Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.


Introduction
Empirical covariance matrices are natural random matrix models in applied mathematics and their study goes back at least to the work of Wishart [35]. In the large dimensional regime, where both the size of the observations and of the sample go to infinity at the same speed, Marčenko and Pastur provided in the seminal paper [25] the first description of the limiting spectral distribution for such matrices, see also [31].
For instance, this limiting distribution has a continuous density on (0, ∞); its support is compact if the spectral norm of the population covariance matrix is bounded; it may include the origin and may also present several connected components.
Afterwards, attention turned to the local behaviour of the random eigenvalues near points of interest in the limiting spectrum, like positive endpoints (soft edges), see e.g. [20,22,3,13,17], interior points were the density vanishes (cusp points) [26], or the origin when it belongs to the spectrum (hard edge) [15,17]. Complex correlated Wishart matrices, namely covariance matrices with complex Gaussian entries, play a particular role in such investigations since their random eigenvalues form a determinantal point process. Indeed, for determinantal point processes, a local asymptotic analysis can often be performed by using tools from complex analysis such as saddle point analysis or Riemann-Hilbert techniques. In the more general setting of non-necessarily Gaussian entries, one then typically shows that the local behaviours are the same as in the Gaussian case by comparison or interpolation methods, see e.g. [23,24].
For complex correlated Wishart matrices, a fairly complete picture of the local fluctuations at every edges of the limiting spectrum has been obtained in the recent work [17], provided that a regularity condition is satisfied. This condition essentially warrants the local fluctuations to follow the usual laws from random matrix theory. For instance, if one considers a soft edge of the limiting spectrum, then this regularity condition ensures that the limiting density vanishes like a square root at the edge, and the fluctuations of the associated extremal eigenvalues follow the Tracy-Widom law involving the Airy kernel. As for the hard edge, when it is present, the fluctuations are described instead by means of the Bessel kernel.
The aim of this work is twofold. First, we investigate the local behaviour of the eigenvalues near a cusp point which satisfies the regularity condition: We show that the limiting density vanishes like a cube root near the cusp point (hence justifying the name) and, under an extra assumption on the decay of a speed parameter, we establish that the 2 Statement of the main results

The matrix model and assumptions
The random matrix model of interest here is the N × N matrix where X N is a N × n matrix with independent and identically distributed (i.i.d.) entries with zero mean and unit variance, and Σ N is a n × n deterministic positive definite Hermitian matrix. The random matrix M N thus has N non-negative eigenvalues, but which may be of different nature: The smallest N − min(n, N ) eigenvalues are deterministic and all equal to zero, whereas the other min(n, N ) eigenvalues are random. The problem is then to describe the asymptotic behaviour of the random eigenvalues of M N , as the size of the matrix grows to infinity. As for the asymptotic regime of interest, we let both the number of rows and columns of X N grow to infinity at the same speed: We assume n = n(N ) and n, N → ∞ so that lim N →∞ n N = γ ∈ (0, ∞) .

(2.2)
This regime will be simply referred to as N → ∞ in the sequel. Let us mention that the n × n random covariance matrix which is also under consideration, has exactly the same random eigenvalues as M N , and hence results on the random eigenvalues can be carried out from one model to the other immediately.
Our first assumption is that the entries of X N are complex Gaussian. As we shall state later on, this assumption is fundamental for our local eigenvalue behaviour analysis, but not for our results on the limiting density behaviour, see Remark 2.3. Considering now the matrix Σ N , we denote by 0 < λ 1 · · · λ n its eigenvalues and by ν N = 1 n n j=1 δ λj (2.3) its spectral measure. We make the following assumption: (a) For N large enough, the eigenvalues of Σ N stay in a compact subset of (0, +∞) independent of N , i.e.  f (x)ν(dx) (2.5) for every bounded and continuous function f .
Again, Assumption 2(a) is necessary for our results on the local eigenvalue behaviour, but our results on the limiting density behaviour require a weaker assumption, see Remark 2.3.
We now turn to the description of the asymptotic eigenvalue distribution.

Limiting eigenvalue distribution
Consider the empirical distribution of the eigenvalues (x i ) of M N , namely Since the seminal work of Marčenko and Pastur [25], it is known that this measure almost surely (a.s.) converges weakly towards a limiting probability measure µ with compact support, provided that Assumption 2 holds true: for every bounded and continuous function f . As a probability measure, µ is characterized by its Cauchy-Stieltjes transform, which is the holomorphic function defined by m(z) = 1 z − λ µ(dλ), z ∈ C + = z ∈ C : Im(z) > 0 . (2.7) Marčenko and Pastur proved that m(z) is the unique solution m ∈ C − = {z ∈ C : Im(z) < 0} of the fixed-point equation where we recall that γ has been introduced in (2.2) and ν is the weak limit of ν N , see (2.5). Thanks to this equation, Silverstein and Choi then showed in [31] that µ({0}) = (1 − γ) + and lim z∈C+→x m(z) ≡ m(x) exists for every x ∈ R * = R−{0}. Consequently, the function m(z) can be continuously extended to C + ∪ R * and, furthermore, µ has a density on (0, ∞) given by We therefore have the representation (2.10) Silverstein and Choi also obtained that ρ(x) is real analytic wherever it is positive, and they moreover characterized the (compact) support Supp(ρ) of the measure ρ(x)dx by building on ideas from [25]. More specifically, one can see that the function m(z) has an explicit inverse (for the composition law) on m(C + ) given by then the map g analytically extends to C + ∪ C − ∪ D. It was shown in [31] that Equipped with the definitions of m, g and D, we are now able to state our results concerning the behaviours of the limiting density ρ(x) near a cusp point or at the hard edge. a . Figure 1: Plot of ρ(x) with parameters γ 0.336 and ν = 0.7δ 1 + 0.3δ 3 with a cusp point a.

Density behaviour near a cusp point
As stated in the introduction, we define a cusp point a as an interior point where the density vanishes, namely a ∈ int(Supp(ρ)) such that ρ(a) = 0. In particular, c = m(a) ∈ R by virtue of (2.9). Our first result states that the density ρ(x) behaves like a cube root near a cusp point, provided that c ∈ D.
We prove Propositions 1 and 2 in sections 3.1 and 3.2 respectively. Their proofs are based on the fact that there is a strong relation between the property that a = g(c) is a cusp point and the local behaviour of g near c. For an illustration of these propositions, we refer to Figure 2 where we displayed the graph of the map g associated with the density from Figure 1.  Figure 2: Plot of function x → g(x) on D for γ 0.336 and ν = 0.7δ 1 + 0.3δ 3 . The vertical dotted lines are g's asymptotes at x = 1/3 and x = 1. The thick segment on the vertical axis represents Supp(ρ) who is delimited by the local extrema of g (see Eq. (2.13)). The point a is a cusp point.
We now turn to the hard edge setting.

Density behaviour near the hard edge
As usual in random matrix theory, the hard edge refers here to the origin when it belong to the limiting spectrum. In general, the limiting eigenvalue distribution may not display a hard edge, like in Figure 1. In fact, this is always the case when γ = 1, see [17, Proposition 2.4 (a),(c)]. Our next result states that there is a hard edge, namely 0 ∈ Supp(ρ), if and only if γ = 1, and that in this case ρ(x) blows up like an inverse square root at the origin. We furthermore relate the presence of a hard edge to the behaviour of g near ∞. More precisely, since Supp(ν) ⊂ (0, ∞) by Assumption 2, one can see from the definition (2.11) of g that the map g(1/z) is holomorphic at the origin. Thus, we have the analytic expansion as z → 0 Clearly, g(∞) = 0 and the coefficients g (∞) and g (∞) are respectively given by the first and second derivative of the map z → g(1/z) evaluated at z = 0.
Proposition 3 (Limiting density at the hard edge). The following three assertions are equivalent: Proposition 3 is proven in Section 3.3.

Remark 2.2.
There is an analogous statement for any left edge a > 0 of the spectrum satisfying c = m(a) ∈ D which follows from [31]; see also [18,Section 2] for further information. Indeed, in this case we have g(c) = a, g (c) = 0, g (c) < 0, and furthermore, as x → a + , (1)) .
By analogy with this equation, the preimage c ∈ D corresponding to the hard edge is c = ∞. The fact that it actually belongs to D follows from Assumption 2(a).

Remark 2.3.
As we shall see in Section 3, proofs of Propositions 1, 2 and 3 only rely on the properties of the limiting eigenvalue distribution µ, which do not depend on whether the entries of X N are Gaussian or not. More precisely, the exact assumptions required for these propositions are that the entries of X N are i.i.d centered random variables with variance one, Assumption 2-(b), and that µ({0}) = 0 (which follows from Assumption 2-(a)), see [25,31].

The Pearcey kernel and fluctuations near a cusp point
Our next result essentially states that the random eigenvalues of M N , properly scaled near a regular cusp point, asymptotically behave like the determinantal point process associated with the Pearcey kernel, provided that an extra condition on a speed parameter is satisfied. In order to state this result, we first introduce this limiting point process. Next, we define what we mean by regular, and provide the existence of appropriate scaling parameters. After that, we finally state our result for the fluctuations near a cusp point.

Determinantal point processes
A point process on R (or in a subset therein), namely a probability distribution P over the locally finite discrete subsets (y i ) of R, is determinantal if there exists an appropriate kernel K(x, y) : R × R → R which characterizes the correlation functions in the following way: For every k 1 and every compactly supported Borel function Φ : In particular, the gap probabilities can be expressed as Fredholm determinants. Namely, given any interval J ⊂ R, the probability that the point process avoids J reads dy 1 · · · dy k , (2.18) and the right hand side is the Fredholm determinant det(I − K) L 2 (J) of the integral operator acting on L 2 (J) with kernel K(x, y), provided that it makes sense. For instance, if one assumes that J is a compact interval, which is enough for the purpose of this work, then det(I − K) L 2 (J) is well-defined and finite as soon as K J = sup x,y∈J |K(x, y)| < ∞. Moreover, the map K(x, y) → det(I − K) L 2 (J) is Lipschitz with respect to · J when restricted to the kernels satisfying K J < ∞ (see e.g. [2, Lemma 3.4.5]). We refer the reader to [19,21,2] for further information on determinantal point processes.

The Pearcey kernel
Given any τ ∈ R, consider the Pearcey-like integral functions where the contour Σ has two non-intersecting components, one which goes from e iπ/4 ∞ to e −iπ/4 ∞, whereas the other one goes from e −3iπ/4 ∞ to e 3iπ/4 ∞. More precisely, we parametrize here this contour by (2.19) with the orientation as shown in Figure 3.
It follows from their definitions that the functions φ and ψ satisfy the respective differential equations The Pearcey kernel is then defined for x, y ∈ R by

20)
. Σ Σ . Figure 3: The contour Σ. see for instance [7]. One can alternatively represent this kernel as a double contour integral from which one can easily see the symmetry K (τ ) Pe (−x, −y) by performing the changes of variables z → −z and w → −w.
The Pearcey kernel first appeared in the works of Brézin and Hikami [8,7] when studying the eigenvalues of a specific additive perturbation of a GUE random matrix near a cusp point. Subsequent generalizations have been considered by Tracy and Widom [34], and a Riemann-Hilbert analysis has been performed by Bleher and Kuijlaars [5] as well. This kernel also arises in the description of random combinatorial models, such as certain plane partitions [27]. Furthermore, it has been established that the gap probabilities for the associated point process satisfy differential equations. For instance, log det(I − K (τ ) Pe ) L 2 (s,t) satisfies PDEs with respect to the variables s, t and τ , see [34,4,1], which should be compared to the connection between the Tracy-Widom distribution and the Painlevé II equation.

The regularity condition
We start with the following definition.
The regularity condition (2.22) has been considered in [17] when dealing with soft edges, to ensure the appearance of the Tracy-Widom distribution. Similarly here, as we soon shall see, this condition enables the Pearcey kernel to arise at a cusp point.
Moreover, the behaviour of ρ(x) at such cusp points is well described by Proposition 1, as explained in the next remark. Remark 2.5. If a is a regular cusp point, then it follows from the weak convergence ν N → ν and the definition of D that necessarily c = m(a) ∈ D. Thus, a satisfies the hypothesis of Proposition 1. In particular, we have a = g(c) and ρ(x) behaves like a cube root near a.
Finally, the regularity assumption yields the existence of natural scaling parameters for the eigenvalue local asymptotics. Consider the counterpart of the map g introduced in (2.11) after replacing ν by ν N , see (2.3), and γ by n/N , namely (2.23) The map g N is the inverse Cauchy transform of a probability measure µ N usually referred to as deterministic equivalent for the random eigenvalues distribution of M N , see [18,Section 3.2]. According to Section 2.2, µ N has a decomposition of the form (2.10). In particular, it has a density ρ N on (0, +∞) which is analytic wherever it is positive.
The next proposition provides an appropriate sequence of finite-N approximations of c, which we will use in the definition of the scaling parameters.  N converge uniformly to g (k) on a neighbourhood of c for every k 0. The proposition then follows by applying Hurwitz's theorem to g N since c is a simple root for g , according to Proposition 2.14.
Let us emphasize that there is however an important difference with regular soft edges as described in [17], where it was shown that if a = g(c) is a regular soft edge for µ, then g N (c N ) is a soft edge for µ N . Remark 2.6. A regular cusp point may not be the limit of finite-N cusp points. More precisely, if a = g(c) is a regular cusp point, then in particular g (c) = g (c) = 0. However, this only ensures the existence of a sequence (c N ) such that g N (c N ) = 0. A priori, g N (c N ) converges to zero as N → ∞ but might not be equal to zero. In fact, it is not hard to show we have the following alternatives: We are finally in position to state our result concerning the eigenvalue behaviour at a regular cusp point.

Fluctuations around a cusp point
Thanks to Assumption 1, the random eigenvalues (x i ) of M N form a determinantal point process with respect to a kernel K N (x, y), see [3,28]. An explicit formula for this kernel is provided in Section 4. The main result of this section is the local uniform convergence of this kernel, properly scaled, towards the Pearcey kernel.
Theorem 5 (Fluctuations near a cusp point). Let a = g(c) be a regular cusp point. Let (c N ) be the sequence associated to c coming from Proposition 4. Assume moreover that the following decay assumption holds true: There exists κ ∈ R such that uniformly for x, y in compact subsets of R.
This result was obtained by Mo [26] in the special case where the matrix Σ N has exactly two distinct eigenvalues (each with multiplicities proportional N ), by means of a Riemann-Hilbert asymptotic analysis.
determinantal with the kernel given by the left hand side of (2.26). Thus, having in mind Section 2.5.1, a direct consequence of this theorem is the convergence of the compact gap probabilities.

Corollary 2.7.
Under the assumptions of Theorem 5, we have for every s, t ∈ R, We now make a few comments on the assumption (2.24).
Remark 2.8. The decay assumption (2.24) roughly states that the cusp point a = g(c) appears fast enough. More precisely, for a cusp point a = g(c), one has in particular g (c) = 0.
• If κ > 0, then g N (c N ) > 0 for large N . According to Remark 2.6, the density ρ N is positive near g N (c N ) and will converge to zero to asymptotically give birth to a cusp point. The family of densities ρ N display a sharp non-negative minimum at g N (c N ) converging to zero which may be thought of as the erosion of a valley, see the thin curve in Figure 4.
• If κ < 0, then g N (c N ) < 0 for large N and the density ρ N vanishes in a vicinity of g N (c N ). However, this interval will shrink and asymptotically disappear. Thus, two connected components of the support of µ N move towards one another (moving cliffs), see the dotted curve in Figure 4.
The assumption (2.24) is an indication on the speed at which the bottom of the valley reaches zero (κ > 0), or at which the two cliffs approach one another (κ < 0). See [18] for a more in-depth discussion.
Remark 2.9. When the assumption (2.24) is not satisfied, namely when g N (c N ) goes to zero as N → ∞ (which is always true by Proposition 4) slow enough so that diverges to plus or minus infinity, we do not expect the Pearcey kernel to arise. See [18] for further discussion.
The proof of Theorem 5 is provided in Section 4.

Asymptotic expansion at the hard edge
Our last result concerns the behaviour of the smallest random eigenvalue of M N when the hard edge is present. Recall that n/N → γ. By Proposition 3, the limiting density displays a hard edge if and only if γ = 1. With this respect, we restrict ourselves here to the case n = N + α where α ∈ Z is fixed and does not depend on N .

The Bessel kernel
The Bessel function of the first kind J α with parameter α ∈ Z is defined by x > 0 , (2.28) with the convention 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). The Bessel kernel is then defined for x, y > 0 by One can alternatively express it as a double contour integral, Be L 2 (0,s) , s > 0, (2.31) where the right hand side stands for the Fredholm determinant of the restriction to Be . According to (2.18), F α (s) is the probability that the smallest particle of the determinantal point process associated with the Bessel kernel is larger than s. Tracy and Widom [33] established that certain simple transformations of log F α (s) satisfy Painlevé equations (Painlevé III and Painlevé V are involved).

Correction for the smallest eigenvalue's fluctuations
We denote by x min the smallest random eigenvalue of M N , namely Our last result is stated as follows.
Theorem 6 (Hard edge expansion). Assume n = N + α where α ∈ Z is fixed and does not depend on N . Set Then, for every s > 0, we have as N → ∞, The convergence towards F α (s) has been first observed by Forrester [15] when Σ N is the identity. As for the general Σ N case, it has been established by the authors in [17]. An explicit formula for the 1/N -correction term was conjectured when Σ N is the identity by Edelman, Guionnet and Péché [12], a conjecture proved true soon after by Perret and Schehr [29] and Bornemann [6], with different techniques. We thus generalize this formula to the general Σ N case. The strategy of the proof is rather similar to Bornemann's one: It relies on an identity involving the resolvent of K for every L 1 as N → ∞. Although we are able to provide a close formula for the coefficient C (α) N,1 (s) (as stated in Theorem 6) thanks to a formula due to Tracy and Widom, to the best of our knowledge the next order coefficients do not seem to benefit from such a simple representation.
We prove Theorem 6 in Section 5.

Proofs of the limiting density behaviours
This section is devoted to the proofs of Proposition 1, 2 and 3. We first recall a few facts stated in Section 2.2 that we shall use in the forthcoming proofs: The map m is the Cauchy-Stieltjes transform (2.7); it is analytic on C + and extends continuously to C + ∪ R * . Moreover, Im(m(x)) = −πρ(x) for every x ∈ R * . The map g defined in (2.11) is analytic on C + ∪ C − ∪ D. In particular, g has isolated zeroes on D. Moreover, after noticing that m(C + ) ⊂ C − , we have the identity g(m(z)) = z for every z ∈ C + .
We start with a simple but useful fact, which follows by taking the limit z ∈ C + → x in the previous identity and using the continuity of m and g on their respective domains: We will also use the following property.
and notice it is continuous on its definition domain by dominated convergence. It follows from the definition (2.11) of g that G(m) = g (m) when m ∈ D, and moreover that for every m ∈ C − we have the identity Im(g(m)) = Im(m)G(m).
By using the fact that g(m(z)) = z on C + , we thus obtain for every z ∈ C + ,

Density behaviour near a cusp point: proof of Proposition 1
Proof. Assume that a ∈ int(Supp(ρ)) and ρ(a) = 0. Set c = m(a) and assume moreover that c ∈ D. Thus, the facts that g(c) = a and g (c) = 0 directly follows from Lemma 3.1 and Lemma 3.2.
First, we prove that g (c) = 0. To do so, we show that g > 0 on (c − η, c + η) − {c} for some η > 0. Since g (c) = 0 this would indeed yield that c is a local extremum for g . We proceed by contradiction: Assume there exists a sequence (x k ) in D − {c} such that x k → c and g (x k ) 0. Since g has isolated zeroes on D, necessarily g (x k ) < 0 for every k large enough. It then follows from (2.13) that g(x k ) ∈ R − Supp(ρ) and, since g(x k ) → g(c) = a, this contradicts the assumption that a ∈ int(Supp(ρ)).
Next, we similarly show that there exists η > 0 such that We will use (3.3) later on in this proof. Assume there exists a sequence (x k ) in R * − {a} such that x k → a and ρ(x k ) = 0. Since a ∈ int(Supp(ρ)) and m(a) ∈ D by assumption, then x k ∈ Supp(ρ) and m(x k ) ∈ D for every k large enough. Moreover, we have m(x k ) → m(a), but since Lemma 3.2 then yields g (m(x k )) = 0, this contradicts that g has isolated zeroes on D and (3.3) follows. Now, we show that g (3) (c) > 0 by direct computation. Recalling the definition (2.11) of g, the equation g (c) = 0 reads As a consequence, we obtain We finally turn to the proof of the cube root behaviour (2.14). Since g (c) = g (c) = 0 and g (3) (c) > 0, there exists an analytic map ϕ defined on a complex neighbourhood of c such that g(m) − a = ϕ(m) 3 and ϕ (c) = 0, see e.g. [30,Theorem 10.32]. In particular, we have ϕ (c) 3 = g (3) (c)/6 = C > 0. Moreover, the inverse function theorem yields that ϕ has a local inverse ϕ −1 , defined on a neighbourhood of zero, such that ϕ −1 (0) = c and and, by taking the cube root (principal determination), applying ϕ −1 , and performing a Taylor expansion, we obtain Finally, (2.14) follows by taking the imaginary part in (3.4). Proof of Proposition 1 is therefore complete.

Identification of a cusp point: proof of Proposition 2
The main part of the proof consists in showing the following lemma. Proof of Proposition 2. Assume that c ∈ D satisfies g (c) = g (c) = 0 and set a = g(c).
We now prove the lemma.
By reorganizing this equation as we see the lemma would follow by taking the limit z ∈ C + → g(c) ∈ R * assuming that for some constant C > 0. To show (3.6), we start by writing Recalling the definition (3.1) of G, we see from (3.2) that G(m(z)) < 0 on C + , and this yields lim sup z∈C+→g(c) γ Φ(λm(z)) 2 ν(dλ) 1 .  Finally, we have  The lower bound (3.6) then follows by combining (3.7)-(3.11) and the proof is complete.

Density behaviour near the hard edge: proof of Proposition 3
The key to study the hard edge is to study the map g near infinity, which is holomorphic there. More precisely, we have the analytic expansion as z → 0, The reason why one should do so is that |m(x)| goes to ∞ as x 0: Lemma 3.4. Assume 0 ∈ Supp(ρ). Then, there exists η > 0 such that ρ(x) > 0 on (0, η). Moreover, as x > 0 decreases to zero, we have |m(x)| → +∞.
We finally turn to the proof of the lemma.
Proof of Lemma 3.4. Let z = x + iy with x, y > 0 and assume that ρ(x) > 0. The fixed point equation (2.8) yields where G is defined as in (3.1). When y → 0, since ρ(x) > 0, we have m(z) → m(x) ∈ C − and moreover G(m(z)) → 0 for the same reason as in proof of Lemma 3.2. Thus, we obtain, (3.14) We now show that ρ(x) > 0 for every x ∈ (0, η) for some η > 0 by contradiction: Assume there exists a sequence (x k ) such that x k > 0, x k → 0 as k → ∞, and ρ(x k ) = 0. Since 0 ∈ Supp(ρ), without loss of generality one can assume that x k ∈ Supp(ρ). As a consequence, using moreover that m is continuous on R * , one can construct a sequence (y k ) satisfying y k > 0, y k → 0 as k → ∞, ρ(x k ) > 0, and |m(x k ) − m(y k )| 1. Because Assumption 2 yields min Supp(ν) > 0, we see from (3.14) that necessarily |m(y k )| → +∞, and hence |m(x k )| → +∞, as k → ∞. In particular, m(x k ) ∈ D for k large enough and Lemma 3.2 then yields g (m(x k )) = 0. Thus, (1/m(x k )) is a vanishing sequence as k → ∞ of zeroes of the derivative of g(1/z). But since g(1/z) is holomorphic near zero, and hence so does its derivative, this contradicts the isolated zero principle and our claim follows.

Fluctuations around a cusp point: proof of Theorem 5
In this Section, we study the local behaviour of the random eigenvalues (x i ) of M N near a regular cusp point and prove Theorem 5. Our asymptotic analysis is based on that, thanks to Assumption 1, the random eigenvalues form a determinantal point process with an explicit kernel K N (x, y), see [3,28]. The kernel has the following double contour integral formula where the q ∈ R is a free parameter and we recall that the λ j 's are the eigenvalues of Σ N . Γ and Θ are disjoint closed contours such that Γ encloses all the λ −1 j 's whereas Θ encloses the origin, see for instance Figure 5.

Notations and conventions:
• All the contours we consider are simple and oriented counterclockwise.
• We denote by B(z, ρ) the open disc in C with center z ∈ C and radius ρ > 0.
• By convention, we shall use at several instances a constant C which depends neither on N nor on x, y ∈ [−s, s], but may depend on s, and whose exact value may change from one ligne to another.
• If a contour Γ is parametrized by γ : I → Γ for some interval I ⊂ R, then for every when it does make sense. In particular, Γ |dz| is the length of a closed contour Γ.

Remark 4.1.
In [3,28], it is assumed that q satisfies an extra restriction so that the associated operator K N is trace class on the semi-infinite intervals (s, +∞).  In the remaining of this section, we assume that the framework (assumptions and notations) of Theorem 5 holds true. We then set for every fixed s > 0.

Preparation
It follows from (4.1) and (4.2), by setting q = c N , that   . . , λ −1 n }. However, one needs to specify an appropriate determination of the complex logarithm in order to define f N properly. By Assumption 2 and the regularity condition (2.22), there exists ε > 0 such that λ −1 j ∈ (0, +∞) \ B(c, ε) for every 1 j n and every N large enough. As a consequence, if we introduce the compact set  As a consequence, we have for every k 1, N (z) . In particular, since a N = g N (c N ) by definition, the decay assumption (2.24) and Proposi- By performing a Taylor expansion of f N near c N in (4.4), one can already guess from (4.9)-(4.12) and a change of variables that the Pearcey kernel should appear in the large N limit, at least if one restricts the contours Γ and Θ to a neighbourhood of c N (see also [18] for a more detailed heuristic, which may serve as a guideline for the forthcoming proof). In a first step, we provide precise estimates in order to prove that claim, see Section 4.2. In a second step, we prove that the contribution coming from the pieces of contour away from c N are exponentially negligible, see Section 4.3. To do so, we establish the existence of appropriate contours in the same fashion as in [17]. This will enable us to conclude. Notice that one of the key arguments to assert the existence of appropriate contours is the maximum principle for subharmonic functions. This argument only appears in the proof of Lemma 4.9, which is similar to the proof of [17,Lemma 4.11] and hence omitted. The interested reader may refer to [17] for more details.

Step 1: local analysis around c N
We start with a quantitative Taylor expansion for f N near c N .
In particular, since g N (c N ) is real, for every z ∈ B(c N , ρ 0 ), Proof. Choose ρ 0 such that B(c, 2ρ 0 ) ⊂ C \ K. The convergence c N → c then yields B(c N , ρ 0 ) ⊂ B(c, 2ρ 0 ) ⊂ C \ K for every N large enough. By using (4.8)-(4.11) and performing a Taylor expansion for f N around c N , we obtain provided that z ∈ B(c N , ρ 0 ) and N is sufficiently large. Moreover, since f converges uniformly on B(c, 2ρ 0 ) to g (4) which is bounded there, the existence of ∆ = ∆(ρ 0 ) independent of N follows.
From now, we let ρ > 0 be small enough so that 0 < ρ < ρ 0 and g (3) We introduce the contours  The orientations of these contours are specified by letting the parameters θ and t increase in their definition domains. We also introduce Similarly, the orientation of contour Ξ * is specified by letting t increase in its definition domain.
We now establish the following estimate which essentially allows to replace f N by its Taylor expansion around c N in the double integral over the contours Υ * and Ξ * .
Proof. Let s > 0 be fixed, and set for convenience , so that it amounts to prove that vanishes as N → ∞.
First, by definition of the contours (4.14)-(4.19), we have the bound for every z ∈ Υ * and w ∈ Ξ * . As a consequence, Next, we use the following elementary inequality, which holds true for every u, v ∈ C. By specializing it to where z ∈ Υ * and w ∈ Ξ * , so that Lemma 4.
As a consequence, |dw|. (4.24) We now handle the integrals over each piece of contour separately.
First, consider the integrals over the contour Ξ * . By definition w ∈ Ξ * if and only if there exists t ∈ [−ρ, ρ] such that w = c N + it. Thus, For N large enough this yields, together with (4.13), where we set for convenience ξ = 1 2 For the last estimate, we have used the decay assumption Next, we turn to the integrals over the contours Υ + × and Υ − × . By definition z ∈ Υ + × , resp. z ∈ Υ − × , if and only if there exists t ∈ [N −1/4 , ρ] such that z = c N + e ±iπ/4 t, resp. z = c N − e ±iπ/4 t, so that in both cases we have As a consequence, for every x ∈ [−s, s], where the last inequality follows from the fact that lim inf N σ N > 0. Similarly, As a consequence, for every x ∈ [−s, s], where for the last estimate, we used the fact that lim inf N σ N > 0, the decay assumption √ N g N (c N ) → κ ∈ R and the convergence g (4.30) By gathering (4.24)-(4.30), we thus obtain the estimate Combined together with (4.22), this finally yields Proof. Let s > 0 be fixed and recall the definition (2.21) of K (τ ) Pe (x, y). We first show that, uniformly in x, y ∈ [−s, s], as N → ∞, Indeed, keeping in mind that the sequence (σ N ) is bounded, it follows from the definition (2.19) of Σ together with simple estimates that for every x, y ∈ [−s, s], from which (4.31) follows.
As a consequence, by performing the changes of variables z → N 1/4 (z − c N )/σ N and w → N 1/4 (w − c N )/σ N in the right hand side of (4.31) and using the definition (2.25) of σ N , we obtain In order to complete the proof of the proposition, we now prove the estimate (4.32) still holds true after replacement of the constant − √ N τ /σ 2 N by N g N (c N ). This amounts to showing that converges to zero uniformly in x, y ∈ [−s, s], which we now establish by using the same type of arguments as in the proof of Lemma 4.3. To do so, we set and observe that, since we have the convergences g for every z ∈ Υ * and w ∈ Ξ * , (4.33) is bounded from above by (4.34) Next, we use inequality (4.23) with , in order to obtain that (4.34) is bounded from above by |dw|.
By performing essentially the same estimates as in (4.25)-(4.30), we then prove that Pe (x, y) uniformly in x, y ∈ [−s, s] as N → ∞.

Step 2: existence of appropriate contours
In this section, we prove the existence of appropriate completions for the contours and Ξ * into closed contours, on which the contribution coming from e N (f N (z)−f N (w)) in the kernel K N (x, y) will bring an exponential decay as N increases to infinity. More precisely, we establish the following.

Proposition 8.
For every ρ > 0 small enough, there exist N -dependent contours Υ − , Υ + and Ξ which satisfy for every N large enough the following properties.
(1) (a) Υ − encircles the λ −1 j 's smaller than c N (b) Υ + encircles all the λ −1 j 's larger than c N (c) Ξ encircles all the λ −1 j 's smaller than c N and the origin The contours Υ − , Υ + and Ξ lie in a bounded subset of C independent of N (b) The lengths of Υ − , Υ + and Ξ are uniformly bounded in N .
In order to provide a proof for Proposition 8, we use the same approach as in [17,Section 4.4], from which we borrow a few lemmas. Introduce the asymptotic counterpart where we recall that K has been introduced in (4.6). We shall use the following property.  We now turn to a qualitative analysis for the map Ref . To do so, introduce the sets The regions ∆ k are shown on Figure 6.
Proof. Let η > 0 be small enough so that B(c, 2η) ⊂ C \ K. In particular, one can choose a determination of the logarithm such that the map Figure 6: Preparation of the saddle point analysis. The two dotted paths in Ω − correspond to Υ + res and Υ − res while the dotted path in Ω + corresponds to Ξ res .
is well-defined and holomorphic on B(c, 2η), and its real part is given by (4.35). Moreover, we have f (z) = g(z) − a. Since a = g(c) and g (c) = g (c) = 0, a Taylor expansion of f around c then yields for every z ∈ B(c, η), Since Re(z − c) 4 = (−1) k r 4 when z = c + re ikπ/4 , and because g (3) (c) > 0, the lemma follows by choosing η and then θ small enough.
Let Ω 2 +1 be the connected component of Ω − which contains ∆ 2 +1 . Similarly, let Ω 2 be the connected component of Ω + which contains ∆ 2 . We now prove that the following holds true. (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 (4.41) (2) We have Ω 2 = Ω −2 , the interior of Ω 2 is connected, and there exists R > 0 such that To prove the lemma, we use the following key properties. ∈ Ω * , then the interior of Ω * is connected.
Proof of Lemma 4.8. We first prove (2). Since Ω 2 is by definition a connected subset of Ω + , Lemma 4.9-(1) yields that its interior is connected (since Ω 2 is open). Next, we show by contradiction that Ω 2 is unbounded. If Ω 2 is bounded, then Lemma 4.9-(1) shows We now turn to (1). Since Ω 2 is unbounded, then Ω 1 does not contain the origin and it follows from Lemma 4.9(2a)-(2c) that Ω 1 is unbounded and has a connected interior. Proof of Proposition 8. Given any ρ > 0 small enough, it follows from the convergence of c N to c that for every N 0 large enough the points c N0 + ρe iπ/4 and c N0 + ρe −iπ/4 belong to ∆ 1 and ∆ −1 respectively. Thus both points belong to Ω 1 by Lemma 4.8-(1). As a consequence, we can complete the path Υ + o ∪ Υ + × into a closed contour with a path Υ + res (N 0 ) lying in the interior of Ω 1 . Since Υ + res (N 0 ) lies in the interior of Ω 1 , the convergence c N → c moreover yields that we can perform the same construction for all N N 0 with Υ + res (N ) in a closed tubular neighbourhood T ⊂ Ω 1 of Υ + res (N 0 ). By Lemma 4.8-(1) again, we can moreover choose Υ + 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 + 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 Ref Similarly, the same conclusion for Ξ follows from the same lines but by using Ω 2 instead of Ω 1 and Lemma 4.8- (2).
We now turn to the contour Υ − . Given any ρ > 0 small enough, for every N 0 large enough the points c N0 − ρe −iπ/4 and c N0 − ρe iπ/4 belong to ∆ 3 and ∆ −3 respectively. Thus both points belong to Ω 3 by Lemma 4.8-(3), which contains the origin in its interior. As a consequence, we can complete the path Υ − o ∪ Υ − × into a closed contour with a path Υ − res (N 0 ) lying in the interior of Ω 3 , which encircles all the λ −1 j 's smaller than c N for every N large enough, since we assumed lim inf N →∞ λ 1 > 0. Then, we can follow the same construction as we did for Υ + in order to prove the existence of a contour Υ − satisfying the requirements of Proposition 8, except for the point (4).
Finally, the item (4) of Proposition 8 is clearly satisfied by construction since the sets Ω − and Ω + are disjoint, and the proof of the proposition is therefore complete.
Equipped with Proposition 8, we are now in position to establish the remaining estimate towards the proof of Theorem 5.
Proof. Let s > 0 be fixed. This amounts to showing that   Next, Proposition 8 (5-b) yields the existence of L > 0 independent of N such that, for every z ∈ Υ ∪ Ξ, we have |Re(z − c N )| L. Since by using moreover Proposition 8 (4) and then Proposition 8 (3-a), we readily obtain Because of Proposition 8 (5-b) and (4.46), this yields the exponential decay of (4.43) to zero uniform in x, y ∈ [−s, s]. The same holds true for (4.44) by using the same arguments, up to the replacement of Proposition 8 (3-a) by (3-b), and hence the proof of Proposition 9 is complete.
Finally, we can now easily conclude.

Conclusion
Proof of Theorem 5. Recalling the integral representation (4.4) for K N (x, y), we split the contour Γ into two pieces Γ − and Γ + , where Γ − encircles the λ −1 j 's smaller than c N , and Γ + the λ −1 j 's larger than c N . Then, we deform Θ so that it encircles Γ − . This does not modify the value of the kernel. Indeed, there is no pole at w = λ −1 j and the residue picked at w = z reads and thus vanishes by Cauchy's theorem since the integrand is analytic. Now, we can deform by analyticity Γ + , Γ − and Θ into the respective contours Υ + , Υ − and Ξ provided by Proposition 8. Hence, K N (x, y) equals to and the uniform convergence (4.3) follows for every s > 0 from Proposition 7 and Proposition 9. The proof of Theorem 5 is therefore complete.

Hard edge expansion: proof of Theorem 6
In this section, we use basic properties of Fredholm determinants and trace class operators. We refer the reader to [32,16] for comprehensive introductions, see also [17,Section 4.2] for a quick overview. We also use well-known formulas and identities for the Bessel function that can be found in [14].

Preparation and proof of Theorem 6
Recalling the definition (2.33) of σ N and the definition (4.1) of the kernel K N , we consider here the scaled kernel x, y ∈ (0, s) , (5.1) and denote by K N the associated integral operator acting on L 2 (0, s). Thus, if x min is the smallest random eigenvalue of M N (see (2.32)), we have P N 2 σ N x min s = det I − K N L 2 (0,s) .
Recalling the definition (2.29) of the Bessel kernel, [17, Proposition 6.1] states that, for every s > 0, we have uniformly in x, y ∈ (0, s) as N → ∞, Notice that the limiting kernel appearing in the right hand side is exactly (2.30) without the pre-factor (y/x) α/2 , and hence is bounded on (0, s) × (0, s). The integral operator associated with this kernel is EK   To prove Theorem 6, the first step is to improve (5.3) by making explicit the 1/Ncorrection term, and showing the remainder is of order 1/N 2 . More precisely, with ζ N defined as in (2.33), we prove the following kernel expansion.

Proposition 10.
For every s > 0, we have uniformly in x, y ∈ (0, s) as N → ∞, Be (x, y) The proof of the proposition is deferred to Section 5.2. If Q is the integral operator on L 2 (0, s) with kernel Be (x, y) , (5.7) then Proposition 10 yields the operator expansion (for the trace class norm on L 2 (0, s)),  Since the right hand side of (5.9) equals Tr (I − K It is now easy to prove Theorem 6. Proof of Theorem 6. The identity (5.2) and the operator expansion (5.8) yield (5.10) By plugging the identity xJ α (x) = αJ α (x) − xJ α+1 (x) into (2.29), we obtain K (α) . (5.11) Combined with the asymptotic behaviour (which follows from the definition (2.28)), it is then easy to check from (5.7) that both EQ and QE −1 are trace class and thus, by  We now turn to the proof of the proposition.

Proof of the kernel expansion
Proof of Proposition 10. First, by using the representation (5.11) of the Bessel kernel and then the identity 2αJ α (x) = xJ α+1 (x) + xJ α−1 (x), we have Be (x, y) Thus, to prove Proposition 10 is equivalent to show uniformly in x, y ∈ (0, s) as N → ∞. To do so, let 0 < r < R < lim inf N λ 1 /2 and introduce the map Recall that Assumption 2 yields lim inf N λ 1 > 0 and thus one can choose a determination of the logarithm so that G N is well-defined and holomorphic on {z ∈ C , |z| < R + 1} for every N large enough. In the proof of [ , and hence we have the Taylor expansion see for instance [17,Eq. (164) and (165) N, (s) when 2. It would be interesting to identify these coefficients in terms of F α (s) if it is possible.