 + α < arg(z) < π 2 -α ⊂ Ω -, (4.30
)

Ω R + = z ∈ C : |z| > R, π 2 + α < arg(z) < 3π 2 -α ⊂ Ω + . ( 4
Ω R -⊂ Ω -, Ω R + ⊂ Ω + . (4.32)
Next, we compute for any

A ∈ R \ {0} d dt Re f (t + iA) = -b + t t 2 + A 2 + γ (x -1 -t) (x -1 -t) 2 + A 2 ν(dx).
Since b > 0 and Supp(ν) is a compact subset of (0, +∞), there exists A 0 > 0 such that for any A satisfying |A| A 0 the map t → d dt Re f (t + iA) is negative, namely t → Re f (t + iA) is decreasing. Assume there exists an other unbounded connected component of Ω -than the one containing Ω R -. By (4.32), this unbounded connected component then lies in C \ (Ω R -∪ Ω R + ) and thus there exists z 0 in this component satisfying |Im(z 0 )| A 0 . Since the half line {Re(z 0 ) + t + iIm(z 0 ) : t 0} then belongs to Ω -and eventually hits Ω R -, we obtain a contradiction. That Ω + has a unique unbounded connected component follows by using the same line of arguments.

Next, we describe the behavior of Re f at the neighborhood of d. Taking advantage of Lemma 4.7-(c) which encodes that Re f (z) -Re f (d) behaves like Re[(zd) 3 ] around d, we describe in the following lemma subdomains of Ω ± of interest. Lemma 4.9. There exist η > 0 and θ > 0 small enough such that, if we set

∆ k = z ∈ C : 0 < |z -d| < η, arg(z -d) -k π 3 < θ for -2 k 3, then ∆ 2k+1 ⊂ Ω -, ∆ 2k ⊂ Ω + , k ∈ {-1, 0, 1}.
The regions ∆ k are shown on Figure 7.

Proof. Recall Lemma 4.7-(c) and let η < ρ 0 as defined there. Then

Re

f (z) -Re f (d) -g ′′ (d) Re[(z -d) 3 ]/6 ∆(ρ 0 )|z -d| 4
for every z ∈ B(d, η). Notice that Re[(zd) 3 ] = (-1) k if z = d + e ikπ/3 for consecutive integers k. Since g ′′ (d) > 0, the lemma follows by choosing η small enough.

We denote by Ω 2k+1 the connected component of Ω -which contains ∆ 2k+1 . Similarly, Ω 2k stands for the connected component of Ω + which contains ∆ 2k . We now describe these sets by using the maximum principle for subharmonic functions, in the same spirit than in [29, Sec. 6.1], see also [START_REF] Delvaux | A phase transition for nonintersecting Brownian motions, and the Painlevé equation[END_REF]Section 2.4.2], although the setting is more involved here; such a use of the maximum principle has been communicated to us by Steven Delvaux.

Recall that if G is an open subset of C, a function u : G → R ∪ {-∞} is subharmonic if u is upper semicontinuous and, for every closed disk B(z, δ) contained in G, we have the inequality where the compactly supported probability measure τ is the image of ν by x → x -1 . The assumptions on ν imply that log x is ν-integrable. Now, it is a standard fact from potential theory that, given a positive Borel measure η on C with compact support, the map z → log |z -x| η(dx) is subharmonic on C and harmonic on C \ Supp(η), see e. (

u(z) 1 2π 2π 0 u(z + δe iθ )dθ. A function u : G → R ∪ {+∞} is superharmonic if -u
) Let Ω * be a connected component of Ω -such that Ω * ⊂ R. (a) If Ω * is bounded, then 0 ∈ Ω * . (b) If Ω * is bounded, then its interior is connected. (c) If 0 / ∈ Ω * , 2
f (z) = Re f (d), z ∈ ∂Ω * \ K. (4.33) Let us fix ε 0 > 0 such that Ω * ∩ {Im(z) ε 0 } = ∅ and pick z 0 ∈ Ω * satisfying Im(z 0 ) ε 0 and Re f (z 0 ) < Re f (d).
Our goal is to construct a bounded domain which contains z 0 but not the origin and where Re f > Re f (z 0 ) on its boundary. Indeed, this would lead to a contradiction via the minimum principle for superharmonic functions since Re f is superharmonic on C\{0} as stated in Lemma 4.10. First, notice that if dist(Ω * , R) > 0 then Re f is harmonic on Ω * , Re f = Re f (d) on ∂Ω * and Re f < Re f (d) on Ω * which is a bounded domain. But this contradicts the minimum principle for (super)harmonic functions, and thus dist(Ω * , R) = 0. Because dist(Ω * , R) = 0 and Ω * ∩ {Im(z) > 0} is open and non-empty, for every ε > 0 small enough Ω * ∩ {Im(z) = ε} = U + iε where U is a non-empty open subset of the real line. Thus, we can write

Ω * ∩ {Im(z) = ε} = j∈J u (j) min (ε), u (j) max (ε) + iε
where J is a countable set satisfying Card(J) 1, and the u

(j 2 ) min (ε), u (j 2 )
max (ε)) are disjoint as soon as j 1 = j 2 . Notice that by symmetry,

Ω sym * ∩ {Im(z) = -ε} = j∈J u (j) min (ε), u (j) max (ε) -iε.
By construction, for every j ∈ J, both u

(z) = Re f (z), Re f u (j) min (ε) ± iε = Re f u (j) max (ε) ± iε = Re f (d), j ∈ J. (4.34)
Since by assumption 0 / ∈ Ω * ⊂ Ω -and Re f (z) → -∞ as |z| → 0, there exists δ > 0 such that B(0, δ) ∩ Ω * = ∅. Because Ω * is moreover bounded by assumption, |u stay in a compact subset of (0, +∞) independent on ε and j ∈ J as ε → 0. As a consequence, we can choose ε ∈ (0, ε 0 ) small enough so that max

ε 2 u (j) min (ε) 2 , ε 2 u (j) max (ε) 2 < min Re f (d) -Re f (z 0 ), 1 2 , j ∈ J. (4.35)
If we moreover consider for any j ∈ J the open rectangle

R j (ε) = u + iv ∈ C : u (j) min (ε) < u < u (j) max (ε), |v| < ε ,
then we can also assume that ε is small enough so that 0 / ∈ R j (ε) for every j ∈ J. Let us fix j ∈ J. Let η ∈ R be such that |η| ε and set

z η = u (j) min (ε) + iη. Since |1 -xz η | |1 -xz ε | for every x ∈ R, it follows that log |1 -xz η | ν(dx) log |1 -xz ε | ν(dx)
and, together with (4.34), that

Re f (z η ) Re f (z ε ) + log z η z ε = Re f (d) + log z η z ε . (4.36)
Next, we have log

z η z ε = 1 2 log u (j) min (ε) 2 + η 2 u (j) min (ε) 2 + ε 2 = 1 2 log 1 - ε 2 -η 2 u (j) min (ε) 2 + ε 2 1 2 log 1 - ε 2 u (j) min (ε) 2 - ε 2 u (j) min (ε) 2 , ( 4.37)
where for the last inequality we used that log(1-x) -2x for any x ∈ [0, 1/2]. By combining (4.35)-(4.37), we have shown that Re f u

(j) min (ε) + iη > Re f (z 0 ), |η| ε, j ∈ J. (4.38)
The same line of arguments also shows that

Re f u (j) max (ε) + iη > Re f (z 0 ), |η| ε, j ∈ J. (4.39)
Now, consider the set

Ω * = z ∈ Ω * : Im(z) ε ∪ z ∈ Ω sym * : Im(z) -ε ∪ j∈J R j (ε)
and notice it is a bounded open set containing z 0 (since Im(z 0 ) ε 0 > ε), but which may not be connected, and which does not contain the origin. Let Ω * (z 0 ) be the connected component of Ω * which contains z 0 . Since 0 / ∈ Ω * (z 0 ), Re f is superharmonic on the bounded domain Ω * (z 0 ). It follows from (4.33), (4.38), (4.39) and the symmetry Re f (z) = Re f (z) that Re f > Re f (z 0 ) on ∂ Ω * (z 0 ). This yields a contradiction with the minimum principle for superharmonic functions and (2a) follows.

We now turn to (2b) and (2c) and, again, argue by contradiction. Let Ω * ⊂ R be connected component of Ω -such that its interior int(Ω * ) is not connected. Notice that since Re f is continuous on C \ K, we have int(Ω * ) \ K = Ω * \ K and in particular (4.33) yields

Re f (z) = Re f (d), z ∈ ∂ int(Ω * ) \ K.
If Ω * is bounded, then by (2a) we have 0 ∈ Ω * and moreover, since Re f (z) → -∞ as z → 0, 0 ∈ int(Ω * ). Let Ω ′ * be a connected component of int(Ω * ) which does not contain the origin. It is then a bounded domain on which Re f < Re f (d) and Re f = Re f (d) on ∂Ω ′ * \ K. By picking z 0 ∈ Ω ′ * ∩ {Im(z) > 0} and by performing the same construction than in the proof of (2a) but replacing Ω * by Ω ′ * , we obtain a bounded domain Ω ′ * (z 0 ) containing z 0 in its interior, on which Re f is superharmonic, and such that Re f > Re f (z 0 ) on its boundary. The minimum principle for superharmonic functions shows this is impossible and (2b) follows.

To show (2c), assume now that 0 / ∈ Ω * , so that Ω * is necessarily unbounded by (2a). By using that int(Ω * ) \ K = Ω * \ K where K is a compact set, that Ω -has a unique unbounded connected component by Lemma 4.8, and that by assumption int(Ω * ) is not connected, it follows that at least one connected component of int(Ω * ), say Ω ′ * , is bounded. Since by assumption 0 / ∈ Ω ′ * , the same argument than in the proof of (2b) yields a contradiction and (2c) is proved.

Finally, we use the previous lemmas to describe the sets Ω k 's. Lemma 4.12.

(1) We have Ω 1 = Ω -1 , the interior of Ω 1 is connected, and for every 0 < α < π/2 there exists R > 0 such that

z ∈ C : |z| > R, - π 2 + α < arg(z) < π 2 -α ⊂ Ω 1 . (4.40)
(2) We have Ω 2 = Ω -2 , the interior of Ω 2 is connected, and there exists R > 0 such that

z ∈ C : |z| > R, π 2 + α < arg(z) < 3π 2 -α ⊂ Ω 2 . (4.41)
(3) The interior of Ω 3 is connected and there exists δ > 0 such that B(0, δ) ⊂ Ω 3 .

Proof. We first prove (2). Since Ω 2 is by definition a connected subset of Ω + , Lemma 4.11 [START_REF] Bai | Random matrix theory and its applications[END_REF] yields that its interior is connected (since it is open). Next, we show by contradiction that Ω 2 is unbounded. If Ω 2 is bounded, then Lemma 4.11 [START_REF] Bai | Random matrix theory and its applications[END_REF] shows there exists x ∈ Supp(ν) such that

x -1 ∈ Ω 2 . If x -1 < d (resp. x -1 > d), then it follows from the symmetry Re f (z) = Re f (z)
that Ω 2 completely surrounds Ω 3 (resp. Ω 1 ). As a consequence, Ω 3 ⊂ R (resp. Ω 1 ⊂ R) is a bounded connected component of Ω -which does not contain the origin, and Lemma 4.11 (2a) shows this is impossible. The symmetry Re f (z) = Re f (z) moreover provides that Ω -2 is also unbounded, and (2) follows from the inclusion (4.31) and the fact that Ω + has a unique unbounded connected component, see Lemma 4.8. We now turn to [START_REF] Bai | Random matrix theory and its applications[END_REF]. Since Ω 2 is unbounded, then Ω 1 does not contain the origin and it follows from Lemma 4.11 (2a)-(2c) that Ω 1 is unbounded and has a connected interior. Then, (1) follows from symmetry Re f (z) = Re f (z), the inclusion(4.30) and that Ω -has a unique unbounded connected component (cf. Lemma 4.8).

Finally, since Ω 3 is bounded as a byproduct of Lemma 4.12-( 2), it has a connected interior (Lemma 4.11 (2b)) and contains the origin (Lemma 4.11 (2a)). Moreover, since Re f (z) → -∞ as z → 0, (3) follows.

. res (N 0 ). The dotted path at the left is its counterpart for Θ. The closed contour at the left of d is Υ (0) . We are finally in position to prove Proposition 4.6.

∆ 1 ∆ 2 ∆ -1 ∆ -2 Ω - Ω + ← radius η Ω - d ∆ 0 ∆ 3 .
Proof of Proposition 4.6. Given any ρ > 0 small enough, it follows from the convergence of d N to d that for all N 0 large enough the points d N 0 + ρe iπ/3 and d N 0 + ρe -iπ/3 belong to ∆ 1 and ∆ -1 respectively. Thus both points belong to Ω 1 by Lemma 4.12-(1). As a consequence, we can complete the path {d N 0 + te ±iπ/3 : t ∈ [0, ρ]} into a (closed) contour with a path Υ [START_REF] Bai | Random matrix theory and its applications[END_REF] res (N 0 ) lying in the interior of Ω 1 (see Figure 7). Since Υ [START_REF] Bai | Random matrix theory and its applications[END_REF] res (N 0 ) lies in the interior of Ω 1 , the convergence d N → d moreover yields that we can perform the same construction for all N N 0 with Υ

(1)

res (N ) in a closed tubular neighborhood T ⊂ Ω 1 of Υ (1)
res (N 0 ). By Lemma 4.12-(1) again, we can moreover choose Υ [START_REF] Bai | Random matrix theory and its applications[END_REF] res (N 0 ) in a way that it has finite length and only crosses the real axis at a real number lying on the right of K. By construction, this yields that the set T is compact and that the Υ This finally yields that Re(f N (z)f N (d N )) -K for all z ∈ T and proves the existence of a contour Υ (1) satisfying the requirements of Proposition 4.6, except for the point (4). Similarly, the same conclusion for Θ follows from the same lines but by using Ω 2 instead of Ω 1 and Lemma 4.12-(2).

As a consequence of Lemma 4.12-(3), there exists a contour in the interior of Ω 3 surrounding {x ∈ K : 0 < x < d} but staying in {z ∈ C : Re z > 0} and which intersects exactly twice the real axis in R \ K with finite length, see Figure 7. Using again Lemma 4.7-(a), the existence of Υ (0) with the properties provided in the statement of Proposition 4.6 follows.

Finally, the item (4) of Proposition 4.6 is clearly satisfied by construction since the sets Ω -and Ω + are disjoint, and the proof of the proposition is therefore complete.

Asymptotic analysis for the right edges and proof of Theorem 3-(b)

Recall that K N = K (0) N + K (1)
N . We now analyze the asymptotic behavior of K (0) N in the next section and then investigate K N and its associated contours Γ (0) and Θ, cf. Figure 6. The aim of this section is to establish the following statement, which asserts that K (0)

N will have no impact on the asymptotic analysis in the large N limit. Proposition 4.13. Let Assumptions 1 and 2 hold true, then for every ε > 0 small enough

lim N →∞ 1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) 2 = 0 , (4.42)
lim

N →∞ Tr 1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) = 0 . (4.43)
Notation: If a contour Γ is parametrized by γ : I → Γ for some interval I ⊂ R, then for every map h : Γ → C we set

Γ h(z)|dz| = I h • γ(t) |γ ′ (t)|dt
when it does make sense. In particular, Γ |dz| is the length of the contour Γ.

Proof. Recall that by definition of K (0) N (x, y), see (4.23), we have

K (0) N (x, y) = N 1/3 (2iπ) 2 δ N Γ (0) dz Θ dw 1 w -z e -N 1/3 x(z-d N )/δ N +N 1/3 y(w-d N )/δ N +N f N (z)-N f N (w) , (4.44
) where Θ and Γ (0) are as in Figure 6. We now deform the contours Θ and Γ (0) so that Θ = Θ and Γ (0) = Υ (0) where Θ and Υ (0) are given by Proposition 4.6. As a consequence of Proposition 4.6-(4), we have the upper bound

K (0) N (x, y) N 1/3 d(2π) 2 δ N Υ (0) e -N 1/3 x Re(z-d N )/δ N +N Re(f N (z)-f N (d N )) |dz| × Θ e N 1/3 y Re(w-d N )/δ N -N Re(f N (w)-f N (d N )) |dw|. (4.45)
Recall that Υ (0) does not depend on N . By Proposition 4.6-(5b), the contour Θ lies in a compact set. Hence there exists L > 0 independent of N such that | Re(zc N )| L for z ∈ Υ (0) or z ∈ Θ. Together with Proposition 4.6 (3a), we obtain that for all x s

Υ (0) e -N 1/3 x Re(z-d N )/δ N +N Re(f N (z)-f N (d N )) |dz| e -N K+ N 1/3 δ N (L(x-s)+L|s|) Υ (0) |dz| . (4.46)
Similarly, by splitting Θ into Θ res and Θ * , we get from Proposition 4.6-(3c) for every y s

Θ e N 1/3 y Re(w-d N )/δ N -N Re(f N (w)-f N (d N )) |dw| e N 1/3 L(y-s)/δ N +N 1/3 L|s|/δ N e -N K Θres |dw| + Θ * e -N Re(f N (w)-f N (d N )) |dw| . (4.47)
The definition of Θ * and Lemma 4.7-(b) then yield

Θ * e -N Re(f N (w)-f N (d N )) |dw| Θ * e -N g ′′ N (d N ) Re(w-d N ) 3 +N ∆|w-d N | 4 |dw| Θ * e -N g ′′ N (d N ) Re(w-d N ) 3 +N ρ∆|w-d N | 3 |dw| = 2 ρ 0 e -N t 3 (g ′′ N (d N )-ρ∆) dt 2ρ (4.48)
provided that ρ is chosen small enough so that g ′′ N (d N ) -ρ∆ > 0. By combining (4.45)-(4.48), we thus obtained that there exist constants C 0 , C 1 > 0 independent of N such that for every x, y s and every N large enough

K (0) N (x, y) C 0 e -C 1 N + N 1/3 δ N 2L(x+y) . (4.49)
Since by (4.3),

1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) 2 = εN 2/3 δ N s εN 2/3 δ N s K (0) N (x, y) 2 dxdy 1/2
, we obtain from (4.49) the rough estimate

1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) 2 C 0 (εN 2/3 δ N -s)e -N (C 1 -4εL)
from which (4.42) follows provided that we chose ε small enough. Similarly, by (4.4)

Tr 1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) = εN 2/3 δ N s K (0) N (x, x)dx,
and (4.49) yields the estimate

| Tr 1 (s, εN 2/3 δ N ) K (0) N 1 (s, εN 2/3 δ N ) | εN 2/3 δ N s |K (0) N (x, x)|dx C 0 (εN 2/3 σ N -s)e -N (C 1 -4εL)
which proves (4.43) as soon as ε is small enough. Proof of Proposition 4.13 is therefore complete.

Asymptotic analysis for K

(1)

N and proof of Theorem 3-(b)

We now investigate the convergence of K

(1)

N towards K Ai and thereafter complete the proof of Theorem 3-(b). Proposition 4.14. For every ε > 0 small enough, we have

lim N →∞ 1 (s,εN 2/3 δ N ) (K (1) N -K Ai )1 (s,εN 2/3 δ N ) 2 = 0 , (4.50) lim N →∞ Tr 1 (s,εN 2/3 δ N ) (K (1)
N -K Ai )1 (s,εN 2/3 δ N ) = 0 . (4.51)
First, we represent the Airy kernel as a double complex integral. To do so, we introduce for some δ > 0, which will be specified later, the contours

Γ ∞ = d N +δe iπθ : θ ∈ [-π/3, π/3] ∪ d N + te ±iπ/3 : t ∈ [δ, ∞) , (4.52) Θ ∞ = d N +δe iπθ : θ ∈ [2π/3, 4π/3] ∪ d N -te ±iπ/3 : t ∈ [δ, ∞) , (4.53)
and prove the following.

Lemma 4.15. For every δ > 0 and x, y ∈ R, we have

K Ai (x, y) = N 1/3 (2iπ) 2 δ N Γ ∞ dz Θ ∞ dw 1 w -z e -N 1/3 x(z-d N ) δ N + N 6 g ′′ N (d N )(z-d N ) 3 × e N 1/3 y(w-d N ) δ N -N 6 g ′′ N (d N )(w-d N ) 3 .
Proof. First, it easily follows from the differential equation satisfied by the Airy function, namely Ai ′′ (x) = xAi(x), and an integration by part that

K Ai (x, y) = ∞ 0 Ai(x + u)Ai(y + u)du. (4.54)
The Airy function admits the following complex integral representation (see e.g. [52, p.53])

Ai(x) = - 1 2iπ Ξ e -xz+z 3 /3 dz = 1 2iπ Ξ ′ e xw-w 3 /3 dw, ( 4.55)
where Ξ and Ξ ′ are disjoint unbounded contours, and Ξ goes from e iπ/3 ∞ to e -iπ/3 ∞ whereas Ξ ′ goes from e -2iπ/3 ∞ to e 2iπ/3 ∞. By plugging (4.55) into (4.54) and by using the Fubini theorem, we obtain

K Ai (x, y) = - 1 (2iπ) 2 Ξ dz Ξ ′ dwe -xz+z 3 /3+yw-w 3 /3 ∞ 0 e u(w-z) du , = 1 (2iπ) 2 Ξ dz Ξ ′ dw 1 w -z e -xz+z 3 /3+yw-w 3 /3 , (4.56)
since Re(wz) < 0 for all z ∈ Ξ and w ∈ Ξ ′ . Lemma 4.15 then follows after the changes of variables z → N 1/3 (zd N )/δ N and w → N 1/3 (wd N )/δ N , the mere definition δ 3 N = 2/g ′′ N (d N ) and an appropriate deformation of the contours.

We now turn to the proof of Proposition 4.14.

Proof of Proposition 4.14. Recall that K

(1)

N (x, y) = N 1/3 (2iπ) 2 δ N Γ (1) dz Θ dw 1 w -z e -N 1/3 x(z-d N ) δ N +N 1/3 y(w-d N ) δ N +N f N (z)-N f N (w) .
(4.57) The key step in the analysis is to deform the contours Γ (1) and Θ into Υ (1) and Θ of Proposition 4.6, but since the later intersect in d N , we need to slightly modify them.

Let ρ 0 be fixed so that Lemma 4.7 holds true, fix ρ ρ 0 and recall the definitions of Υ (1) = Υ * ∪ Υ (1) res and Θ = Θ * ∪ Θ res (4.58) as provided by Proposition 4.6. Since Υ * ∩ Θ * = {d N }, we deform them to make them disjoint. Set δ = N -1/3 (4.59)

and from now until the end of the proof, denote (with a slight abuse of notation)

Υ * = d N +δe iθ : θ ∈ - π 3 , π 3 ∪ d N + te ±i π 3 : t ∈ [δ, ρ] , (4.60)
:= Υ * ,1 ∪ Υ * ,2 (4.61)
Θ * = d N +δe iθ : θ ∈ 2π 3 , 4π 3 ∪ d N -te ±i π 3 : t ∈ [δ, ρ] . (4.62)
Notice in particular that this deformation provides now the control min |w -z| : z ∈ Υ * , w ∈ Θ * δ .

Now, let Γ (1) = Υ (1) and Θ = Θ. We can also express the Airy contours Γ ∞ and Θ ∞ as

Γ ∞ = Υ * ∪ Γ ∞ res with Γ ∞ res = d N + te ±iπ/3 : t ∈ [ρ, ∞) , Θ ∞ res = Θ * ∪ Θ ∞ res with Θ ∞ res = d N -te ±iπ/3 : t ∈ [ρ, ∞) .
It follows from Proposition 4.6-( 4) and the definition of the contours that there exists d ′ such that for any

(Ξ, Ξ ′ ) ∈ (Γ * , Θ res ), (Γ res , Θ * ), (Γ res , Θ res ), (Γ * , Θ ∞ res ), (Γ ∞ res , Θ * ), (Γ ∞ res , Θ ∞ res ) we have min |w -z| : z ∈ Ξ, w ∈ Ξ ′ d ′ .
As a consequence, by using (4.57), (4.59), Lemma 4.15 and by splitting contours into their different components, we obtain that K

(1)

N (x, y) -K Ai (x, y) N 2/3 (2π) 2 δ N E 0 + N 1/3 d ′ (2π) 2 δ N E 1 + E 2 + E 3 + E 4 + E 5 + E 6 (4.63)
where, setting for convenience

F N (x, z) = e -N 1/3 x(z-d N ) δ N +N (f N (z)-f N (d N )) , F Ai (x, z) = e -N 1/3 x(z-d N ) δ N + N 6 g ′′ N (d N )(z-d N ) 3 , G N (y, w) = e N 1/3 y(w-d N ) δ N -N (f N (w)-f N (d N )) , G Ai (y, w) = e N 1/3 y(w-d N ) δ N -N 6 g ′′ N (d N )(w-d N ) 3 ,
we introduced

E 0 = Υ * |dz| Θ * |dw| F N (x, z)G N (y, w) -F Ai (x, z)G Ai (y, w) , (4.64
)

E 1 = Υ * F N (x, z) |dz| Θres G N (y, w) |dw| , (4.65
)

E 2 = Υres F N (x, z) |dz| Θ * G N (y, w) |dw| , (4.66)
E 3 = Υres F N (x, z) |dz| Θres G N (y, w) |dw| , (4.67)
E 4 = Υ * F Ai (x, z) |dz| Θ ∞ res G Ai (y, w) |dw| , (4.68) E 5 = Γ ∞ res F Ai (x, z) |dz| Θ * G Ai (y, w) |dw| , (4.69)
E 6 = Γ ∞ res F Ai (x, z) |dz| Θ ∞ res G Ai (y, w) |dw| . ( 4

.70)

Convention: In the rest of the proof, C, C 0 , C 1 , . . . stand for positive constants which are independent on N or x, y, but which may change from one line to an other.

Step 1: Estimates for E 0 . We rely on the following elementary inequality,

|e u -e v | = e Re(v) |e (u-v) -1| e Re(v) k 1 |u -v| k k! |u -v|e Re(v)+|u-v| , (4.71)
which holds for every u, v ∈ C. By combining this inequality for

u = N (f N (z) -f N (d N )) -N (f N (w) -f N (d N )) , v = N g ′′ N (d N ) 6 (z -d N ) 3 -(w -d N ) 3
together with Lemma 4.7-(b), we obtain

|F N (x, z)G N (y, w) -F Ai (x, z)G Ai (y, w)| ∆N (|z -d N | 4 + |w -d N | 4 ) e -N 1/3 x Re(z-d N ) δ N +N 1/3 y Re(w-d N ) δ N ×e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 - N g ′′ N (d N ) 6 Re(w-d N ) 3 +N ∆|w-d N | 4
provided that z, w ∈ B(d N , ρ). This yields with (4.64)

E 0 ∆ Υ * N |z -d N | 4 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| × Θ * e N 1/3 x Re(w-d N ) δ N e -N g ′′ N (d N ) 6 Re(w-d N ) 3 +N ∆|w-d N | 4 |dw| + ∆ Υ * e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| × Θ * N |w -d N | 4 e N 1/3 x Re(w-d N ) δ N e -N g ′′ N (d N ) 6 Re(w-d N ) 3 +N ∆|w-d N | 4 |dw|. (4.72)
We first handle the integrals over the contour Υ * = Υ * ,1 ∪ Υ * ,2 , see (4.61), and consider separately the two different portions of the contour. First, let z ∈ Υ * ,1 and recall that x s by assumption. Since

δ 2 Re(z -d N ) |z -d N | δ and δ = 1 N 1/3 , we have |z -d N | 4 = N -4/3 and the following estimates e -N 1/3 x Re(z-d N ) δ N e -x-s 2δ N + |s| δ N , e N g ′′ N (d N ) Re(z-d N ) 3 +N ∆|z-d N | 4 e g ′′ N (d N )+ ∆ N 1/3 .

This immediately yields

Υ * ,1 N |z -d N | 4 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| 1 N 1/3 e -x-s 2δ N + |s| δ N e g ′′ N (d N )+ ∆ N 1/3 2π 3N 1/3 C N 2/3 e -x-s 2δ N (4.73)
where 2π/3N 1/3 accounts for the length of Υ * ,1 . Similarly

Υ * ,1 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| C N 1/3 e -x-s 2δ N . (4.74)
Consider now the situation where z ∈ Υ * ,2 . In this case,

Re(z -d N ) = t 2 , Re(z -d N ) 3 = -t 3 , |z -d N | 4 = t 4 ,
with N -1/3 t ρ and thus

e -N 1/3 x Re(z-d N ) δ N e -tN 1/3 x-s 2δ N +N 1/3 |s|t 2δ N e -x-s 2δ N +N 1/3 |s|t 2δ N , e N g ′′ N (d N ) Re(z-d N ) 3 +N ∆|z-d N | 4 e -N (g ′′ N (d N )-ρ∆)t 3 .
Assuming that we chose ρ small enough so that g ′′ (d) -ρ∆ > 0 and recalling that g ′′ N (d N ) → g ′′ (d), this provides for every N large enough the inequalities

Υ * ,2 N |z -d N | 4 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| 2e -x-s 2δ N ρ N -1/3 N t 4 e N 1/3 |s|t 2δ N -N (g ′′ N (d N )-ρ∆)t 3 dt 2 N 2/3 e -x-s 2δ N ∞ 1 u 4 e |s|u 2δ N -(g ′′ N (d N )-ρ∆)u 3 du C N 2/3 e -x-s 2δ N . (4.75)
Similarly,

Υ * ,2 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| C N 1/3 e -x-s 2δ N . (4.76)
Gathering (4.73)-(4.76), we finally obtain estimates over the whole contour Υ * :

Υ * N |z -d N | 4 e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| C N 2/3 e -x-s 2δ N , Υ * e -N 1/3 x Re(z-d N ) δ N e N g ′′ N (d N ) 6 Re(z-d N ) 3 +N ∆|z-d N | 4 |dz| C N 1/3 e -x-s 2δ N . (4.77)
The same line of arguments also yields equivalent estimates for the integrals over Θ * . Namely,

Θ * N |w -d N | 4 e N 1/3 y Re(w-d N ) δ N e -N g ′′ N (d N ) 6 Re(w-d N ) 3 +N ∆|w-d N | 4 |dw| C N 2/3 e -y-s 2δ N , Θ * e N 1/3 y Re(w-d N ) δ N e -N g ′′ N (d N ) 6 Re(w-d N ) 3 +N ∆|w-d N | 4 |dw| C N 1/3 e -y-s 2δ N . (4.78)
Combining (4.77)-(4.78), we have shown that

E 0 C N e -x+y-2s 2δ N . (4.79)
Step 2: Estimates for the remaining E i 's. Using the same estimates as in Step 1, we can prove that

Υ * |F N (x, z)| |dz| C N 1/3 e -x-s 2δ N , Υ * F Ai (x, z) |dz| C N 1/3 e -x-s 2δ N , (4.80)
Θ * G N (y, w) |dw| C N 1/3 e -y-s 2δ N , Θ * G Ai (y, w) |dw| C N 1/3 e -y-s 2δ N . (4.81)
The definitions of the paths and Proposition 4.6 yield that there exists

L > 0 independent of N such that | Re(z -c N )| L , z ∈ Υ * ∪ Θ * ∪ Υ res ∪ Θ res .
This estimate, together with Proposition 4.6 (3b), (3c) and (5c) yields that for every x, y s

Υres F N (x, z) |dz| Ce -N K+N 1/3 L x-s δ N +N 1/3 L|s| δ N , (4.82)
Θres G N (y, w) |dw| Ce -N K+N 1/3 L y-s δ N +N 1/3 L|s| δ N . (4.83)
Combining (4.80)-(4.83), we readily obtain

E 1 + E 2 + E 3 Ce -C 1 N +C 2 N 1/3 x+y δ N .
We now handle

Γ ∞ res F Ai (x, z) |dz| and Θ ∞ res G Ai (y, w) |dw| .
We have

Γ ∞ res F Ai (x, z) |dz| = Θ ∞ res G Ai (y, w) |dw| = 2 ∞ ρ e -N 1/3 xt 2δ N - N g ′′ N (d N ) 6 t 3 dt 2 ∞ ρ e N 1/3 |s|t 2δ N - N g ′′ N (d N ) 6 t 3
dt .

Let now N large enough so that

3 g ′′ N (d N )N 6 ρ 2 - N 1/3 |s| 2δ N ρ
(beware that such a condition only depends on s). Then

2 ∞ ρ e N 1/3 |s|t 2δ N - N g ′′ N (d N ) 6 t 3 dt 2 ρ ∞ ρ 3 g ′′ N (d N )N 6 t 2 - N 1/3 |s| 2δ N e N 1/3 |s|t 2δ N - N g ′′ N (d N ) 6 t 3 dt 2 ρ -e N 1/3 |s|t 2δ N - N g ′′ N (d N ) 6 t 3 ∞ ρ = 2 ρ e N 1/3 |s|ρ 2δ N - N g ′′ N (d N ) 6 ρ 3
and we hence obtain the estimate

Γ ∞ res F Ai (x, z) |dz| = Θ ∞ res G Ai (y, w) |dw| Ce -C 1 N (4.84)
We can now easily handle E 4 , E 5 and E 6 and finally obtain

6 k=1 E k Ce -C 1 N +C 2 N 1/3 x+y δ N . (4.85)
Step 3: Conclusions. By combining (4.63), (4.79) and (4.85), we have shown for every x, y s and N large enough that K

N (x, y) -K Ai (x, y) C N 1/3 e -x+y-2s 2δ N + C 1 e -C 2 N +C 3 N 1/3 x+y δ N . (1)
As a consequence,

Tr 1 (s, εN 2/3 δ N ) (K (1)
N -K Ai )1 (s, εN 2/3 δ N ) εN 2/3 δ N s K (1)
N (x, x) -K Ai (x, x) dx δ N C N 1/3 + (εN 2/3 δ N -s)C 1 e -N (C 2 -2εC 3 )
and (4.51) follows provided ε is chosen small enough. Similarly,

1 (s, εN 2/3 δ N ) (K (1) N -K Ai )1 (s, εN 2/3 δ N ) 2 2 = εN 2/3 δ N s εN 2/3 δ N s K (1) N (x, y) -K Ai (x, y) 2 dxdy δ N C N 1/3 2 + (εN 2/3 δ N -s) 2 C ′ 1 e -N (C 2 -2εC 3 )
where C ′ 1 > 0 is independent on N . This yields (4.50) as soon as ε is chosen small enough and thus completes the proof of Proposition 4.14.

We are finally in position to prove Theorem 3 (b).

Proof of Theorem 3 (b)

. First, we check that the Airy operator K Ai is trace class and Hilbert-Schmidt on L 2 (s, ∞) for every s ∈ R. Indeed, the representation (4.54) provides the factorization K Ai = A 2 s of operators on L 2 (s, ∞), where A s is the integral operator having for kernel A s (x, y) = Ai(x + ys). The fast decay as x → +∞ of the Airy function (see [52, p.394

]) Ai(x) e -2 3 x 3/2 2π 1/2 x 1/4 , x > 0, (4.86)
then show that both A s and K Ai are Hilbert-Schmidt, and moreover that K Ai is trace class being the product of two Hilbert-Schmidt operators. Next, by using again the upper bound (4.86), it follows that for every ε > 0 lim

N →∞ 1 (s,εN 2/3 δ N ) K Ai 1 (s,εN 2/3 δ N ) -1 (s,∞) K Ai 1 (s,∞) 2 = 0, lim N →∞ Tr 1 (s,εN 2/3 δ N ) K Ai 1 (s,εN 2/3 δ N ) = Tr 1 (s,∞) K Ai 1 (s,∞) .
Together with Proposition 4.14, this yields lim

N →∞ 1 (s,εN 2/3 δ N ) K (1)
N 1 (s,εN 2/3 δ N ) -1 (s,∞) K Ai 1 (s,∞) 2 = 0, lim N →∞ Tr 1 (s,εN 2/3 δ N ) K (1)
N 1 (s,εN 2/3 δ N ) = Tr 1 (s,∞) K Ai 1 (s,∞)
and, combined moreover with Proposition 4.13 and (4.24), we obtain lim

N →∞ 1 (s,εN 2/3 δ N ) K N 1 (s,εN 2/3 δ N ) -1 (s,∞) K Ai 1 (s,∞) 2 = 0, (4.87)
lim

N →∞ Tr 1 (s,εN 2/3 δ N ) K N 1 (s,εN 2/3 δ N ) = Tr 1 (s,∞) K Ai 1 (s,∞) , (4.88)
provided we chose ε small enough. Finally, it follows from (4.15)-(4.16), (4.87)-(4.88), and Proposition 4.1 that, for every s ∈ R, lim

N →∞ P N 2/3 δ N xφ(N) -b N s = det I -K Ai L 2 (s,∞) .
Proof of Theorem 3 (b) is therefore complete.

In the next section, we provide a proof for Theorem 3 (a), and thus complete the proof for Theorem 3. We shall see that we can recover the setting of the proof of Theorem 3 (b); the only task left is to prove the existence of appropriate contours for the saddle point analysis which differ from the case of a right edge.

Asymptotic analysis for the left edges and proof of Theorem 3-(a)

This section is devoted to the end of the proof of Theorem 3. We precisely recall the setting for the analysis of a left regular soft edge a; we state and prove the counterparts of Proposition 4.6 (i.e. the existence of appropriate contours for the asymptotic analysis), that is Proposition 4.16 for the case where c > 0 with a = g(c), and Proposition 4.17 for the case where c < 0.

The remaining of the asymptotic analysis is omitted since we show it is essentially the same than in Section 4.5.

Let a be a left regular soft edge; recall the definitions of g, c, (c N ) as provided by Proposition 2.11 and set

a N = g N (c N ) , σ N = - 2 g ′′ N (c N ) 1/3 . (4.89)
Recall moreover that

g ′ N (c N ) = 0 , lim N →∞ c N = c , lim N →∞ a N = a , lim N →∞ σ N = - 2 g ′′ (c) 1/3 . (4.90)
In particular, for N large enough, -g ′′ N (c N ) and σ N are positive numbers and c N and c have the same sign.

Reduction to the right edge setting

The definition of the extremal eigenvalue xϕ(N) , see Theorem 2, and Proposition 4.4 yield that for every ε > 0 small enough

P N 2/3 σ N a N -xϕ(N) s = det I -K N L 2 (a N -ε, a N -s/(N 2/3 σ N )) + o(1) (4.91)
as N → ∞. We then write

det I -K N L 2 (a N -ε, a N -s/(N 1/3 σ N )) = det I -1 (s, N 2/3 εc N ) K N 1 (s, N 2/3 εc N ) L 2 (s,∞)
where the scaled operator K N has for kernel

K N (x, y) = - 1 N 2/3 σ N K N a N - x N 2/3 σ N , a N - y N 2/3 σ N ,
and where K N (x, y) was introduced in (4.19) (with d N replaced by c N ). If we introduce the map

f * N (z) = a N (z -c N ) -log(z) + 1 N n j=1 log(1 -λ j z), (4.92)
which differs from f N defined in (4.18) by a minus sign and by the fact that b N is replaced by a N , then we have

K N (x, y) = - N 1/3 (2iπ) 2 σ N Γ dz Θ dw 1 w -z e N 1/3 x(z-c N )/σ N -N 1/3 y(w-c N )/σ N -N f * N (z)+N f * N (w) .
Set moreover K * N (x, y) = K N (y, x), then it follows by exchanging z and w in the last integral that

K * N (x, y) = N 1/3 (2iπ) 2 σ N Θ dz Γ dw 1 w -z e -N 1/3 x(z-c N )/σ N +N 1/3 y(w-c N )/σ N +N f * N (z)-N f * N (w) .
(4.93) Note that, as a consequence of the definition of f * N and (4.90), we have

(f * N ) ′ (c N ) = (f * N ) ′′ (c N ) = 0, (f * N ) (3) (c N ) = -g ′′ N (c N ) > 0. (4.94)
Proposition 4.16. For every ρ > 0 small enough, there exists a contour Υ (0) independent of N and two contours Υ (1) = Υ (1) (N ) and Θ = Θ(N ) which satisfy for every N large enough the following.

(1) (a) Υ (0) encircles the λ -1 j 's larger than c N , (b) Υ (1) encircles the λ -1 j 's smaller than c N , (c) Θ encircles the λ -1 j 's smaller than c N and the origin.

(

) (a) Υ (1) = Υ * ∪ Υ 2
res where

Υ * = {c N -te ±iπ/3 : t ∈ [0, ρ]}. (b) Θ = Θ * ∪ Θ res where Θ * = {c N + te ±iπ/3 : t ∈ [0, ρ]} .
(3) There exists K > 0 independent of N such that

(a) Re f N (w) -f N (c N ) K for all w ∈ Υ (0) , (b) Re f N (w) -f N (c N ) K for all w ∈ Υ (1) res , (c) Re f N (z) -f N (c N )
-K for all z ∈ Θ res .

(4) There exists d > 0 independent of N such that

inf |z -w| : z ∈ Υ (0) , w ∈ Θ d , inf |z -w| : z ∈ Υ (1) * , w ∈ Θ res d , inf |z -w| : z ∈ Υ res , w ∈ Θ * d , inf |z -w| : z ∈ Υ res , w ∈ Θ res d .
(5) (a) The contours Υ (0) and Θ are disjoint, so are Υ (0) and Υ (1) , but Θ ∩ Υ (1) = {c N }.

(b) The contours Υ (1) and Θ lie in a compact subset of C, independent of N .

(b) The lengths of Υ (1) and Θ are uniformly bounded in N .

Although the proof uses the same type of arguments than in the proof of Proposition 4.6, the analytical setting is not identical. Thus, although we shall provide less details than in the proof of Proposition 4.6, we shall emphasize on the required changes. Figure 8 may help as a visual support for the argument.

Proof. The regularity assumption yields ε > 0 such that λ -1 j ∈ (0, +∞) \ B(c, ε) for every 1 j n and every N large enough. We then introduce the compact set K defined by

K = inf N 1 λ n , sup N 1 λ 1 \ B(c, ε) ∪ {0} (4.101)
and notice that by construction {x ∈ R : x -1 ∈ Supp(ν N )} ⊂ K for every N large enough, and also that {x ∈ R : x -1 ∈ Supp(ν)} ⊂ K. If we introduce the map

f * (z) = a(z -c) -log(z) + γ log(1 -xz)ν(dx) ,
then, given any simply connected subset of C \ K, we can choose a determination of the logarithm such that both the maps f * N and f * are well-defined and holomorphic there for every N large enough. Notice that the definition of Re f * does not depend on the determination of the logarithm. Moreover, the proof of shows that Re f * N converges locally uniformly on C \ K toward Re f * , and moreover Re f * N (c N ) → Re f * (c) as N → ∞. Next, we perform a qualitative analysis for Re f * and introduce the sets

Ω -= z ∈ C : Re f * (z) < Re f * (c) , Ω + = z ∈ C : Re f * (z) > Re f * (c) .
Since a > 0, the asymptotic behavior Re f * (z) = a Re(zc) + O(log |z|) as z → ∞ shows that for every α ∈ (0, π/2) there exisits R > 0 large enough such that

z ∈ C : |z| > R, - π 2 + α < arg(z) < π 2 -α ⊂ Ω + (4.102) and z ∈ C : |z| > R, π 2 + α < arg(z) < 3π 2 -α ⊂ Ω -. (4.103)
Notice that the role of Ω + and Ω -has been exchanged compared to the setting of a right edge. Moreover, the arguments of the proof of Lemma 4.8 show that both Ω + and Ω -have a unique unbounded connected component.

As for the behavior of Re f * around c, because a = g(c) it follows from the definition of f * that (f * ) ′ (z) = g(c)g(z). Thus, by Proposition 2.11, we have (f * ) ′ (c) = (f * ) ′′ (c) = 0 and (f * ) (3) (c) = -g ′′ (c) > 0. As a consequence, the same proofs than those of Lemmas 4.9 and 4.7-(b),(c) show there exist η > 0 and 0 < θ < π/2 small enough such that

∆ 2k+1 ⊂ Ω -, ∆ 2k ⊂ Ω + , k ∈ {-1, 0, 1},
where we introduced as in Section 4.4

∆ k = z ∈ C : 0 < |z -c| < η, arg(z -c) -k π 3 < θ .
Notice that the role of Ω -and Ω + is the same than in the right edge setting. We then denote by Ω 2k+1 the connected component of Ω -which contains ∆ 2k+1 , and similarly Ω 2k the connected component of Ω + which contains ∆ 2k . The proof of Lemma 4.10 yields that Re f * is subharmonic in C\{0} and is superharmonic in C \ {x ∈ R : x -1 ∈ Supp(ν)}. As a consequence, it follows from the proof of Lemma 4.11 that we obtain a similar statement as in Lemma 4.11 for Re f * after having exchanged the role of Ω + and Ω -(to furthermore convince the reader, notice that Re f * (z)a Re(zc) = -Re f (z)b Re(zd) and that both the maps z → a Re(zc) and z → b Re(zd) are harmonic). Namely, [START_REF] Bai | Random matrix theory and its applications[END_REF] If Ω * is a connected component of Ω -, then Ω * is open and, if Ω * is moreover bounded, there exists x ∈ Supp(ν) such that x -1 ∈ Ω * .

(2) Let Ω * be a connected component of Ω + such that Ω * ⊂ R. Equipped with the previous observations we are now in position to provide the counterpart of Lemma 4.12 in the present setting, namely to prove that the following statements hold true.

(A) We have Ω 1 = Ω -1 , the interior of Ω 1 is connected, and for every 0 < α < π/2 there exists R > 0 such that

z ∈ C : |z| > R, π 2 + α < arg(z) < 3π 2 -α ⊂ Ω 1 .
(B) The interior of Ω 0 is connected and, for every 0 < α < π/2, there exists

R > 0 such that z ∈ C : |z| > R, - π 2 + α < arg(z) < π 2 -α ⊂ Ω 0 .
(C) We have Ω 2 = Ω -2 , the interior of Ω 2 is connected, and there exists δ > 0 such that B(0, δ) ⊂ Ω 2 .

Let us first prove (A). Since by definition Ω 1 is a connected component of Ω -, its interior is connected by [START_REF] Bai | Random matrix theory and its applications[END_REF]. Let us prove by contradiction that Ω 1 is unbounded, from which (A) will follow by using the symmetry Re f * (z) = Re f * (z), the inclusion (4.103) and that Ω - has a unique unbounded connected component. Assume Ω 1 is bounded. Then (1) yields the existence of x ∈ Supp(ν) such that x -1 ∈ Ω 1 . If x -1 < c (resp. x -1 > c), it then follows from the symmetry Re f * (z) = Re f * (z) that Ω 1 surrounds Ω 2 (resp. Ω 0 ) so that Ω 2 ⊂ R (resp. Ω 0 ⊂ R) is a bounded connected component of Ω + which does not contain the origin. This yields with (2a) a contradiction and our claim follows.

Since we just proved that Ω 1 is unbounded, the origin does not belong to Ω 0 . As a consequence, (2a) and (2c) yield respectively that Ω 0 is unbounded and has a connected interior. Using moreover the inclusion (4.102) and that Ω + has a unique unbounded connected component, (B) follows.

As a byproduct of (A), Ω 2 is bounded. Thus Ω 2 contains the origin by (2a) and has a connected interior by (2b). By using the symmetry Re f * (z) = Re f * (z) and that Re f * (z) → +∞ as z → 0, (C) is proved.

Finally, as a consequence of (A), (B) and (C), the existence of the contour Υ (0) , resp. Υ (1) , resp. Θ, in Proposition 4.16 is proved by choosing Υ (0) in the interior of Ω 0 encircling {x ∈ K : x > c} and intersecting the real axis exactly twice in R \ K with finite length, resp. by completing {c Nte ±iπ/3 : t ∈ [0, ρ]} for ρ small enough and N large enough so that both the points c Nρe iπ/3 and c Nρe -iπ/3 lie in Ω 2 into a closed contour with a path lying in the interior of Ω 2 but staying in {z ∈ C : Re(z) > 0} and intersecting the real line exactly once at the left of K with finite length, resp. by completing {c N + te ±iπ/3 : t ∈ [0, ρ]} for ρ small enough and N large enough so that both the points c N + ρe iπ/3 and c N + ρe -iπ/3 belongs to Ω 1 into a closed contour with a path lying in the interior of Ω 1 and crossing the real axis exactly once at the left of the origin with finite length, and then by using the local uniform convergence of Re f * N → Re f * on C \ K; see the proof of Proposition 4.6 for the details. (3) There exists K > 0 independent of N such that . c Proof. We use the notations, definitions and properties used in the proof of Proposition 4.16, except for K that we define by

(a) Re f N (w) -f N (c N ) K for all w ∈ Υ res (b) Re f N (z) -f N (c N ) -K for all z ∈ Θ res (
∆ 1 ∆ 2 ∆ -2 ∆ -1 Ω + Ω - ∆ 0 ∆ 3 .
K = inf N 1 λ n , sup N 1 λ 1 ∪ {0} .
If instead log |1x/x 0 |ν(dx) > -∞, then by upper semicontinuity there exists η 0 ∈ (0, η 1 ) small enough to that

γ log |1 -x(x -1 0 + iη 0 )|ν(dx) γ log |1 -x/x 0 |ν(dx) + K. (4.106)
Let η 0 be defined as above and consider a compact tubular neighborhood T of Ξ\{x -1 0 +iη : |η| < η 0 } small enough so that T lies in C \ K and Re f * -Re f * (c) -3K there (the latter is possible since Re f * is upper semicontinuous on C \ {0}). Notice that by construction the interior of T contain both the points c N + ρe iπ/3 and c N + ρe -iπ/3 for every N large enough, and the points x for every z ∈ T . As a consequence, for every N large enough, we can construct the path Θ res is the following way: it goes from c N + ρe -iπ/3 to x -1 0 + iη 0 staying in T , then follows the segment {x -1 0 + iη : 0 η η 0 }, and is finally completed by symmetry with respect to the real axis. As for what is happening on {x -1 0 + iη : |η| < η 0 }, since a priori Re f * N does not converge uniformly toward Re f * there, we need an extra argument to complete the proof of Proposition 4.17. Namely, we need to show that for every N large enough, uniformly in If we assume log |1x/x 0 |ν(dx) = -∞, then for every N large enough, uniformly in

|η| < η 0 , Re f * N (x -1 0 + iη) -Re f * N (c N ) -K . ( 4
|η| < η 0 , Re f * N (z η ) -Re f * N (c N ) sup N a N (x -1 0 -c N ) -Re f * (c N ) -log |z η | + n N log |1 -xz η | ν N (dx) sup N a N (x -1 0 -c N ) -Re f * (c N ) + log(x 0 ) + n N log |1 -xz η 0 | ν N (dx) -K,
where for the last inequality we used (4.108) and (4.105). Now, assume instead that log |1-x/x 0 |ν(dx) > -∞. By using the convergences a N → a, c N → c and Re f * N (c N ) → Re f * (c), we obtain for every N large enough (and independently on η)

a N Re(z η -c N ) -Re f * N (c N ) a Re(z η -c) -Re f * (c) + K.
Proposition 5.1. Consider the setting of Theorem 4. Then, for every ε > 0 small enough and for every sequences (η i,N ) N , (χ j,N ) N of positive numbers growing with N to infinity, it holds that

P η i,N a i,N -x ϕ i (N ) s i , χ j,N x φ j (N ) -b j,N t j , (i, j) ∈ I × J = det I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) + o(1)
as N → ∞, where

A i = (a i,N -ε, a i,N -s i /η i,N ), B j = (b j,N + t j /χ j,N , b j,N + ε).
Since the proof is very similar to the one of Proposition 4.4, we omit it. Now, if we specify the previous proposition to η i,N = N 2/3 σ i,N and χ j,N = N 2/3 δ j,N , then we have as N → ∞ lim

N →∞ P N 2/3 σ i,N a i,N -x ϕ i (N ) s i , N 2/3 δ j,N x φ j (N ) -b j,N t j , (i, j) ∈ I × J = det I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) + o(1) (5.1)
where

A i = a i,N -ε, a i,N -s i /(N 2/3 σ i,N ) , B j = b j,N + t j /(N 2/3 δ j,N ), b j,N + ε .
For every i ∈ I and j ∈ J, we introduce the maps

f * i,N (z) = a i,N (z -c i,N ) -log(z) + 1 N n k=1 log(1 -λ k z) (5.2) f j,N (z) = -b j,N (z -d j,N ) + log(z) - 1 N n k=1 log(1 -λ k z) (5.3)
and the multiplication operators E * i and E j acting on L 2 (A i ) and L 2 (B j ) respectively by

E * i h(x) = e N f * i,N (c i,N )+N xc i,N h(x), h ∈ L 2 (A i ), E j h(x) = e -N f j,N (d j,N )+N xd j,N h(x), h ∈ L 2 (B j ).
The next proposition is the key to obtain Theorem 4.

Proposition 5.2. For every ε small enough, the following holds true.

(a) For every (i, j) ∈ J × J such that i = j, we have

lim N →∞ 1 B i E i K N E -1 j 1 B j 2 = 0.
(5.4) (b) For every (i, j) ∈ I × I such that i = j, we have

lim N →∞ 1 A i E * i K N (E * j ) -1 1 A j 2 = 0. ( 5.5)
(c) For every (i, j) ∈ I × J, we have lim

N →∞ 1 A i E * i K N E -1 j 1 B j 2 = 0 (5.6)
and lim

N →∞ 1 B j E j K N (E * i ) -1 1 A j 2 = 0. ( 5.7)
Before proving Proposition 5.2, let us show how does it lead to the asymptotic joint independence of the extremal eigenvalues:

Proof of Theorem 4. Our purpose is to show that for large N , the determinant at the right hand side of (5.1) converges to a product of Fredholm determinants involving the Airy kernel. Assume that N is large enough so that all the A i 's and B j 's are disjoint sets. Then, as shown in [START_REF] Bornemann | On the numerical evaluation of Fredholm determinants[END_REF] (see also [START_REF] Gohberg | Traces and determinants of linear operators[END_REF]Chap. 6]), the Fredholm determinant det

I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) admits the operator matrix representation det I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) = det   I -    K i,j II (i,j)∈I×I K i,j IJ (i,j)∈I×J K i,j JI (i,j)∈J×I K i,j JJ (i,j)∈J×J       i∈I L 2 (A i ) ⊕ j∈J L 2 (B j ) (5.8)
where K i,j II :

L 2 (A j ) → L 2 (A i ) denotes the integral operator K i,j II h(x) = A j K N (x, y)h(y)dy, x ∈ A i ,
and similarly the operators K i,j IJ : L 2 (B j ) → L 2 (A i ), K i,j JI : L 2 (A j ) → L 2 (B i ) and K i,j JJ : L 2 (B j ) → L 2 (B i ) are defined by restricting K N on appropriate subspaces of L 2 (R). Consider now the diagonal operator

E = i∈I E * i ⊕ j∈J E j acting on i∈I L 2 (A i ) ⊕ j∈J L 2 (B j ) .
Since the A i 's and B j 's are compact sets and K N is locally trace class, the identity (4.1) then yields det

  I -    K i,j II (i,j)∈I×I K i,j IJ (i,j)∈I×J K i,j JI (i,j)∈J×I K i,j JJ (i,j)∈J×J       i∈I L 2 (A i ) ⊕ j∈J L 2 (B j ) = det   I -E    K i,j II (i,j)∈I×I K i,j IJ (i,j)∈I×J K i,j JI (i,j)∈J×I K i,j JJ (i,j)∈J×J    E -1    i∈I L 2 (A i ) ⊕ j∈J L 2 (B j ) = det   I -    E * i K i,j II (E * j ) -1 (i,j)∈I×I E * i K i,j IJ E -1 j (i,j)∈I×J E i K i,j JI (E * j ) -1 (i,j)∈J×I E i K i,j JJ E -1 j (i,j)∈J×J       i∈I L 2 (A i ) ⊕ j∈J L 2 (B j ) = det   I -    1 A i E * i K N (E * j ) -1 1 A j (i,j)∈I×I 1 A i E * i K N E -1 j 1 B j (i,j)∈I×J 1 B i E i K N (E * j ) -1 1 A j (i,j)∈J×I 1 B i E i K N E -1 i 1 B j (i,j)∈J×J       L 2 (R) ⊕(|I|+|J|)
(5.9)

where |I| and |J| stand for the cardinalities of I and J respectively. By using the definition (4.2) of det 2 , it follows from (5.8) and (5.9) that det

I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) = i∈I e Tr(1 A i E * i K N (E * i ) -1 1 A i ) j∈J e Tr(1 B j E j K N E -1 j 1 B j ) × det 2   I -    1 A i E * i K N (E * j ) -1 1 A j (i,j)∈I×I 1 A i E * i K N E -1 j 1 B j (i,j)∈I×J 1 B i E i K N (E * j ) -1 1 A j (i,j)∈J×I 1 B i E i K N E -1 i 1 B j (i,j)∈J×J       L 2 (R) ⊕(|I|+|J|)
.

(5.10)

Let us inspect the diagonal elements of the matrix valued operator in the Fredholm determinant at the right hand side of the previous identity. In Section 4, we have precisely shown that for every i ∈ I and j ∈ J,

lim N →∞ 1 A i E * i K N (E * i ) -1 1 A i -1 (s i ,∞) K Ai 1 (s i ,∞) 2 = 0, lim N →∞ 1 B j E j K N E -1 j 1 B j -1 (t j ,∞) K Ai 1 (t j ,∞) 2 = 0 and lim N →∞ Tr(1 A i E * i K N (E * i ) -1 1 A i ) = Tr(1 (s i ,∞) K Ai 1 (s i ,∞) ), lim N →∞ Tr(1 B j E j K N E -1 j 1 B j ) = Tr(1 (t j ,∞) K Ai 1 (t j ,∞) ).
Proposition 5.2 then yields that the Hilbert-Schmidt norms of the off diagonal entries of the matrix valued operator in the Fredholm determinant at the right hand side of (5.10) converge to zero. Recalling that det 2 is continuous with respect to the Hilbert-Schmidt norm, we obtain from (5.10) that lim

N →∞ det I -K N L 2 ( i∈I A i ) ∪ ( j∈J B j ) = i∈I e Tr(1 (s i ,∞) K Ai 1 (s i ,∞) ) det 2 I -1 (s i ,∞) K Ai 1 (s i ,∞) L 2 (R) × j∈J e Tr(1 (t j ,∞) K Ai 1 (t j ,∞) ) det 2 I -1 (t j ,∞) K Ai 1 (t j ,∞) L 2 (R) = i∈I det I -K Ai L 2 (s i ,∞) j∈J det I -K Ai L 2 (t j ,∞) ,
and Theorem 4 is proved.

Now we turn to the proof of Proposition 5.2.

To do so, we shall deform the contours Γ and Θ in the integral representation of K N to appropriate contours for the asymptotic analysis, as provided by the propositions 4.6, 4.16 and 4.17. The problem is that since Θ and Γ will be associated to different critical points c N 's or d N 's, it is a possibility that they intersect. This raises a problem related to the presence of the factor (wz) -1 in the integral representation of K N . This problem can be avoided by using the following alternative expression of the kernel K N , that was established in [START_REF] Bleher | Integral representations for multiple Hermite and multiple Laguerre polynomials[END_REF]; since the proof is short, we provide it for the sake of completeness. Lemma 5.3. For every x = y we have

K N (x, y) = N (2iπ) 2 (x -y) Γ dz Θ dw e -N xz+N yw C N (z, w) z w N n i=1 1 -λ i w 1 -λ i z , (5.11)
where

C N (z, w) = 1 zw - 1 N n j=1 λ 2 j (1 -λ j z)(1 -λ j w) .
(5.12)

Proof. Starting from (4.5) with q = 0 and following [17, Section 3.3], we obtain by integrations by parts

xK N (x, y) = 1 (2iπ) 2 Γ dz Θ dw e -N xz+N yw 1 w -z z w N n i=1 1 -λ i w 1 -λ i z ×   1 w -z + N z - n j=1 λ j 1 -λ j z   and yK N (x, y) = 1 (2iπ) 2 Γ dz Θ dw e -N xz+N yw 1 w -z z w N n i=1 1 -λ i w 1 -λ i z ×   1 w -z + N w - n j=1 λ j 1 -λ j w   .
The main point here is that, since i = j, there exists C > 0 independent of N , x and y such that

N 1/3 (2iπ) 2 (b i,N -b j,N + x/(N 2/3 δ i,N ) -y/(N 2/3 δ j,N )) CN 1/3 . (5.16)
Then, as we did in Section 4.3 and 4.4, we replace the contour Γ by Υ (0) ∪ Υ (1) where the contours Υ (0) and Υ (1) are specified by Proposition 4.6 with d N = d i,N (if Υ (0) does not exist, we just deform Γ to Υ (1) ). Similarly, we deform the contour Θ and replace it with the contour Θ specified by Proposition 4.6 with d N = d j,N . We then deform the contours Υ (1) and Θ around the saddle points similarly as it was done in Section 4.5.2. More precisely, Υ (1) = Υ * ∪ Υ (1) res and Θ = Θ * ∪ Θ res where we introduced

Υ * = {d i,N + N -1/3 e iπθ : θ ∈ [-π/3, π/3]} ∪ {d i,N + te ±iπ/3 : t ∈ [N -1/3 , ρ]} Θ * = {d j,N + N -1/3 e iπθ : θ ∈ [2π/3, 4π/3]} ∪ {d j,N -te ±iπ/3 : t ∈ [N -1/3 , ρ]},
with ρ chosen small enough so that Lemma 4.7-(b) applies for both f i,N and f j,N . In addition, Proposition 4.6 provides K > 0 independent of N such that

Re(f i,N (z) -f i,N (d i,N )) -K, z ∈ Υ (0) (5.17) Re(f i,N (z) -f i,N (d i,N )) -K, z ∈ Υ (1) res (5.18) Re(f j,N (w) -f j,N (d j,N )) K, w ∈ Θ res . (5.19)
Note that the contours Υ (0) and Θ may now intersect, and the contours Υ (1) and Θ as well, since the contours are associated with different edges. This raises no problem since C N (z, w) is analytic on C \ K. More precisely, since by construction the contours Υ (0) , Υ (1) and Θ lie inside a compact subset of C \ K which does not dependent on N , there exists C ′ > 0 independent of N such that

|C N (z, w)| C ′ , z ∈ Υ (0) ∪ Υ (1) , w ∈ Θ. (5.20) Next, Lemma 4.7-(b) yields Re(f i,N (z) -f i,N (d i,N )) g ′′ N (d i,N ) Re(z -d i,N ) 3 /6 + ∆|z -d i,N | 4 , z ∈ Υ * Re(f j,N (w) -f j,N (d j,N )) g ′′ N (d j,N ) Re(w -d j,N ) 3 /6 -∆|w -d j,N | 4 , w ∈ Θ * ,
where ∆ > 0 is independent of N . We moreover assume we chose ρ small enough so that

g ′′ N (d i,N ) -ρ∆ > 0, g ′′ N (d j,N ) -ρ∆ > 0, (5.21)
for all N large enough. Then, by using the same estimates as in Sections 4.5.1 and 4.5.2, we obtain for every x, y s and N large enough

Υ (0) e -N 1/3 x Re(z-d i,N ) δ i,N +N Re(f i,N (z)-f i,N (d i,N )) |dz| C 1 e -C 2 N +C 3 N 1/3 x δ i,N , Υ (1)
e

-N 1/3 x Re(z-d i,N ) δ i,N +N Re(f i,N (z)-f i,N (d i,N )) |dz| C N 1/3 e -x-s 2δ i,N + C 1 e -C 2 N +C 3 N 1/3 x δ i,N , Θ e N 1/3 y Re(w-d j,N ) δ j,N -N Re(f j,N (w)-f j,N (d j,N )) |dw| C N 1/3 e -y-s 2δ j,N + C 1 e -C 2 N +C 3 N 1/3 y δ j,N , and
K (a i ,a j ) N (x, y) = N 1/3 (2iπ) 2 (a i,N -a j,N -x/(N 2/3 σ i,N ) + y/(N 2/3 σ j,N )) Γ dz Θ dw C N (z, w) × e N 1/3 x(z-c i,N )/σ i,N -N (f * i,N (z)-f * i,N (c i,N
)) e -N 1/3 y(w-c j,N )/σ j,N +N (f * j,N (w)-f * j,N (c j,N )) . (5.25)

For the kernel (5.24), we split the contour Γ into Υ (0) and Υ (1) where these contours are specified by Proposition 4.6 for d N = d i,N (again, if Υ (0) does not exist, we just deform Γ into Υ (1) ). We also deform Θ to obtain the contour Θ as in Proposition 4.16 or Proposition 4.17 with c N = c j,N , depending on whether c j > 0 or not. For the kernel (5.25), we similarly split the contour Γ into Υ (0) and Υ (1) and take these contours as in Proposition 4.16 for c N = c i,N if c i > 0, and deform Γ into Υ as in Proposition 4.17 otherwise. Moreover, Θ is replaced by Θ as specified in Proposition 4.16 or Proposition 4.17 with c N = c j,N depending on whether c j > 0 or not. The same line of arguments than in the proof of (a) then shows that (b) and (c) hold true, except when c j,N < 0. Indeed, in the latter case the contour Θ coming with Proposition 4.17 does cross by construction the set K at a point x -1 0 where x 0 ∈ Supp(ν). Thus we cannot use the bound (5.20) anymore.

To overcome this technical point, having in mind the definition (5.3) of C N (z, w), observe that since by construction Υ (0) ∪ Υ (1) or Υ lie in a compact subset of C \ K, the map z → (1zλ ℓ ) -1 is bounded there uniformly in 1 ℓ n and N large enough. Since moreover by construction Θ lies in C \ {0}, the map (z, w) → (zw) -1 is bounded on the contours uniformly in N large enough. Observe furthermore that for every 1 ℓ n, we have

e N f * j,N (w) 1 -λ ℓ w = e N f * [ℓ] j,N (w)
where

f * [ℓ] j,N (w) = a j,N (w -c j,N ) -log(w) + 1 N n k=1 k =ℓ log(1 -λ k w). (5.26)
Namely, the pole at w = λ ℓ introduced by C N (z, w) is actually cancelled by e N f * j,N (w) . Thus, the items (b) and (c) of the proposition follow provided that the previous estimates continue to hold, uniformly in 1 ℓ n, after the replacement of e N f * j,N by e N f * [ℓ] j,N . But this is not hard to obtain because, as a consequence of the definitions (5.2) and (5.26), for every k ∈ N and compact subset B ⊂ C \ K there exists C B,k > 0 independent of N such that sup w∈B max

1 ℓ n f * [ℓ] j,N (k) (w) -f * j,N (k) (w) C B,k N .
The proof of Proposition 5.2 is therefore complete.

6 Proof of Theorem 5: Fluctuations at the hard edge

In this section, we provide a proof for Theorem 5.

Let us fix s > 0 and α ∈ Z. We set n = N +α and define σ N as in (3.7). The representation for the gap probabilities of determinantal point processes as Fredholm determinants yields

P N 2 σ N x min s = det I -K N L 2 (0,s/(N 2 σ N )) ,
where

x min = x 1 = xα+1 if α 0, x 1-α = x1 if α < 0.
If we introduce the integral operator K N acting on L 2 (0, s) with kernel

K N (x, y) = 1 N 2 σ N K N x N 2 σ N , y N 2 σ N , (6.1)
then it follows from a change of variables that

P N 2 σ N x min s = det I -K N L 2 (0,s) . (6.2)
We recall that K Be,α (x, y) has been introduced in (3.6) and also define the operator E and E -1 acting on L 2 (0, s) by Eh(x) = x α/2 h(x) and E -1 h(x) = x -α/2 h(x). Notice that when α 0 (resp. α < 0) the operator E (resp. E -1 ) is well-defined on L 2 (0, s), but E -1 (resp. E) is not defined on the whole space. Nevertheless, in the following these operators will always arise premultiplied or post-multiplied by an appropriate operator so that the product is well-defined on L 2 (0, s), see below. The aim of this section is to prove the following.

J α ( √ x ) = 1 α! √ x 2 α 1 + O(x 2 ) , if α 0, J α ( √ x ) = (-1) α |α|! √ x 2 |α| 1 + O(x 2 ) , if α < 0,
which is provided by the series representation (3.5) of J α , then shows that B s , B s E -1 and K Be,α E -1 when α 0, EB s and EK Be,α when α < 0, and EK Be,α E -1 are well-defined and Hilbert-Schmidt operators. Moreover, E and K Be,α E -1 when α 0, E -1 and EK Be,α when α < 0, and EK Be,α E -1 are trace class being products of two Hilbert-Schmidt operators.

Since [0, s] is compact, it follows from Proposition 6.1 that lim N →∞ 1 (0,s) K N -EK Be,α E -1 1 (0,s) 2 = 0 and lim

N →∞
Tr 1 (0,s) K N 1 (0,s) = Tr 1 (0,s) EK Be,α E -1 1 (0,s) .

We then obtain from Proposition 4.1 that lim N →∞ det I -K N L 2 (0,s) = det I -EK Be,α E -1 L 2 (0,s) , which shows together with (6.2) and (4.1) that lim N →∞ P N 2 σ N x min s = det I -K Be,α L 2 (0,s) .

Finally, that det I -K Be,0 L 2 (0,s) = e -s has been observed in [START_REF] Forrester | The spectrum edge of random matrix ensembles[END_REF], and the proof of Theorem 5 is complete.

We now focus on the proof of Proposition 6.1.

The Bessel kernel

We first provide a double complex integral formula for the Bessel kernel. We recall that, by convention, all contours of integrations are oriented counterclockwise, and thus the notation |z|=r is unambiguous.

Proof. The Laurent series generating function for the Bessel functions with integer parameters reads, see [33, 7.2.4 (25)],

e x 2 (z-1 z ) = α∈Z J α (x)z α , z ∈ C \ {0}.
This yields for every x, r > 0 and α ∈ Z,

J α ( √ x ) = 1 2iπ |z|= r z -α e √ x 2 (z-1 z ) dz z .
After the changes of variables z → 2 √ xz and w → 1/(2 √ yw), this provides for every x, y > 0, 0 < r < R and α ∈ Z We continue the computation by mean of integrations by parts, as explained to us by Manuella Girotti while we discussed a similar formula appearing in her work [START_REF] Girotti | Riemann-hilbert approach to gap probabilities for the bessel process[END_REF]. Indeed, since -ye -yw = ∂ ∂w e -yw , a first integration by parts provides Equipped with Corollary 6.3, we are now in position to establish Proposition 6.1.

for some ∆ > 0 independent of N . Finally, by using Corollary 6. for some C(s) > 0 independent of N and 0 < x, y s, and Proposition 6.1 follows.

The proof of Theorem 5 is therefore complete.

A Proof of Proposition 2.4

The proof of Proposition 2. the dominated convergence theorem implies that xg(x) → 1γ < 0 as x → -∞. It results that g(x) → 0 + as x → -∞, and g(x) reaches a positive maximum on (-∞, 0). By [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF]Th. 4.3 and 4.4], we obtain that the function g(x) shows the behavior described in the statement, and its maximum coincides with a.

To prove (b), recalling the expression of xg(x) and observing that

x 2 g ′ (x) = -1 + γ xλ 1 -xλ 2 ν(dλ),
we deduce that when γ 1, the function g is negative and decreasing on (-∞, 0).

We now show (c). For x > 2/η and λ ∈ Supp(ν), we have |1-xλ| xη -1 > 1. Therefore, g(x) → 0 and x 2 g ′ (x) → γ -1 < 0 as x → +∞ by the dominated convergence theorem. This shows that g(x) has a positive supremum on (1/η, ∞) and it decreases to zero as x → +∞. By [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF]Th. 4.3 and 4.4], we obtain that the function g(x) shows the behavior described in the statement, and its supremum coincides with a.

Turning to (d), assume that [c, ∞) ⊂ D. Then by Proposition 2.3-(a) there exists ε > 0 such that g ′ (x) < 0 on (cε, c) and g ′ (c) = 0. It is furthermore clear that g(x) → 0 as x → ∞. Since b = g(c) > 0, we get that there exists an interval in (c, ∞) over which g is decreasing. But this contradicts [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF]Th. 4.3].

To show (e) we observe that m(x), being the Cauchy-Stieltjes transform of a probability measure, decreases from c = lim x↓b m(x) to 0 as x increases over the interval (b, ∞). Proposition 2.1 shows then that g decreases from +∞ to b as x increases from zero to c, and that (0, c) ⊂ (0, 1/ξ). Theorem 4.3 of [START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] shows that g decreases nowhere on (c, 1/ξ).

B Deformed Tracy-Widom fluctuations

We consider here a situation where the fluctuations of the extremal eigenvalue associated with a non-regular positive edge follow a deformation of the Tracy-Widom law introduced by Baik, Ben Arous and Péché in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF]. The proof of this fact consists in a simple adaptation of our previous analysis.

More precisely, we consider the following setting:

(1) Assumptions 1 and 2 hold true.

( has been first introduced in [9, Eq. 17]; to see that the definitions actually match, notice that the product of the operators associated with [9, Eq. (120) and Eq. ( 122)] has for kernel K where the integral operator K N is associated with the kernel

K N (x, y) = N 1/3 (2iπ) 2 δ N Γ dz Θ dw 1 w -z k j=1 w -ζ j -1 z -ζ j -1
× e -N 1/3 x(z-d N )/δ N +N 1/3 y(w-d N )/δ N +N f N (z)-N f N (w) . (B.4) By following the proof of Lemma 4.7, we can see that Re f N similarly converges locally uniformly towards (4.27) on an appropriate subset of the complex plane containing d, and this yields the existence of appropriate contours as in Proposition 4.6 by using the same exact proof. Since by assumption the ζ j -1 's stay in an arbitrary small neighborhood of d for every N large enough, the product over the ζ j 's in the integrand K N (x, y) is bounded away from that neighborhood. As a consequence, we can show as in Section 4. × e -N 1/3 x(z-d N )/δ N +N 1/3 y(w-d N )/δ N +N f N (z)-N f N (w) (B.5) up to negligible terms, in the sense that the remaining terms do not contribute in the large N limit. Moreover, by proceeding similarly as in Lemma 4.15 and Step 2 of the proof of Proposition 4.14, we have that K

  ρ(x)dx > 0 .

Theorem 1 .

 1 Let Assumptions 1 and 2 hold true. Then (a) Extremal eigenvalues: Given a regular right (resp. left) edge, there are perfectly located maximal (resp. minimal) eigenvalues which converge a.s. towards this edge as N → ∞; these eigenvalues are called extremal eigenvalues. (b) Tracy-Widom fluctuations: Given a regular right (resp. left) soft edge, the associated extremal eigenvalue, properly rescaled, converge in law to the Tracy-Widom distribution (resp. reversed Tracy-Widom distribution) at the scale N 2/3 . (c) Asymptotic independence: Given a finite family of regular soft edges, the associated extremal eigenvalues, properly rescaled, are asymptotically independent as N → ∞. (d) Hard-edge Tracy-Widom fluctuations: In the case where γ = 1, the bulk displays a hard edge at 0. If n = N + α with α ∈ Z independent of N , then the smallest eigenvalue, properly rescaled, is shown to converge to the hard-edge Tracy-Widom distribution with parameter α ∈ N at the scale N 2 .

Proposition 2 . 1 (

 21 Silverstein & Choi [60]). For any x ∈ R \ Supp(µ), let p = m(x). Then p ∈ D, x = g(p), and g ′ (p) < 0. Conversely, let p ∈ D such that g ′ (p) < 0. Then x = g(p) ∈ R \ Supp(µ) and p = m(x).

( a )

 a There exists a unique d ∈ D such that b = g(d), g ′ (d) = 0 and g ′′ (d) > 0. (b) There exists a unique d ∈ ∂D such that (dε, d) ⊂ D for some ε > 0 small enough, the function g is decreasing on (dε, d), and b = lim x↑d g(x). In that case, we write b = g(d). Conversely, for any point d satisfying one of these properties, b = g(d) is a right edge of the measure µ.

Definition 2 . 5 .Remark 2 . 6 .

 2526 (Regular edge) Recall that the λ i 's are the eigenvalues of matrix Σ N ; a soft edge a = g(c) is regular if lim inf (a) If a = g(c) is a regular soft edge, then the weak convergence ν N → ν stated in Assumption 2 rules out the options labelled (b) in Proposition 2.3.

  Our purpose is now to locate the eigenvalues of M N , or equivalently those of M N (denoted by x1 • • • xn ), that converge to a prescribed edge. The reader may refer to Figure1to better visualize the results of this proposition: Remark 2.8. (Convention) In the remaining, we shall systematically use the notational convention λ 0 = x0 = 0 and λ n+1 = xn+1 = ∞. Proposition 2.9 (Bai & Silverstein [3, 4]). Let Assumptions 1 and 2 hold true. Assume that [u, v] with u > 0 lies in an open interval outside Supp(µ(γ N , ν N )) for N large enough and recall the definition (2.1) of the fixed-point solution m. Then the following facts hold true: (a) If γ > 1, then xn-N+1 → a almost surely as N → ∞, where a > 0 is the leftmost edge of the bulk. (b) In any of the two cases i) γ 1 or ii) γ > 1 and [u, v] ⊂ [0, a], it holds that m(v) > 0.

3 .

 3 (a) If γ > 1 and a is the leftmost edge of the bulk, then set ϕ(N ) = n -N + 1. Otherwise, let a = g(c) be a regular soft left edge and let ϕ(N ) = min{j : λ -1 j < c}. Then, almost surely, lim N →∞ xϕ(N) = a and lim inf N →∞ (axϕ(N)-1 ) > 0 . (b) Let b = g(d) be a regular right edge and let φ(N ) = max{j : λ -1 j > d}. Then, almost surely lim N →∞ xφ(N) = b and lim inf N →∞ (x φ(N )+1b) > 0 . Eigenvalues xϕ(N) and xφ(N) are called extremal eigenvalues.

Proposition 2 . 11 (

 211 Left regular soft edges). Let Assumption 2 hold true. Let a be a left edge.

( a )

 a Consider first the case where a is the leftmost edge.

.

Figure 1 : 3 ) ∪ ( 1 3 , 1 )

 1331 Figure 1: Plot of g : D → R for γ = 0.1 and ν= 0.7δ 1 + 0.3δ 3 . In this case, D = (-∞, 0) ∪ (0, 1 3 ) ∪ ( 1 3 , 1) ∪ (1, ∞). The two thick segments on the vertical axis represent Supp(µ). The right edge b of the measure µ satisfies property (a) of Proposition 2.3.

Finally, there exists

  a deterministic sequence (ϕ(N )) such that almost surely, lim N →∞ xϕ(N) = a and lim inf N →∞ (axϕ(N)-1 ) > 0 .

Proposition 2 . 12 (

 212 Right regular soft edges). Let Assumption 2 hold true and assume that b is a regular right soft edge. Then b = g(d) with g ′ (d) = 0 g ′′ (d) > 0 and d > 0 .

  the regularity condition still holds for b, and Tracy-Widom fluctuations around b N = g N (d N ) will be observed. Let us say a few words on the fluctuations of the outliers. Notice that ζ incurs the presence of a local minimum and a new local maximum in g N which are absent from g, see Figure 2. Considering e.g. the minimum reached at, say d

2 .

 2 The spike ζ satisfies g ′ (1/ζ) > 0. The case where 1/ζ > d is shown on Figure 3. Here, the spike ζ does not create an outlier and the regularity condition on b is still satisfied. Tracy-Widom fluctuations around b N = g N (d N ) will be also observed here.

Theorem 4 .

 4 Let Assumptions 1 and 2 hold true and let I and J be finite sets of indices. Denote by (a i ) i∈I left regular soft edges and by (b j ) j∈J right regular soft edges.

Figure 2 :

 2 Figure 2: Plot of g N (x) for n = 300, γ N = 0.1 and ν N = 1 300 δ 1.7 + 209 300 δ 1 + 90 300 δ 3 . The spike ζ = 1.7 produces an outlier. Asymptote at 1/ζ not shown for better visibility.

Figure 3 :

 3 Figure 3: Plot of g N (x) for n = 300, γ N = 0.1 and ν N = 1 300 δ 1.1 + 209 300 δ 1 + 90 300 δ 3 . The spike ζ = 1.1 does not produce an outlier. Asymptote at 1/ζ not shown for better visibility.

Theorems 3 and 4 )

 4 together with the a.s. convergence x 1 → a and the convergences a N → a, b N → b, δ N → δ and σ N → σ (cf. Prop. 2.7), one can conclude.

  are well-defined and finite. The maps A → Tr(A) and A → det(I -A) are continuous on (J 1 , • 1 ). If both AB and BA are trace class, then we have the useful identity det(I -AB) = det(I -BA). (4.1)

Figure 4 :

 4 Figure 4: The contours of integration

Proposition 4 . 4 .

 44 Consider the setting of Theorem 3 and recall that by convention x0 = 0 and xn+1 = +∞. Then the following facts hold true:

  2-(b) and the convergence bN → b that det I -K N L 2 (b N +s/η N , b N +ε) = P η N xφ(N)b N s , xφ(N)+1 b N + ε + o(1). (4.11)Finally, (4.7) follows by combining (4.8) and (4.11).

Figure 6 :

 6 Figure 6: The new contours for the kernel K N .

27 )andLemma 4 . 7 .

 2747 Re f N are well-defined. The following properties of Re f and Re f N around d and d N will be of constant use in the sequel. Let Assumption 2 hold true and let K be as in (4.25). Then (a) The function Re f N converges locally uniformly to Re f on C \ K. Moreover, lim N →∞ Re f N (d N ) = Re f (d).(4.28) (b) There exists ρ 0 > 0 and ∆ = ∆(ρ 0 ) > 0 independent of N such that for every N large enough, B(d N , ρ) ⊂ C \ K for every ρ ∈ (0, ρ 0 ] and, whatever the analytic representation of f N on B(d N , ρ), we have

.

  Fix an open ball B of C \ K. By definition of K, one can chose a determination of the logarithm such that f N is well-defined and holomorphic there for N large enough. Indeed, there exists an analytic determination of the logarithm on every simply connected domain of C \ {0}. Use the same determination for f , which is then also well-defined and holomorphic on B. By weak convergence of ν N to ν, f N converges pointwise to f on B. Similarly as in the proof of Proposition 2.7, the sequence of holomorphic functions (f N ) N is uniformly bounded on B and thus has compact closure by the Montel theorem, which upgrades the pointwise convergence f N → f to the uniform one on B. The uniform convergence of Ref N to Re f on B follows since | Re f N (z) -Re f (z)| |f N (z)f (z)|for all z ∈ B. Now since d N → d and d N , d ∈ C \ K for all N large enough by the regularity assumption, (4.28) follows from the local uniform convergence Re f N → Re f on C \ K and (a) is proved. It follows from Proposition 2.7 that for ρ 0 > 0 small enough and every N large enough we have B(d N , ρ 0 ) ⊂ B(d, 2ρ 0 ) ⊂ C \ K. Using the same determination of the log as previously yields that f N is well-defined and holomorphic on B(d N , ρ 0 ). Since (4.14) and (4.21) yield f

. 31 )

 31 Proof. Recall the expression (4.27) of Re f (z) which yields that Re f (z) = -b Re(zd) + O(log |z|) as |z| → ∞. Since b > 0, it follows that for any fixed α ∈ (0, π/2) there exists R > 0 large enough such that

Lemma 4 . 10 .

 410 The function Re f is subharmonic on C \ {x ∈ R : x -1 ∈ Supp(ν)} and superharmonic on C \ {0}. Proof. It will be enough to establish the result for the map z → log |z|γ log |1 -xz| ν(dx) = log |z|γ log |z -x| τ (dx)γ log x ν(dx)

  g. [58, Chapter 0]. Consequently, z → log |z| is harmonic on C \ {0} and subharmonic on C, and z → γ log |z -x| τ (dx) is harmonic on C\Supp(τ ) and subharmonic on C. The result follows. Equipped with Lemma 4.10, we can obtain more information concerning the connected components of Ω ± . Lemma 4.11. (1) If Ω * is a connected component of Ω + , then Ω * is open and, if Ω * is moreover bounded, there exists x ∈ Supp(ν) such that x -1 ∈ Ω * .

  ) + iε belong to ∂Ω * \ R. In particular, by (4.33) and the symmetry Re f

  (j) min (ε)| and |u

Figure 7 :

 7 Figure 7: Preparation of the saddle point analysis for a right edge. The dotted path at the right is Υ (1)

( 1 )

 1 res (N )'s can be chosen with a uniformly bounded length as long as N N 0 . Since Ω 1 ⊂ Ω -there exists K > 0 such that Re f (z) Re f (d) -3K on T . Since moreover Re f N uniformly converges to Re f on T and Re f N (d N ) → Re f (d) according to Lemma 4.7-(a), we can choose N 0 large enough such that Re f N Re f + K on T and Re f (d) Re f N (d N ) + K.

  Recall the definition (4.23) of the kernel K (0)

  (a) If Ω * is bounded, then 0 ∈ Ω * . (b) If Ω * is bounded, then its interior is connected. (c) If 0 /∈ Ω * , then the interior of Ω * is connected.

  (b) Θ = Θ * ∪ Θ res whereΘ * = {c N + te ±iπ/3 : t ∈ [0, ρ]}.

4 )

 4 There exists d > 0 independent of N such that inf |z -w| : z ∈ Υ * , w ∈ Θ res d inf |z -w| : z ∈ Υ res , w ∈ Θ * d inf |z -w| : z ∈ Υ res , w ∈ Θ res d (5) (a) Υ and Θ lie in a bounded subset of C independently of N (b) The lengths of Υ and Θ are uniformly bounded in N .

Figure 9 :

 9 Figure 9: Preparation of the saddle point analysis for a left edge with c < 0. The path Θ res is close to the inner dotted path. The path Υ res is close to the outer dotted path. The thick segment represents the support of the image of ν by the map x → x -1 .

- 1 0

 1 + iη 0 and x -1 0iη 0 as well. Using the local uniform convergence of Re f * N to Re f * on C \ K and the convergence Re f * N (c N ) → Re f (c), we can show as in the proof of Proposition 4.6 that for every N large enough we have Re f * N (z) -Re f * N (c N ) -K

  .107) Let us set for convenience z η = x -1 0 + iη for any |η| η 0 . First, since the map x → log |1xz η 0 | is bounded and continuous on any compact subset of R, the weak convergence ν N → ν and the convergence n/N → γ yield that for any N large enough n N log |1xz η 0 |ν N (dx) γ log |1xz η 0 |ν(dx) + K. (4.108)

  y)∈(0,s]×(0,s] K N (x, y) -EK Be,α E -1 (x, y) = 0.Let us first show how Theorem 5 follows from this proposition.Proof of Theorem 5. The relation xJ′ α (x) = αJ α (x) -xJ α+1 (x), see [33, 7.2.8 (54)], provides K Be,α (x, y) = √ xJ α+1 ( √ x )J α ( √ y ) -√ yJ α+1 ( √ y )J α ( α ( √ yu )du,and, after the change of variables u → u/s, this yields the factorization K Be,α = B 2 s as operators of L 2 (0, s) where B s has for kernel B s (x, y) = J α ( xy/s)/(2 √ s). The asymptotic behavior as x → 0

Lemma 6 . 2 .

 62 With K Be,α (x, y) defined in(3.6), for every 0 < r < R and x, y > 0 we have K Be,α (x, y)

6 )

 6 By plugging (6.5) and (6.6) into (6.3), we obtain (xy)K Be,α (x, y)

Corollary 6 . 3 .

 63 follows after the change of variables z → -1/z and w → -1/w. For every 0 < r < R and x, y > 0, we haveEK Be,α E -1 (x, y) = 1 (2iπ) 2 |z|= r dz z |w|= R

= z -w 4 ,

 4 3 and the inequality (4.71) withu = -N F N (z) -F N (0) + N F N (w) -F N (0)and vwe obtain for every 0 < x, y sK N (x, y) -EK Be,α E -1 (x, y) ∆r α-1 2π 2 R α+1 (Rr)N |z|= r e -x Re(1/z)+Re(z)/4+∆/N |dz| |w|= Re y Re(1/w)-Re(w)/4+∆/N |dw| C(s) N

  4 makes use of[START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] Th. 4.3 and 4.4]. In a word,[START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] Th. 4.3] says that on any connected component of D, there is at most one interval on which the function g is decreasing while[START_REF] Silverstein | Analysis of the limiting spectral distribution of large-dimensional random matrices[END_REF] Th. 4.4] says that on any two disjoint open intervals of D where g is decreasing, the images of the closures of these intervals by g are disjoint.Proof of Proposition 2.4. Let us prove (a). Assume γ > 1. Since m(z) is the Cauchy-Stieltjes transform of a probability measure supported by [0, +∞), the function m(x) decreases from zero as x increases from -∞ to the origin. Hence its inverse g(x) decreases to -∞ as x increases to zero. Since xg(x) = 1 + γ xλ 1xλ ν(dλ),

( 3 ) 1 j( 4 )

 314 ) b is a right edge and b = g(d) with d ∈ D, see Section 2 (the positive left edge setting is similar). Let k be a fixed integer and assume there exist eigenvalues ζ 1 , . . . , ζ k ∈ {λ 1 , . . . , λ n } satisfying ζ j -1 → d as N → ∞ for every 1 j k. As regards the other eigenvalues, they satisfy lim inf N →∞ min j=1,...,n, λ j =ζ 1 ,...,ζ k dλ -The ζ j -1 's satisfy the condition (B.2) below. Thus, b is not regular as soon as k 1. However, we shall see that situation can be tackled by using similar arguments than in Section 4. Indeed, if we introduce the probability measure νN = n nk ν N -1 n k j=1 δ ζ j , then we still have the weak convergence νN → ν. Because of the assumption (B.1), the edge b = g(d) is regular with respect to the measure νN . Thus, if we replace ν N by νN and γ N = n/N by (nk)/N in the equation (2.6) defining the function g N , Section 2 yields the existence of appropriate sequences d N and xφ(N) who play the same role as in Section 4. We then assume that the ζ j 's satisfy the supplementary condition lim 3 /3+yw-w 3 /3 , where the contours Ξ and Ξ ′ are the same as in the proof of Lemma 4.15. Notice that when k = 0, we recover the usual Airy kernel (4.56). The distribution F k (s) = det(I -K (k) Ai ) L 2 (s,∞)

  (x, y). We claim that if we set b N = g N (d N ) andδ N = (2/g ′′ N (d N )) 1/3 , then lim N →∞ P N 2/3 δ N xφ(N)b N s = det I -K (k) Ai L 2 (s,∞) , s ∈ R. (B.3)We start by introducing the mapf N (z) = -b N (zd N ) + log(z) -nk N log(1xz)ν N (dx),which is the counterpart of f N from Section 4. From Proposition 4.4 and a change of variables, we have as N → ∞P N 2/3 δ N xφ(N)b N s = det I -1 (s, εN 2/3 δ N ) K N 1 (s, εN 2/3 δ N ) L 2 (s,∞) + o(1),

  5 and in Step 2 of the proof of Proposition 4.14 that, with Υ * and Θ * respectively defined in (4.60) and (4.62),K N (x, y) = N 1/3 (2iπ) 2 δ N Υ * dz

×

  e -N 1/3 x(z-d N )/δ N +N g ′′ N (d N )(z-d N ) 3 /6+N 1/3 y(w-d N )/δ N -N g ′′ N (d N )(w-d N ) 3 /6 (B.6)up to negligible terms. Finally, to conclude we need to estimate the difference between the right hand sides of (B.5) and (B.6), which is the counterpart of Step 1 in the proof of Proposition 4.14; we claim that similar estimates can be performed with minor modifications provided that (B.2) holds true.

  Description of the limiting bulk . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Regularity condition and its consequences . . . . . . . . . . . . . . . . . . . . 2.3 Extremal eigenvalues and their convergence . . . . . . . . . . . . . . . . . . . 2.4 Summary of the properties of regular edges . . . . . . . . . . . . . . . . . . .

	1 Introduction
	2 Bulk description, regularity and extremal eigenvalues
	2.1

  is subharmonic; in particular it is lower semicontinuous. Moreover, if u : G → C is subharmonic, it satisfies a maximum principle: For any bounded domain (i.e. connected open set) U ⊂ C where u is subharmonic, if for some κ ∈ R it holds that

	lim sup z→ζ, z∈U	u(z) κ,	ζ ∈ ∂U,

then u κ on U . Similarly, superharmonic functions satisfy a minimum principle.

The use of the maximum principle for subharmonic functions is made possible here because of the following observation.

  Let us show (1). Since Re f (z) → -∞ as |z| → 0, then Ω + ⊂ C \ {0}. By Lemma 4.10, Re f is then lower semicontinuous on Ω + , which shows that Ω + is an open set. As a consequence, ∂Ω + ⊂ C\Ω + and thus Re f Re f (d) on ∂Ω + . If Ω * is a connected component of Ω + , then Ω * is open and ∂Ω * ⊂ ∂Ω + ; in particular Re f Re f (d) on ∂Ω * . If Ω * is moreover bounded, then we have Re f > Re f (d) on the bounded domain Ω * and Re fRe f (d) on its boundary. Since subharmonic functions satisfies a maximum principle, Re f cannot be subharmonic on the whole set Ω * and (1) follows from Lemma 4.10.We now turn to (2a). We argue by contradiction and assume that Ω * ⊂ R is a bounded connected component of Ω * which does not contains the origin. The assumption Ω * ⊂ R shows that at least one of the sets Ω * ∩ {Im(z) > 0} and Ω * ∩ {Im(z) < 0} is non-empty.

	Consider the set
	Ω sym *

then the interior of Ω * is connected. Proof. = z ∈ C : z ∈ Ω * and notice it is also a connected component of Ω -because of the symmetry Re f (z) = Re f (z). Without loss of generality, assume that Ω * ∩ {Im(z) > 0} = ∅ (otherwise switch the role of Ω * and Ω sym * in what follows). Notice that since Re f is continuous on C \ K by Lemma 4.10, the set Ω * ∩ {Im(z) > 0} is open and moreover Re

By Gaussian-like, we mean that the largest eigenvalue of MN , when correctly centered et rescaled and when associated to a large perturbation of the identity ΣN of finite multiplicity k, asymptotically converges to the distribution of the largest eigenvalue of a fixed k × k GUE.

Note that our definition of the Cauchy-Stieltjes transform differs by a sign from the one in[START_REF] Marcenko | Distribution of eigenvalues in certain sets of random matrices[END_REF] but will turn out to be more convenient in the sequel.

In view of Remark 2.10, one can relax the Gaussianity assumption and replace it by the fact that XN 's entries are extrated from a doubly infinite array of i.i.d. random variables.

See for instance[START_REF] Gohberg | Traces and determinants of linear operators[END_REF] Theorem 8.2] for sufficient conditions on K to be trace class.

Thus, by comparing (4.93) with (4.20) and (4.94) with (4.22), we recover the setting of the proof of , except that we exchanged x and y, the role of Γ and Θ as well, and that we replaced f N by f * N . Since the Airy kernel is symmetric, see (3.1), it is enough to show that In the case of left regular soft edges, the analysis substantially changes whether c (cf. Prop. 2.11) is positive or not and we consider separately the two cases in the sequel.

The case where c is positive

We first consider the case where c > 0, which is always the case except if a is the leftmost edge and γ > 1, see Proposition 2.4. In particular, c N > 0 for all N large enough. We then split Γ into two disjoint contours Γ (0) and Γ (1) in the following way: Γ (0) encloses the λ -1 j 's which are larger that c N , while Γ (1) encloses the λ -1 j 's which are smaller that c N . Proposition 2.4-(e) applied to the measure ν N shows that the set {j , 1 j n : λ -1 j < c N } is not empty and thus the contour Γ (1) is always well-defined. If c N is actually larger than all the λ -1 j 's, as it is the case when dealing with the smallest eigenvalue when γ < 1, then set Γ (1) = Γ, K

N = K * N ; any later statement involving Γ (0) will be considered as empty. Otherwise, Γ (0) is well-defined and we introduce for α ∈ {0, 1} the kernels

N (x, y). We similarly have for the associated operators that K

N . Observe moreover that we can deform Θ in K

N (x, y) so that it encloses the origin and Γ (1) since the residue we pick at z = w vanishes.

In order to establish (4.95) and (4.96), it is then enough to prove that lim

The exact same estimates as in the proof of the Propositions 4.13 and 4.14 show that (4.97)-(4.100) hold true, provided we can show the existence of appropriate contours similarly as in Proposition 4.6. More precisely, it is enough to establish the next proposition in order to prove Theorem 3-(a), in the case where c > 0.

.

The case where c is negative

Here we consider the case where c is negative, which only happens if we are looking at the leftmost edge a when γ > 1, and thus c N < 0 for all N large enough. We recall that

Note that the λ -1 j 's are zeros for e f * N , and that 0 is a zero for e -f * N . Thus, since the residue picked at w = z vanishes, we can deform Θ and Γ is a way that Γ encircles Θ and all the λ -1 j 's, whereas Θ encircles the origin and possibly some λ -1 j 's. It is enough to establish the next proposition in order to obtain (4.95) and (4.96) in the case where c < 0, and thus to complete the proof of Theorem 3-(a), since the same estimates as in the proof of Proposition 4.14 can be used after setting K

(1) N = K * N and Γ (1) = Γ. The reader may refer to Figure 9 to better visualize the results of the next proposition as well as the proof argument. Proposition 4.17. For every ρ > 0 small enough, there exist contours Υ = Υ(N ) and Θ = Θ(N ) which satisfy for every N large enough the following.

(1) (a) Υ encircles Θ, the origin and all the λ -1 j 's. (b) Θ encircles the origin (and possibly some λ -1 j 's).

(2) (a) Υ = Υ * ∪ Υ res where

Clearly {x ∈ R : x -1 ∈ Supp(ν N )} ⊂ K for every N and moreover {x ∈ R : x -1 ∈ Supp(ν)} ⊂ K. We now prove that the following facts hold true.

(A) We have Ω 1 = Ω -1 , the interior of Ω 1 is connected, and there exists x 0 ∈ Supp(ν) and δ > 0 such that B(x -1 0 , δ) ⊂ Ω 1 .

(B) We have Ω 2 = Ω -2 , the interior of Ω 2 is connected, and for every 0 < α < π/2 there exists R > 0 such that

The proof will mainly use properties ( 1) and ( 2)-(a)/(b)/(c) from the proof of Proposition 4.16. Let us show (A). First, Ω 1 has a connected interior by [START_REF] Bai | Random matrix theory and its applications[END_REF]. Let us show by contradiction that Ω 1 is bounded. If Ω 1 is unbounded, then by using the symmetry Re f * (z) = Re f * (z), the inclusion (4.103) and the uniqueness of the unbounded connected component of Ω -, it follows that Ω 2 is bounded without containing the origin, which contradicts (2a). Thus Ω 1 is bounded, and has to contain some x -1 0 with x 0 ∈ Supp(ν) as a consequence of (1). Moreover, since Re f * is upper semicontinuous on an open neighborhood of x -1 0 (because it is subharmonic on C \ {0}), there exists δ > 0 such that B(x -1 0 , δ) ⊂ Ω 1 . As a consequence, together with the symmetry Re f * (z) = Re f * (z), (A) is proved.

Next, since Ω 1 thus surrounds the origin, then Ω 2 has to be unbounded by (2a) and has a connected interior by (2c). Finally, (B) follows from the symmetry Re f * (z) = Re f * (z), the inclusion (4.102) and the uniqueness of the unbounded connected component of Ω + .

To construct Υ satisfying the conditions of Proposition 4.17, by (B) we can complete {c Nte ±iπ/3 : t ∈ [0, ρ]}, for N large enough and ρ small enough so that both the points c Nρe iπ/3 and c Nρe -iπ/3 lie in Ω 2 , into a closed contour with a path lying in the interior of Ω 2 and intersecting the real line exactly once at the right of K with finite length, and then use the local uniform convergence of Re f * N to Re f * on C \ K, see the proof of Proposition 4.6 for the details.

To construct Θ, we need to proceed more carefully since Ω 1 actually crosses K and Re f * N may not converge uniformly to Re f * there. For N large enough and ρ small enough so that the points c N +ρe iπ/3 and c N +ρe -iπ/3 lie in Ω 1 , by (A) we can complete {c N +te ±iπ/3 : t ∈ [0, ρ]} into a closed contour with a path Ξ lying in the interior of Ω 1 and crossing the real axis exactly once at x -1 0 with finite length. Since B(x -1 0 , δ) ⊂ Ω 1 we can moreover assume that Ξ crosses the real axis perpendicularly, namely that there exists η 1 > 0 small enough such that the segment {x -1 0 + iη : |η|

Notice that the map z → log |1 -xz|ν(dx) is upper semicontinuous on C since it is subharmonic (see the proof of Lemma 4.10). As a consequence, if log |1-xx -1 0 |ν(dx) = -∞, then there exists η 0 ∈ (0, η 1 ) small enough so that

Combined with the inequalites (4.104), (4.108) and (4.106), we obtain that for every N large enough and uniformly in |η| < η 0

and this completes the proof of Proposition 4.17.

Proof of Theorem 4: Asymptotic independence

In this section we prove Theorem 4.

Our strategy builds on an approach used by Bornemann [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF]. Indeed, the asymptotic independence for the smallest and largest eigenvalues of an N × N GUE random matrix is established in [START_REF] Bornemann | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble[END_REF] by showing that the trace class norm of the off-diagonal entries of a two by two operator valued matrix goes to zero as N → ∞. Here we obtain that proving the asymptotic joint independence of several extremal eigenvalues leads to consider a larger operator valued matrix. Moreover, we show that it is actually sufficient to establish that the Hilbert-Schmidt norms of the off-diagonal entries go to zero as N → ∞, instead of the trace class norms. The latter can be provided by an asymptotic analysis for double complex integrals as we performed in the previous section.

More generally, our method can be applied to several other determinantal point processes for which a contour integral representation for the kernel and its asymptotic analysis are known, e.g. the eigenvalues of an additive perturbation of a GUE matrix [START_REF] Capitaine | Fluctuations at the edge of the spectrum of the full rank deformed gue[END_REF].

Conventions: In this section, we fix two finite sets I and J of indices, and real numbers (s i ) i∈I and (t j ) j∈J as well. Assume that (a i = g(c i )) i∈I are regular left soft edges and (b j = g(d j )) j∈J are regular right edges. We denote by c i,N and d j,N the sequences associated respectively with a i and b j as specified by Proposition 2.7-(c). We moreover set

and

, where g N has been introduced in (2.6). Finally, ϕ i (N ) (respectively φ j (N )) denotes the sequence associated with a i,N (resp. b j,N ) as in Theorem 2 (see also Propositions 2.11 and 2.12). Finally, we shall consider that the free parameter q introduced in the statement of Proposition 4.2 is zero when dealing with the kernel K N (x, y) , see Remark 4.3.

Our starting point the following proposition.

This provides

and Lemma 5.3 follows.

Equipped with Lemma 5.3, we are now in position to prove Proposition 5.2.

Proof of Proposition 5.2. Since the sets of indices I and J are finite by assumption, the regularity condition provides ε > 0 such that λ -1 j ∈ (0, +∞) \ B for every 1 j n and every N large enough, where

We then set

We start by proving (a). To do so, we essentially use the estimates from the Section 4.5.2. For any (i, j) ∈ J × J such that i = j, we have

By using Lemma 5.3 and performing the changes of variables x → N 2/3 δ i,N (xb i,N ) and y → N 2/3 δ j,N (yb j,N ), we obtain

)) e N 1/3 y(w-d j,N )/δ j,N -N (f j,N (w)-f j,N (d j,N )) . (5.15) for some C, C 1 , C 2 , C 3 > 0 independent on N and x, y. Combined with (5.16) and (5.20), it follows from (5.15) that

where C ′ , C ′ 1 , C ′ 2 , C ′ 3 > 0 are independent on N and x, y. Finally, by mimicking the Step 3 of the proof of Proposition 4.14, we obtain lim

as soon as ε is small enough. We thus have proved (a).

Concerning the points (b) and (c), we proceed similarly as for the point (a) and use Lemma 5.3 and the changes of variables x → N 2/3 σ i,N (a i,Nx) and y → N 2/3 δ j,N (yb j,N ) in order to obtain

)) e N 1/3 y(w-d j,N )/δ j,N -N (f j,N (w)-f j,N (d j,N )) . (

If c i > 0, then we replace the contour Γ by the contour Υ (0) ∪ Υ (1) (if Υ (0) does not exist, we just deform Γ into Υ (1) ) specified by Proposition 4.16 with c N = c i,N , and otherwise deform Γ into Υ as in Proposition 4.17. We moreover deform the contour Θ to obtain the contour Θ specified by Proposition 4.6 with d N = d j,N . The same arguments than in the proof of (a) show that lim

)) e -N 1/3 y(w-c j,N )/σ j,N +N (f * j,N (w)-f * j,N (c j,N )) , (5.24)

Asymptotic analysis

We now perform an asymptotic analysis for the kernel K N (x, y) as in Section 4. The main idea is that when the leftmost edge is a hard edge, the associated critical point c should be at infinity. This leads us to study the integrand of the double integral representation of K N (x, y) in a neighborhood of z = 0 and w = 0 after the changes of variables z → 1/z and w → 1/w.

Proof of Proposition 6.1. By choosing q = 0 in (4.5), which is possible according to Remark 4.3, we obtain with (6.1)

where we recall that the contour Γ encloses the λ -1 j 's whereas the contour Θ encloses the origin and is disjoint from Γ. We deform Γ so that it encloses Θ, which is possible since the integrand is analytic at the origin as a function of z and the residue picked at z = w vanishes. Moreover, since the λ -1 j 's are zeros of the integrand as a function of w, we can deform Θ such that it encloses all the λ -1 j 's. More precisely, we specify the contours to be Γ = {z ∈ C : |z| = N σ N /r} and Θ = {z ∈ C : |z| = N σ N /R} with 0 < r < R < lim inf N λ 1 /2. Notice that for N large enough, Γ = Γ(N ) and Θ = Θ(N ) enclose the λ j 's.

Next, we perform the changes of variables z → N σ N /z and w → N σ N /w in (6.11) in order to get where used the fact that n = N + α and we introduced the map

Note that for every N large enough and z ∈ B(0, R + 1) we have |z|/N σ N lim inf N λ 1 /2δ for some δ > 0. Thus we can choose a branch of the logarithm such that F N is well-defined and holomorphic on B(0, R + 1) for all N sufficiently large. Moreover, recalling that

and observing the identity F ′ N (0) = -1/(4N ), a Taylor expansion of F N around zero yields for every z ∈ B(0, R + 1) and for all N large enough