1 Introduction

It is a natural and central question, in mathematics and physics, to understand how the spectral properties of an operator are altered when the operator is subject to a small perturbation. This question is at the center of perturbation theory and has been studied in many different contexts. We refer the reader to Kato’s book [17] for a thorough account on this subject. In this text, we provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. More explicitly, let \(D_n\) be an \(n\times n\) Hermitian matrix, that, up to a change in basis, we suppose diagonal.Footnote 1 We denote by \(\mu _n\) the empirical spectral distribution of \(D_n\). This matrix is additively perturbed by a random Hermitian matrix \(\varepsilon _n X_n\) whose entries are chosen at random independently and scaled so that the operator norm of \(X_n\) has order one. We are interested in the empirical spectral distribution \(\mu _n^\varepsilon \) of

$$\begin{aligned} D_n^\varepsilon := D_n+\varepsilon _n X_n \end{aligned}$$

in the regime where the matrix size n tends to infinity and \(\varepsilon _n\) tends to 0. We shall prove that, depending on the order of magnitude of the perturbation, several regimes can appear. We suppose that \(\mu _n\) converges to a limiting measure \(\rho (\lambda )\mathrm {d}\lambda \) and that the variance profile of the entries of \( X_n\) has a macroscopic limit \(\sigma _{{\text {d}}}\) on the diagonal and \(\sigma \) elsewhere. We then prove that there is a deterministic function F and a Gaussian random linear form \(\mathrm {d}Z\) on the space of \(\mathcal {C}^6\) functions on \(\mathbb {R}\), both depending only on the limit parameters of the model \(\rho ,\sigma \) and \(\sigma _{{\text {d}}}\) such that if one defines the distribution \(\mathrm {d}F: \phi \longmapsto -\int \phi '(s)F(s)\mathrm {d}s\), then, for large n:

$$\begin{aligned} \mu _n^\varepsilon&\approx \mu _n+\frac{\varepsilon _n}{n}\mathrm {d}Z&\text { if }&\varepsilon _n\ll n^{-1} \end{aligned}$$
(1)
$$\begin{aligned} \mu _n^\varepsilon&\approx \mu _n+\frac{\varepsilon _n}{n}\left( c\mathrm {d}F + \mathrm {d}Z \right)&\text { if }&\varepsilon _n\sim \frac{c}{n} \end{aligned}$$
(2)
$$\begin{aligned} \mu _n^\varepsilon&\approx \mu _n+\varepsilon _n^2\mathrm {d}F&\text { if }&n^{-1}\ll \varepsilon _n\ll 1 \end{aligned}$$
(3)

and if, moreover, \( n^{-1}\ll \varepsilon _n\ll n^{-1/3},\) then convergence (2) can be refined as follows:

$$\begin{aligned} \mu _n^\varepsilon&\approx \mu _n+\varepsilon _n^2\mathrm {d}F+ \frac{\varepsilon _n}{n} \mathrm {d}Z . \end{aligned}$$
(4)

In Sect. 3, several figures show a very good matching of random simulations with these theoretical results. The definitions of the function F and of the process Z are given below in (6) and (7). In many cases, the linear form \(\mathrm {d}F\) can be interpreted as the integration with respect to the signed measure \(F'(x)\mathrm {d}x\). The function F is related to free probability theory, as explained in Sect. 4 below, whereas the linear form \(\mathrm {d}Z\) is related to the so-called one-dimensional Gaussian free field defined, for instance, at [14, Sect. 4.2]. If the variance profile of \(X_n\) is constant, then it is precisely the Laplacian of the Gaussian free field, defined in the sense of distributions.

The transition at \(\varepsilon _n\sim n^{-1}\) is the well-known transition, in quantum mechanics, where the perturbative regime ends. Indeed, one can distinguish the two following regimes:

  • The regime \(\varepsilon _n\ll n^{-1}\), called the perturbative regime (see [15]): the size of the perturbation (i.e. its operator norm) is much smaller than the typical spacing between two consecutive eigenvalues (level spacing), which is of order \(n^{-1}\) in our setting.

  • The regime \(n^{-1}\ll \varepsilon _n\ll 1\), sometimes called the semi-perturbative regime, where the size of the perturbation is not small compared to the level spacing. This regime concerns many applications [1, 19] in the context of covariance matrices and applications to finance.

A surprising fact discovered during this study is that the semi-perturbative regime \(n^{-1}\ll \varepsilon _n\ll 1\) decomposes into infinitely many sub-regimes. In the case \(n^{-1}\ll \varepsilon _n\ll n^{-1/3}\), the expansion of \(\mu _n^\varepsilon -\mu _n\) contains a single deterministic term before the random term \(\frac{\varepsilon _n}{n}\mathrm {d}Z\). In the case \(n^{-1/3}\ll \varepsilon _n\ll n^{-1/5}\), the expansion of \(\mu _n^\varepsilon -\mu _n\) contains two of them. More generally, for all positive integer p, when \(n^{-1/(2p-1)}\ll \varepsilon _n\ll n^{-1/(2p+1)}\), the expansion contains p of them. For computational complexity reasons, the only case we state explicitly is the first one. We refer the reader to Sect. 6.5 for a discussion around this point.

In the papers [1,2,3,4, 23], Wilkinson, Walker, Allez, Bouchaud et al. have investigated some problems related to this one. Some of these works were motivated by the estimation of a matrix out of the observation of its noisy version. Our paper differs from these ones mainly by the facts that firstly, we are interested in the perturbations of the global empirical distribution of the eigenvalues and not of a single one, and secondly, we push our expansion up to the random term, which does not appear in these papers. Besides, the noises they consider have constant variance profiles (either a Wigner-Dyson noise in the four first cited papers or a rotationally invariant noise in the fifth one). The transition at \(\varepsilon _n\sim n^{-1}\) between the perturbative and the semi-perturbative regimes is already present in these texts. They also consider the transition between the perturbative regime \(\varepsilon _n\ll 1\) and the non-perturbative regime \(\varepsilon _n\asymp 1\). As explained above, we exhibit the existence of an infinity of sub-regimes in this transition and focus on \(\varepsilon _n\ll 1\) for the first order of the expansion and to \(\varepsilon _n\ll n^{-1/3}\) for the second (and last) order. The study of other sub-regimes is postponed to forthcoming papers.

The paper is organized as follows. Results, examples and comments are given in Sects. 2–4, while the rest of the paper, including an appendix, is devoted to the proofs, except for Sect. 6.5, where we discuss the sub-regimes mentioned above.

Notations For \(a_n,b_n\) some real sequences, \(a_n\ll b_n\) (resp. \(a_n\sim b_n\)) means that \(a_n/b_n\) tends to 0 (resp. to 1). Also, \({\mathop {\longrightarrow }\limits ^{P}}\) and \({\mathop {\longrightarrow }\limits ^{\mathrm {dist.}}}\) stand, respectively, for convergence in probability and convergence in distribution for all finite marginals.

2 Main Result

2.1 Definition of the Model and Assumptions

For all positive integer n, we consider a real diagonal matrix \(D_n ={\text {diag}}(\lambda _n(1), \ldots , \lambda _n(n))\), as well as a Hermitian random matrix

$$\begin{aligned} X_n=\frac{1}{\sqrt{n}}[x^n_{i,j}]_{1\le i,j\le n} \end{aligned}$$

and a positive number \(\varepsilon _n\). The normalizing factor \( n^{-1/2}\) and our hypotheses below ensure that the operator norm of \(X_n\) is of order one. We then define, for all n,

$$\begin{aligned} D_n^\varepsilon :=D_n+ \varepsilon _n X_n. \end{aligned}$$

We now introduce the probability measures \(\mu _n\) and \(\mu _n^\varepsilon \) as the respective uniform distributions on the eigenvalues (with multiplicity) of \(D_n\) and \(D_n^\varepsilon \). Our aim is to give a perturbative expansion of \(\mu _n^\varepsilon \) around \(\mu _n\).

We make the following hypotheses:

  1. (a)

    the entries \(x^n_{i,j}\) of \(\sqrt{n}X_n\) are independent (up to symmetry) random variables, centered, with variance denoted by \(\sigma _n^2(i,j)\), such that \(\mathbb {E}|x^n_{i,j}|^{8} \) is bounded uniformly on n, i, j,

  2. (b)

    there are \(f,\sigma _{{\text {d}}},\sigma \) real functions defined, respectively, on [0, 1], [0, 1] and \([0,1]^2\) such that, for each \(x\in [0,1]\),

    $$\begin{aligned} \lambda _n(\lfloor nx\rfloor )\underset{n\rightarrow \infty }{\longrightarrow }f(x)\quad \text { and }\quad \sigma _n^2(\lfloor nx\rfloor ,\lfloor nx\rfloor )\underset{n\rightarrow \infty }{\longrightarrow }\sigma _{{\text {d}}}(x)^2 \end{aligned}$$

    and for each \(x\ne y\in [0,1]\),

    $$\begin{aligned} \sigma _n^2(\lfloor nx\rfloor ,\lfloor ny\rfloor )\underset{n\rightarrow \infty }{\longrightarrow }\sigma ^2(x,y). \end{aligned}$$

    We make the following hypothesis about the rate of convergence:

    $$\begin{aligned} \eta _n := & {} \max \{n\varepsilon _n,1\}\times \sup _{1\le i\ne j\le n}( |\sigma _n^2(i,j)\\&-\sigma ^2(i/n,j/n)|+|\lambda _n(i)-f(i/n)|)\quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0. \end{aligned}$$

Let us now make some assumptions on the limiting functions \(\sigma \) and f:

  1. (c)

    the function f is bounded and the push-forward of the uniform measure on [0, 1] by the function f has a density \(\rho \) with respect to the Lebesgue measure on \(\mathbb {R}\) and a compact support denoted by \(\mathcal {S}\),

  2. (d)

    the variance of the entries of \(X_n\) essentially depends on the eigenspaces of \(D_n\), namely there exists a symmetric function \(\tau (\,\cdot \,,\,\cdot \,)\) on \(\mathbb {R}^2\) such that for all \(x\ne y\), \(\sigma ^2(x,y)=\tau (f(x),f(y))\),

  3. (e)

    the following regularity property holds: there exist \(\eta _0>0, \alpha >0\) and \(C<\infty \) such that for almost all \(s\in \mathbb {R}\), for all \(t\in [s-\eta _0, s+\eta _0]\),   \(|\tau (s,t)\rho (t)-\tau (s,s)\rho (s)|\le C|t-s|^\alpha \).

We add a last assumption which strengthens assumption (c) and makes it possible to include the case where the set of eigenvalues of \(D_n\) contains some outliers:

  1. (f)

    there is a real compact set \(\widetilde{\mathcal {S}}\) such that

    $$\begin{aligned}\max _{1\le i\le n}{\text {dist}}(\lambda _n(i), \widetilde{\mathcal {S}})\underset{n\rightarrow \infty }{\longrightarrow }0.\end{aligned}$$

Remark 1

(About the hypothesis that \(D_n\) is diagonal)

  1. (i)

    If the perturbing matrix \(X_n\) belongs to the GOE (resp. to the GUE), then its law is invariant under conjugation by any orthogonal (resp. unitary) matrix. It follows that in this case, our results apply to any real symmetric (resp. Hermitian) matrix \(D_n\) with eigenvalues \(\lambda _n(i)\) satisfying the above hypotheses.

  2. (ii)

    As explained after Proposition 2 below, we conjecture that when the variance profile of \(X_n\) is constant, for \(\varepsilon _n\gg n^{-1}\), we do not need the hypothesis that \(D_n\) is diagonal neither. However, if the perturbing matrix does not have a constant variance profile, then for a non-diagonal \(D_n\) and \(\varepsilon _n\gg n^{-1}\), the spectrum of \(D_n^\varepsilon \) should depend heavily on the relation between the eigenvectors of \(D_n\) and the variance profile, which implies that our results should not remain true.

  3. (iii)

    At last, it is easy to see that the random process \((Z_\phi )\) introduced at (7) satisfies, for any test function \(\phi \),

    $$\begin{aligned} \frac{1}{\varepsilon _n}\sum _{i=1}^n\left( \phi (\lambda _n(i)+\frac{\varepsilon _n}{\sqrt{n}}x_{ii})-\phi (\lambda _n(i))\right) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z_\phi . \end{aligned}$$

Thus, regardless of the variance profile, the convergence of (8) rewrites, informally,

$$\begin{aligned} \mu _n^\varepsilon =\frac{1}{n}\sum _{i=1}^n\delta _{\lambda _n(i)+(\varepsilon _n/\sqrt{n})x_{ii}}+o(\varepsilon _n/n). \end{aligned}$$
(5)

A so simple expression, up to a \(o(\varepsilon _n/n)\) error, of the empirical spectral distribution of \(D_n^\varepsilon \), with some independent translations \(\frac{\varepsilon _n}{\sqrt{n}}x_{ii}\), should not remain true without the hypothesis that \(D_n\) is diagonal or that the distribution of \(X_n\) is invariant under conjugation.

2.2 Main Result

Recall that the Hilbert transform, denoted by H[u], of a function u, is the function

$$\begin{aligned} H[u](s):={\text {p.v.}}\int _{t\in \mathbb {R}}\frac{u(t)}{s-t}\mathrm {d}t \end{aligned}$$

and define the function

$$\begin{aligned} F(s) = -\rho (s) H[\tau (s,\cdot )\rho (\cdot )] (s). \end{aligned}$$
(6)

Note that, by assumptions (c) and (e), F is well defined and supported by \(\mathcal {S}\). Besides, for any \(\phi \) supported on an interval where F is \(\mathcal {C}^1\),

$$\begin{aligned} -\int \phi '(s)F(s)\mathrm {d}s=\int \phi (s)\mathrm {d}F(s), \end{aligned}$$

where \(\mathrm {d}F(s)\) denotes the measure \(F'(s)\mathrm {d}s\).

We also introduce the centered Gaussian field, \((Z_\phi )_{\phi \in \mathcal {C}^6}\), indexed by the set of \(\mathcal {C}^6\) complex functions on \(\mathbb {R}\), with covariance defined by

$$\begin{aligned} \mathbb {E}Z_\phi Z_\psi = \int _0^1\sigma _{{\text {d}}}(t)^2\phi '(f(t)) \psi '(f(t))\mathrm {d}t\qquad \text { and }\qquad \overline{Z_\psi } = Z_{\overline{\psi }} . \end{aligned}$$
(7)

Note that the process \((Z_\phi )_{\phi \in {\mathcal {C}}^6}\) can be represented, for \((B_t)\) is the standard one-dimensional Brownian motion, as

$$\begin{aligned} Z_\phi = \int _0^1 \sigma _{{\text {d}}}(t) \phi '(f(t)) \mathrm {d}B_t. \end{aligned}$$

Theorem 1

For all compactly supported \(\mathcal {C}^6\) function \(\phi \) on \(\mathbb {R}\), the following convergences hold:

  • Perturbative regime if \(\varepsilon _n\ll n^{-1}\), then,

    $$\begin{aligned} n\varepsilon _n^{-1}(\mu _n^\varepsilon -\mu _n)(\phi ) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z_\phi .\end{aligned}$$
    (8)

  • Critical regime if \(\varepsilon _n\sim c/n\), with c constant, then,

    $$\begin{aligned} n\varepsilon _n^{-1}(\mu _n^\varepsilon -\mu _n)(\phi ) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad -c\int \phi '(s)F(s)\mathrm {d}s+Z_\phi .\end{aligned}$$
    (9)

  • Semi-perturbative regime if \( n^{-1}\ll \varepsilon _n\ll 1\), then,

$$\begin{aligned} \varepsilon _n^{-2}(\mu _n^\varepsilon -\mu _n)(\phi )\quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{P}}\quad -\int \phi '(s)F(s)\mathrm {d}s, \end{aligned}$$
(10)

and if, moreover, \( n^{-1}\ll \varepsilon _n\ll n^{-1/3}\), then,

$$\begin{aligned} n\varepsilon _n^{-1}\left( (\mu _n^\varepsilon -\mu _n)(\phi ) + \varepsilon _n^2\int \phi '(s)F(s)\mathrm {d}s\right) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z_\phi . \end{aligned}$$
(11)

Remark 2

(Sub-regimes for \(n^{-1/3}\ll \varepsilon _n\ll 1\)) In the semi-perturbative regime, the reason why we provide an expansion up to a random term, only for \(\varepsilon _n\ll n^{-1/3}\), is that the study of the regime \(n^{-1/3}\ll \varepsilon _n\ll 1\) up to such a precision requires further terms in the expansion of the resolvent of \(D_n^\varepsilon \) that make appear, beside \(\mathrm {d}F\), additional deterministic terms of smaller order, which are much larger than the probabilistic term containing \(Z_\phi \). The computation becomes rather intricate without any clear recursive formula. As we will see in Sect. 6.5, there are infinitely many regimes. Precisely, for any positive integer p, when \(n^{-1/(2p-1)}\ll \varepsilon _n\ll n^{-1/(2p+1)}\), there are p deterministic terms in the expansion before the term in \(Z_\phi \).

Remark 3

(Local law) The approximation

$$\begin{aligned} \displaystyle \mu _n^\varepsilon (I) \approx \mu _n (I)+ \varepsilon _n^2\int _I\mathrm {d}F \end{aligned}$$

of (10) should stay true even for intervals I with size tending to 0 as the dimension n grows, as long as the size of I stays much larger than the right-hand side term of (30), as can be seen from Proposition 5.

Remark 4

The second part of Hypothesis (b), concerning the speed of convergence of the profile of the spectrum of \(D_n\) as well as of the variance of its perturbation, is needed in order to express the expansion of \(\mu _n^\varepsilon -\mu _n\) in terms of limit parameters of the model \( \sigma \) and \(\rho \). We can remove this hypothesis and get analogous expansions where the terms \(\mathrm {d}F\) and \(\mathrm {d}Z\) are replaced by their discrete counterparts \(\mathrm {d}F_n\) and \(\mathrm {d}Z_n\), defined thanks to the “finite n” empirical versions of the limit parameters \(\sigma \) and \(\rho \).

3 Examples

3.1 Uniform Measure Perturbation by a Band Matrix

Here, we consider the case where \(f(x)=x\), \(\sigma _{{\text {d}}}(x)\equiv m\) and \(\sigma (x,y)=\mathbb {1}_{|y-x|\le \ell }\), for some constants \(m\ge 0\) and \(\ell \in [0,1]\) (the relative width of the band). In this case, \(\tau (\,\cdot \,,\,\cdot \,)=\sigma (\,\cdot \,,\,\cdot \,)^2\), hence

$$\begin{aligned} F(s) =\mathbb {1}_{(0,1)}(s) {\text {p.v.}}\int _{t}\frac{\tau (s,t)}{s-t}\mathrm {d}t =- \mathbb {1}_{(0,1)}(s)\log \frac{\ell \wedge (1-s)}{\ell \wedge s} \end{aligned}$$
(12)

and \((Z_\phi )_{\phi \in \mathcal {C}^6}\) is the centered complex Gaussian process with covariance defined by

$$\begin{aligned} {\mathbb {E}}Z_\phi \overline{Z_\psi } = m^2 \int _0^1 \phi '(t) \ \overline{\psi '(t)} \ \mathrm {d}t \qquad \text { and }\qquad \overline{Z_\psi }= Z_{\overline{\psi }}. \end{aligned}$$

Theorem 1 is then illustrated in Fig. 1, where we plotted the cumulative distribution functions.

Fig. 1
figure 1

Deforming the uniform distribution by a band matrix. Cumulative distribution function of \(\varepsilon _n^{-2}(\mu _n^\varepsilon -\mu _n)\) (in blue) and function \(F(\,\cdot \,)\) of (12) (in red). The non-smoothness of the blue curves results of the noise term \(Z_\phi \) in Theorem 1. Each graphic is realized thanks to one single matrix (no averaging) perturbed by a real Gaussian band matrix. a\(n=10^4\), \(\varepsilon _n=n^{-0.4}\), \(\ell =0.2\), b\(n=10^4\), \(\varepsilon _n=n^{-0.4}\), \(\ell =0.8\) (Color figure online)

Full size image

3.2 Triangular Pulse Perturbation by a Wigner Matrix

Here, we consider the case where \(\rho (x)=(1-|x|)\mathbb {1}_{[-1,1]}(x)\), \(\sigma _d \equiv m\), for some real constant m, and \(\sigma \equiv 1\) (what follows can be adapted to the case \(\sigma (x,y)=\mathbb {1}_{|y-x|\le \ell }\), with a bit longer formulas). In this case, thanks to the formula (9.6) of \(H[\rho (\,\cdot \,)]\) given p. 509 of [18], we get

$$\begin{aligned} F(s)=(1-|s|)\mathbb {1}_{[-1,1]}(s)\left\{ (1-s)\log (1-s)-(1+s)\log (1+s)+2s\log |s|\right\} , \end{aligned}$$
(13)

and the covariance of \((Z_\phi )_{\phi \in \mathcal {C}^6}\) is given by

$$\begin{aligned} {\mathbb {E}}Z_\phi \overline{Z_\psi } = m^2 \int _{-1}^1 (1-|t|) \ \phi '(t) \ \overline{\psi '(t)} \ \mathrm {d}t \qquad \text { and }\qquad \overline{Z_\psi }= Z_{\overline{\psi }}. \end{aligned}$$

Theorem 1 is then illustrated in Fig. 2 in the case where \(\varepsilon _n\gg n^{-1/2}\). In Fig. 2, we implicitly use some test functions of the type \(\phi (x)=\mathbb {1}_{x\in I}\) for some intervals I. These functions are not \(\mathcal {C}^6\), and one can easily see that for \(\varepsilon _n\ll n^{-1/2}\), Theorem 1 cannot work for such functions. However, considering imaginary parts of Stieltjes transforms, i.e. test functions

$$\begin{aligned}\displaystyle \phi (x)=\frac{1}{\pi }\frac{\eta }{(x-E)^2+\eta ^2}\qquad (E\in \mathbb {R}, \eta >0)\end{aligned}$$

give a perfect matching between the predictions from Theorem 1 and numerical simulations, also for \(\varepsilon _n\ll n^{-1/2}\) (see Fig. 3, where we use Proposition 4 and (17) to compute the theoretical limit).

Fig. 2
figure 2

Triangular pulse perturbation by a Wigner matrix: density and cumulative distribution function. Top left: cumulative distribution function of \(\varepsilon _n^{-2}(\mu _n^\varepsilon -\mu _n)\)(in blue) and function \(F(\,\cdot \,)\) of (13) (in red). Top right and bottom: density \(\rho \) (red dashed line), histogram of the eigenvalues of \(D_n^\varepsilon \) (in black) and theoretical density \(\rho +\varepsilon _n^2F'(s)\) of the eigenvalues of \(D_n^\varepsilon \) as predicted by Theorem 1 (in blue). Here, \(n= 10^4\) and \(\varepsilon _n=n^{-\alpha }\), with \(\alpha =0.25\) (up left), \(\alpha =0.4\) (up right), 0.25 (bottom left) and 0.1 (bottom right) (Color figure online)

Full size image
Fig. 3
figure 3

Triangular pulse perturbation by a Wigner matrix: Stieltjes transform. Imaginary part of the Stieltjes transform of \(\varepsilon _n^{-2}(\mu _n^{\varepsilon } - \mu _n)\) (in blue) and of the measure \(\mathrm {d}F\) (in red) at \(z=E+\mathrm {i}\) as a function of the real part E for different values of \(\varepsilon _n\). Here, \(n= 10^4\) and \(\varepsilon _n=n^{-\alpha }\), with \(\alpha =0.2\), 0.5 and 0.8 (from left to right) (Color figure online)

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3.3 Parabolic Pulse Perturbation by a Wigner Matrix

Here, we consider the case where \(\rho (x)=\frac{3}{4}(1-x^2)\mathbb {1}_{[-1,1]}(x)\), \(\sigma _d \equiv m\), for some real constant m, and \(\sigma \equiv 1\) (again, this can be adapted to the case \(\sigma (x,y)=\mathbb {1}_{|y-x|\le \ell }\)). Theorem 1 is then illustrated in Fig. 4. In this case, thanks to the formula (9.10) of \(H[\rho (\,\cdot \,)]\) given p. 509 of [18], we get

$$\begin{aligned} F(s)= -\frac{9}{16}(1-s^2)\mathbb {1}_{[-1,1]}(s)\left\{ 2s - (1-s^2) \ln \left| \frac{s-1}{s+1} \right| \right\} \end{aligned}$$
(14)

and the covariance of \((Z_\phi )_{\phi \in \mathcal {C}^6}\) is given by

$$\begin{aligned} {\mathbb {E}}Z_\phi \overline{Z_\psi } = \frac{3m^2}{4} \int _{-1}^1 (1-t^2) \ \phi '(t) \ \overline{\psi '(t)} \ \mathrm {d}t\qquad \text { and }\qquad \overline{Z_\psi }= Z_{\overline{\psi }}. \end{aligned}$$
Fig. 4
figure 4

Parabolic pulse perturbation by a Wigner matrix. Top left: cumulative distribution function of \(\varepsilon _n^{-2}(\mu _n^\varepsilon -\mu _n)\)(in blue) and function \(F(\,\cdot \,)\) of (14) (in red). Top right and bottom: density \(\rho \) (red dashed line), histogram of the eigenvalues of \(D_n^\varepsilon \) (in black) and theoretical density \(\rho +\varepsilon _n^2F'(s)\) of the eigenvalues of \(D_n^\varepsilon \) as predicted by Theorem 1 (in blue). Here, \(n= 10^4\) and \(\varepsilon _n=n^{-\alpha }\), with \(\alpha =0.25\) (up left), \(\alpha =0.4\) (up right), 0.2 (bottom left) and 0.18 (bottom right) (Color figure online)

Full size image

4 Relation to Free Probability Theory

Let us now explain how this work is related to free probability theory. If, instead of letting \(\varepsilon _n\) tend to zero, one considers the model

$$\begin{aligned} D_n^t : = D_n+ \sqrt{t} X_n \end{aligned}$$

for a fixed \(t>0\), then, by [5, 12, 13, 22], the empirical eigenvalue distribution of \(D_n^t\) has a limit as \(n\rightarrow \infty \), that we shall denote here by \(\mu _t\). The law \(\mu _t\) can be interpreted as the law of the sum of two elements in a non-commutative probability space which are free with an amalgamation over a certain sub-algebra (see [22] for more details). The following proposition relates the function F from (6) to the first order expansion of \(\mu _t\) around \(t=0\).

Proposition 2

For any \(z\in \mathbb {C}\backslash \mathbb {R}\), we have

$$\begin{aligned} \frac{\partial }{\partial t}_{|t=0} \int \frac{\mathrm {d}\mu _t(\lambda )}{z-\lambda }=-\int \frac{F(\lambda )}{(z-\lambda )^2}\mathrm {d}\lambda =-\int F(\lambda ) \frac{\partial }{\partial \lambda }\left( \frac{1}{z-\lambda } \right) \mathrm {d}\lambda . \end{aligned}$$

This is related to the fact that in Eqs. (1)–(4), for \(\varepsilon _n\) large enough, the term \(\varepsilon _n^2\mathrm {d}F\) is the leading term.

In the particular case where \(X_n\) is a Wigner matrix, \(\mu _t\) is the free convolution of the measure \(\rho (\lambda )\mathrm {d}\lambda \) with a semicircle distribution and admits a density \(\rho _t\), by [8, Cor. 2]. Then, Theorem 1 makes it possible to formally recover the free Fokker–Planck equation with null potential:

$$\begin{aligned} {\left\{ \begin{array}{ll}\frac{\partial }{\partial t}\rho _t(s)+\frac{\partial }{\partial s}\{\rho _t(s)H[\rho _t ](s)\}=0,\\ \rho _0(s)=\rho (s),\end{array}\right. } \end{aligned}$$

where \(H[\rho _t ]\) denotes the Hilbert transform of \(\rho _t\). This equation is also called McKean-Vlasov (or Fokker–Planck) equation with logarithmic interaction (see [9,10,11]).

Note also that when \(X_n\) is a Wigner matrix, the hypothesis that \(D_n\) is diagonal is not required to have the convergence of the empirical eigenvalue distribution of \(D_n^t\) to \(\mu _t\) as \(n\rightarrow \infty \). This suggests that, even for non-diagonal \(D_n\), the convergence of (10) still holds when \(X_n\) is a Wigner matrix.

Proof of Proposition 2

By [22, Th. 4.3], we have

$$\begin{aligned} \int \frac{\mathrm {d}\mu _t(\lambda )}{z-\lambda }=\int _{x=0}^1C_t(x,z)\mathrm {d}x, \end{aligned}$$
(15)

where \(C_t(x,z)\) is bounded by \(|\mathfrak {Im}z|^{-1}\) and satisfies the fixed-point equation

$$\begin{aligned} C_t(x,z)=\frac{1}{z-f(x)-t\int _{y=0}^1 \sigma ^2(x,y)C_t(y,z)\mathrm {d}y}. \end{aligned}$$

Hence as \(t\rightarrow 0\), \(C_t(x, z)\longrightarrow \frac{1}{z-f(x)}\) uniformly in x. Thus

$$\begin{aligned} C_t(x,z)-\frac{1}{z-f(x)}= & {} \frac{t\int _{y=0}^1\sigma ^2(x,y)C_t(y,z)\mathrm {d}y}{(z-f(x)-t\int _{y=0}^1\sigma ^2(x,y)C_t(y,z)\mathrm {d}y)(z-f(x))}\\= & {} t\frac{1}{(z-f(x))^2} \int _{y=0}^1\sigma ^2(x,y)C_t(y,z)\mathrm {d}y+o(t)\\= & {} t\frac{1}{(z-f(x))^2} \int _{y=0}^1\frac{\sigma ^2(x,y)}{z-f(y)}\mathrm {d}y+o(t) \end{aligned}$$

where each o(t) is uniform in \(x\in [0,1]\). Then, by (15), we deduce that

$$\begin{aligned} \frac{\partial }{\partial t}_{|t=0} \int \frac{\mathrm {d}\mu _t(\lambda )}{z-\lambda }= \int _{(x,y)\in [0,1]^2} \frac{\sigma ^2(x,y)}{(z-f(x))^2(z-f(y))}\mathrm {d}x\mathrm {d}y . \end{aligned}$$

The right-hand side term of the previous equation is precisely the number B(z) introduced at (17) below. Then, one concludes using Proposition 4 from Sect. 6.1. \(\square \)

5 Strategy of the Proof

We shall first prove the convergence results of Theorem 1 for test functions \(\phi \) of the form \(\varphi _z(x):=\frac{1}{z-x}\). This is done in Sect. 6 by writing an expansion of the resolvent of \(D_n^\varepsilon \).

Once we have proved that the convergences hold for the resolvent of \(D_n^\varepsilon \), we can extend them to the larger class of compactly supported \({\mathcal {C}}^6\) functions on \(\mathbb {R}\).

In Sect. 7, we use the Helffer–Sjöstrand formula to extend the convergence in probability in the semi-perturbative regime (10) to the case of compactly supported \({\mathcal {C}}^6\) functions on \(\mathbb {R}\).

In Sect. 8, the convergences in distribution (8), (9) and (11) are proved in two steps. The overall strategy is to apply an extension lemma of Shcherbina and Tirozzi which states that a CLT that applies to a sequence of centered random linear forms on some space can be extended, by density, to a larger space, as long as the variance of the image of these random linear forms by a function \(\phi \) of the larger space is uniformly bounded by the norm of \(\phi \). Therefore, our task is twofold. We need first to prove that the sequences of variables involved in the convergences (8), (9) and (11) can be replaced by their centered counterparts \(n\varepsilon _n^{-1}(\mu _n^\varepsilon (\phi )-\mathbb {E}[\mu _n^\varepsilon (\phi )])\) (i.e. they differ by o(1)). In a second step, we dominate the variance of these latter variables, in order to apply the extension lemma which is precisely stated in Appendix as Lemma 10.

6 Stieltjes Transforms Convergence

As announced in the previous section, we start with the proof of Theorem 1 in the special case of test functions of the type \(\varphi _z:= \frac{1}{z-x}\). We decompose it into two propositions. Their statement and proof are the purpose of the three following subsections. The two last Sects. 6.4 and 6.5 are devoted, respectively, to a local type convergence result and to a discussion about the possibility of an extension of the expansion result to a wider range of rate of convergence of \(\varepsilon _n\), namely beyond \(n^ {-1/3}\).

6.1 Two Statements

Let denote, for \(z\in \mathbb {C}\backslash \mathbb {R}\),

$$\begin{aligned} Z(z):=Z_{\varphi _z}\quad \text { for }\quad \varphi _z(x):=\frac{1}{z-x} \end{aligned}$$
(16)

where \((Z_\phi )_{\phi \in \mathcal {C}^6}\) is the Gaussian field with covariance defined by (7). We also introduce, for \(z\in \mathbb {C}\backslash \mathbb {R}\),

$$\begin{aligned} B(z):=\int _{(s,t)\in [0,1]^2}\frac{\sigma ^2(s,t)}{(z-f(s))^2(z-f(t))}\mathrm {d}s\mathrm {d}t \end{aligned}$$
(17)

and

$$\begin{aligned} \Delta \mathrm {G}_n(z) := (\mu _n^\varepsilon -\mu _n)(\varphi _z)=\frac{1}{n}{\text {Tr}}\frac{1}{z-D_n^\varepsilon }-\frac{1}{n}{\text {Tr}}\frac{1}{z-D_n}. \end{aligned}$$
(18)

Proposition 3

Under Hypotheses (a), (b), (f),

  • if \(\varepsilon _n\ll n^{-1}\), then for all \(z\in \mathbb {C}\backslash \mathbb {R}\),

    $$\begin{aligned} n\varepsilon _n^{-1}\Delta \mathrm {G}_n(z) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z(z)\, \end{aligned}$$
    (19)

  • if \(\varepsilon _n\sim c/n\), with c constant, then for all \(z\in \mathbb {C}\backslash \mathbb {R}\)

    $$\begin{aligned} n\varepsilon _n^{-1}\Delta \mathrm {G}_n(z) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad cB(z)+Z(z) \, , \end{aligned}$$
    (20)

  • if \( n^{-1}\ll \varepsilon _n\ll n^{-1/3}\), then for all \(z\in \mathbb {C}\backslash \mathbb {R}\)

    $$\begin{aligned} n\varepsilon _n^{-1} \left( \Delta \mathrm {G}_n(z)-\varepsilon _n^2B(z)\right) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z(z)\,. \end{aligned}$$
    (21)

  • if \( n^{-1}\ll \varepsilon _n\ll 1\), then for all \(z\in \mathbb {C}\backslash \mathbb {R}\),

    $$\begin{aligned} \varepsilon _n^{-2}\Delta \mathrm {G}_n(z) - B(z) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{P}}\quad 0 \,. \end{aligned}$$
    (22)

Remark

Note that (20) is merely an extension of (21) in the critical regime.

The following statement expresses B(z) as the image of a \(\varphi _z\) by a linear form. So, in the expansion of the previous proposition, both quantities Z(z) and B(z) depend linearly on \(\varphi _z\). Note that as F vanishes at \(\pm \infty \), Proposition 4 does not contradicts the fact that as |z| gets large, \(B(z)=O(|z|^{-3})\).

Proposition 4

Under Hypotheses (c), (d), (e), for any \(z\in \mathbb {C}\backslash \mathcal {S}\), for F defined by (6),

$$\begin{aligned} B(z)=-\int \frac{F(s)}{(z-s)^2}\mathrm {d}s= -\int \varphi _z'(s)F(s)\mathrm {d}s. \end{aligned}$$

6.2 Proof of Proposition 3

The proof is based on a perturbative expansion of the resolvent \(\frac{1}{n} {\text {Tr}}\frac{1}{z-D_n^\varepsilon }\). To make notations lighter, we shall sometimes suppress the subscripts and superscripts n, so that \(D_n^\varepsilon \), \(D_n\), \(X_n\) and \(x_{i,j}^n\) will be, respectively, denoted by \(D^\varepsilon \), D, X and \(x_{i,j}\). Let us fix \(z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}\). We can deduce from the expansion of the resolvent of \(D^\varepsilon \):

$$\begin{aligned} \Delta \mathrm {G}_n(z) = A_n(z)+B_n(z)+C_n(z)+R_n^\varepsilon (z), \end{aligned}$$

with

$$\begin{aligned} A_n(z):= & {} \frac{\varepsilon _n}{n} {\text {Tr}}\frac{1}{z-D}X\frac{1}{z-D}= \frac{\varepsilon _n}{n} \frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{x_{i,i}}{(z-\lambda _n(i))^2} \\ B_n(z):= & {} \frac{\varepsilon _n^2}{n}{\text {Tr}}\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}=\frac{\varepsilon _n^2}{n^2}\sum _{i,j}\frac{|x_{i,j}|^2}{(z-\lambda _n(i))^2(z-\lambda _n(j))} \\ C_n(z):= & {} \frac{\varepsilon _n^3}{n}{\text {Tr}}\frac{1}{z-D} X \frac{1}{z-D}X\frac{1}{z-D}X \frac{1}{z-D} \\= & {} \frac{\varepsilon _n^3}{n^{5/2}} \sum _{i,j,k=1}^n \frac{x_{i,j} \ x_{j,k} \ x_{k,i}}{(z-\lambda _n(i))^2 \ (z-\lambda _n(j)) \ (z-\lambda _n(k))} \\ R_n^\varepsilon (z):= & {} \frac{\varepsilon _n^4}{n}{\text {Tr}}\frac{1}{z-D} X \frac{1}{z-D} X \frac{1}{z-D}X\frac{1}{z-D}X \frac{1}{z-D^\varepsilon }. \end{aligned}$$

The purpose of the four following claims is to describe the asymptotic behavior of each of these four terms.

Claim 1

The finite dimension marginals of the centered process

$$\begin{aligned} (n\varepsilon _n^{-1} A_n(z))_{z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}} \end{aligned}$$

converge in distribution to those of the centered Gaussian process \((Z(z))_{z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}}\). Besides, there is \(C>0\) such that for any \(z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}\),

$$\begin{aligned} \mathbb {E}[ |n\varepsilon _n^{-1}A_n(z)|^2]\le \frac{C}{{\text {dist}}(z, \widetilde{\mathcal {S}})^4}. \end{aligned}$$
(23)

Proof

Estimate (23) follows from

$$\begin{aligned} \mathbb {E}[ |A_n(z)|^2] = \frac{\varepsilon _n^{2}}{n^3} \sum _{i=1}^n \frac{{\mathbb {E}}\left[ |x_{i,i}|^2\right] }{|z-\lambda _n(i)|^4} \le \frac{\varepsilon _n^{2}}{n^3} \sum _{i=1}^n \frac{\sigma _n^2(i,i)}{{\text {dist}}(z,\widetilde{\mathcal {S}})^4} \end{aligned}$$

and from the existence of a uniform upper bound for \(\sigma _n^2(i,i)\) which comes from Hypothesis (a) which stipulates that the 8-th moments of the entries \(x_{i,j}\) are uniformly bounded.

We turn now to the proof of the convergence in distribution of \(n\varepsilon _n^{-1}A_n(z)\) which actually does not depend on the sequence \((\varepsilon _n)\). For all \(\alpha _1,\beta _1,\dots ,\alpha _p,\beta _p \in \mathbb {C}\) and for all \(z_1,\dots ,z_p \in \mathbb {C}\backslash \widetilde{\mathcal {S}}\),

$$\begin{aligned} \sum _{i=1}^p \alpha _i \left( n\varepsilon _n^{-1} A_n(z_i) \right) + \beta _i \overline{\left( n\varepsilon _n^{-1} A_n(z_i) \right) } = \frac{1}{\sqrt{n}} \sum _{j=1}^n x_{j,j} \left( \sum _{i=1}^p \xi _n(i,j) \right) \end{aligned}$$

for \(\displaystyle { \xi _n(i,j) = \frac{\alpha _i}{(z_i-\lambda _n(j))^2} + \frac{\beta _i}{(\overline{z_i}-\lambda _n(j))^2} }\).

On one the hand, by dominated convergence, the covariance matrix of the above two dimensional random vector converges.

On the other hand, \(\mathbb {E}|x_{i,j}|^4\) is uniformly bounded in i, j and n, by Hypothesis (a). Moreover, for n large enough, for all i, j,

$$\begin{aligned} |\xi _n(i,j)|\le 2\max _{1\le i\le p}(|\alpha _i|+|\beta _i|)\times (\min _{1\le i\le p} {\text {dist}}(z_i, \mathcal {S}))^{-1}. \end{aligned}$$

Hence, the conditions of Lindeberg central limit theorem are satisfied and the finite dimension marginals of the process \((n\varepsilon _n^{-1} A_n(z))_{z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}}\) converge in distribution to those of the centered Gaussian process \((Z_z)_{z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}}\) defined by its covariance structure

$$\begin{aligned} \mathbb {E}\left( Z(z) \overline{Z({z'}})\right)= & {} \lim _{n\rightarrow \infty }{\mathbb {E}}\left[ \left( n\varepsilon _n^{-1} A_n(z)\right) . \left( n\varepsilon _n^{-1} \overline{A_n(z')}\right) \right] \\= & {} \lim _{n\rightarrow \infty } \frac{1}{n} \sum _{i,j = 1}^n \frac{{\mathbb {E}}\left[ x_{i,i} \ \overline{x_{j,j}} \right] }{(z-\lambda _n(i))^2 \ (\overline{z'}-\lambda _n(j))^2}\\= & {} \int _0^1 \frac{\sigma _{{\text {d}}}(t)^2}{(z-f(t))^2 \ (\overline{z'}-f(t))^2} \mathrm {d}t \end{aligned}$$

and by the fact that \(\overline{Z(z)} = Z(\overline{z})\) which comes from \(\overline{A_n(z)} = A_n(\overline{z})\). \(\square \)

Claim 2

There is a constant C such that, for \(\eta _n\) as in Hypotheses (b),

  • if \( \varepsilon _n\ll n^{-1}\), then

    $$\begin{aligned} \mathbb {E}[|n\varepsilon _n^{-1}B_n(z)|^2]\le \frac{C(n\varepsilon _n)^2}{{\text {dist}}(z, \widetilde{\mathcal {S}})^6}+ \frac{C\eta _n^2}{{\text {dist}}(z, \widetilde{\mathcal {S}})^8}, \end{aligned}$$

  • if \(\varepsilon _n\sim c/n\) or if \( n^{-1} \ll \varepsilon _n\ll 1\), then

    $$\begin{aligned} \quad \mathbb {E}[| n\varepsilon _n^{-1}(B_n(z)-\varepsilon _n^2B(z))|^2] \le \frac{C \varepsilon _n^2}{{\text {dist}}(z, \widetilde{\mathcal {S}})^6}+\frac{C \eta _n^2 }{{\text {dist}}(z, \widetilde{\mathcal {S}})^8}. \end{aligned}$$

Proof

Remind that,

$$\begin{aligned} B_n(z)=\frac{\varepsilon _n^2}{n^2}\sum _{i,j}\frac{|x_{i,j}|^2}{(z-\lambda _n(i))^2(z-\lambda _n(j))}. \end{aligned}$$

Introduce the variable \(b_n^{\circ }(z)\) obtained by centering the variable \(n\varepsilon _n^{-2} B_n(z)\):

$$\begin{aligned}b_n^{\circ }(z):=n\varepsilon _n^{-2}(B_n(z)-\mathbb {E}B_n(z))=\frac{1}{n}\sum _{i,j}\frac{|x_{i,j}|^2-\sigma _n^2(i,j)}{(z-\lambda _n(i))^2(z-\lambda _n(j))} \end{aligned}$$

and the defect variable

$$\begin{aligned} \delta _n(z):= & {} \varepsilon _n^{-2}\mathbb {E}B_n(z)-B(z) \\= & {} \frac{1}{n^2}\sum _{i,j}\frac{\sigma _n^2(i,j)}{(z-\lambda _n(i))^2(z-\lambda _n(j))}-\int _{(s,t)\in [0,1]^2}\frac{\sigma ^2(s,t)}{(z-f(s))^2(z-f(t))}\mathrm {d}s\mathrm {d}t. \end{aligned}$$

In the two regimes \(\varepsilon _n \ll n^{-1}\) and \(\varepsilon _n \ge c/n\), we want to dominate the \(L^2\) norms, respectively, of \( n\varepsilon _n^{-1}B_n(z)=\varepsilon _nb_n^{\circ }(z)+n\varepsilon _n(\delta _n(z)+B(z))\quad \text {and}\quad n\varepsilon _n^{-1}(B_n(z)-\varepsilon _n^2B(z))=\varepsilon _nb_n^{\circ }+ n\varepsilon _n\delta _n(z).\)

For this purpose, we successively dominate \(b_n^{\circ }\), \(\delta _n(z)\) and B(z).

Using the independence of the \(x_{i,j}\)’s, the fact that they are bounded in \(L^4\) and the fact that z stays at a macroscopic distance of the \(\lambda _n(i)\)’s, we can write for all \(z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}\)

$$\begin{aligned} \mathbb {E}[| b_n^{\circ }(z)|^2]&=\frac{1}{n^2}{\text {Var}}\left( \sum _{ i\le j} \left( |x_{i,j}|^2+\mathbb {1}_{i\ne j}\overline{x_{i,j}}^2\right) \frac{1}{(z-\lambda _n(i))^2(z-\lambda _n(j))}\right) \nonumber \\&=\frac{1}{n^2}\sum _{ i\le j} {\text {Var}}\left( \left( |x_{i,j}|^2+\mathbb {1}_{i\ne j}\overline{x_{i,j}}^2\right) \frac{1}{(z-\lambda _n(i))^2(z-\lambda _n(j))}\right) \nonumber \\&\le C{\text {dist}}(z, \widetilde{\mathcal {S}})^{-6}\, . \end{aligned}$$
(24)

Now, the term \(\delta _n(z)\) rewrites

$$\begin{aligned} \delta _n(z)= & {} O(n^{-1})\\&+\int _{(s,t)\in [0,1]^2}\mathbb {1}_{\lfloor ns\rfloor \ne \lfloor nt\rfloor }\left( \frac{\sigma _n^2(\lfloor ns\rfloor ,\lfloor nt\rfloor )}{(z-\lambda _n(\lfloor ns\rfloor ))^2(z-\lambda _n(\lfloor nt\rfloor ))}\right. \\&\left. -\frac{\sigma ^2(s,t)}{(z-f(s))^2(z-f(t))}\right) \mathrm {d}s \mathrm {d}t. \end{aligned}$$

Since, for \(M_\sigma :=\sup _{0\le x\ne y\le 1}\sigma (x,y)^2\) and for any fixed \(z\notin \widetilde{\mathcal {S}}\), the function

$$\begin{aligned}&\psi _z:(s,\lambda ,\lambda ')\in [0,M_\sigma +1]\times \{x\in \mathbb {R}\,;\, {\text {dist}}(x,\widetilde{\mathcal {S}})\\&\quad \le {\text {dist}}(z,\widetilde{\mathcal {S}})/2\}^2\longmapsto \frac{s}{(z-\lambda )^2(z-\lambda ')} \end{aligned}$$

is \(C{\text {dist}}(z,\widetilde{\mathcal {S}})^{-4}\)-Lipschitz, for C a universal constant, by Hypothesis (b),

$$\begin{aligned} \delta _n(z)=O(n^{-1})+ \frac{O\left( \eta _n \right) }{ \max \{n\varepsilon _n,1\} {\text {dist}}(z,\widetilde{\mathcal {S}})^{4} }. \end{aligned}$$
(25)

Finally, the expression of B(z) given in (17) implies,

$$\begin{aligned} B(z) \le \frac{C}{{\text {dist}}(z,\widetilde{\mathcal {S}})^3} \end{aligned}$$
(26)

Collecting estimations (24), (25) and (26), we conclude. \(\square \)

Claim 3

There is a constant C such that for any \(z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}\),

$$\begin{aligned} \mathbb {E}[ |n\varepsilon _n^{-1}C_n(z)|^2]\le \frac{C\varepsilon _n^4}{{\text {dist}}(z,\widetilde{\mathcal {S}})^{8}}. \end{aligned}$$

Proof

We start by writing for all \(z\in \mathbb {C}\backslash \widetilde{\mathcal {S}}\)

$$\begin{aligned} \mathbb {E}[ |n\varepsilon _n^{-1}C_n(z)|^2 ]= & {} \frac{\varepsilon _n^{4}}{n^3} \mathbb {E}\left[ \left| \sum _{i,j,k=1}^n \frac{x_{i,j} \ x_{j,k} \ x_{k,i}}{(z-\lambda _n(i))^2 \ (z-\lambda _n(j)) \ (z-\lambda _n(k))} \right| ^2 \right] \\= & {} \frac{\varepsilon _n^{4}}{n^3} \sum _{i,j,k,l,m,p=1}^n \frac{{\mathbb {E}}\left( x_{i,j} \ x_{j,k} \ x_{k,i} \ \overline{x_{l,m} \ x_{m,p} \ x_{p,l} }\right) }{(z-\lambda _n(i))^2 \ (z-\lambda _n(j)) \ (z-\lambda _n(k)) \ (\overline{z}-\lambda _n(l))^2 \ (\overline{z}-\lambda _n(m)) \ (\overline{z}-\lambda _n(p))}. \end{aligned}$$

Generically, the set of “edges” \(\{(l,m), (m,p), (p,l)\}\) must be equal to the set \(\{(i,j), (j,k), (k,i)\}\) in order to get a nonzero term. Therefore, the complexity of the previous sum is \(O(n^3)\). Note that other nonzero terms involving third or fourth moments are much less numerous. Hence,

$$\begin{aligned} \mathbb {E}[ |n\varepsilon _n^{-1}C_n(z)|^2 ]\le \frac{\varepsilon _n^{4}}{n^3} \times \frac{O(n^3)}{{\text {dist}}(z,\widetilde{\mathcal {S}})^8} \le \frac{C \varepsilon _n^{4}}{{\text {dist}}(z,\widetilde{\mathcal {S}})^8} \end{aligned}$$

\(\square \)

Claim 4

There is a constant C such that for any \(z\in \mathbb {C}\backslash \mathbb {R}\),

$$\begin{aligned} \mathbb {E}[ |n\varepsilon _n^{-1}R_n^\varepsilon (z)|^2]\le \frac{ O(n^2 \varepsilon _n^6)}{|\mathfrak {Im}(z)|^2 {\text {dist}}(z,\widetilde{\mathcal {S}})^8}. \end{aligned}$$

Proof

Remind that,

$$\begin{aligned} R_n^\varepsilon (z):=\frac{\varepsilon _n^4}{n}{\text {Tr}}\frac{1}{z-D} X \frac{1}{z-D} X \frac{1}{z-D}X\frac{1}{z-D}X \frac{1}{z-D^\varepsilon }. \end{aligned}$$

Hence,

$$\begin{aligned} \mathbb {E}[|n\varepsilon _n^{-1}R_n^\varepsilon (z)|^2]&\le \varepsilon _n^6 \ \mathbb {E}\left[ \left| {\text {Tr}}\frac{1}{z-D} X \frac{1}{z-D} X \frac{1}{z-D}X\frac{1}{z-D}X \frac{1}{z-D^\varepsilon }\right| ^2\right] \\&\le \varepsilon _n^6 \ \mathbb {E}\left[ {\text {Tr}}\left| \left( \frac{1}{z-D}X\right) ^4\right| ^{2} \times {\text {Tr}}\left| \frac{1}{z-D^\varepsilon } \right| ^2 \right] \\&\le \varepsilon _n^6 \ \mathbb {E}\left[ {\text {Tr}}\left( \left( \frac{1}{z-D}X\right) ^4 \left( \overline{\frac{1}{z-D}X}\right) ^4 \right) \frac{n}{|\text {Im}(z)|^2} \right] \\&\le \frac{n\varepsilon _n^6}{|\text {Im}(z)|^2} \ \mathbb {E}\left[ {\text {Tr}}\left( \left( \frac{1}{z-D}X\right) ^4 \left( \overline{\frac{1}{z-D}X}\right) ^4 \right) \right] \\&\le \frac{n\varepsilon _n^6}{|\text {Im}(z)|^2} \frac{O(n^{5})}{n^4 \ {\text {dist}}(z,\widetilde{\mathcal {S}})^8} \le \frac{O(n^2 \varepsilon _n^6)}{|\text {Im}(z)|^2 {\text {dist}}(z,\widetilde{\mathcal {S}})^8}. \end{aligned}$$

The inequality of the last line takes into account that

  • the \(L^8\) norm of the entries of \(\sqrt{n}X\) is uniformly bounded

  • the norm of the entries of X is of order \(n^{-1/2}\)

  • the norm of the coefficients of \((z-D)^{-1}\) is smaller than \({\text {dist}}(z,\widetilde{\mathcal {S}})^{-1}\)

  • the complexity of the sum defining the trace is of order \(O(n^5)\) since its non-null terms are encoded by four edges trees which have therefore five vertices.

\(\square \)

We gather now the results of the previous claims.

For any rate of convergence of \(\varepsilon _n\), Claim 1 proves that the process \(n\varepsilon _n^{-1}A_n(z)\) converges in distribution to the centered Gaussian variable Z(z). Moreover,

  • if \(\varepsilon _n \ll n^{-1}\), then as Claims 2, 3 and 4 imply that the processes \(n\varepsilon _n^{-1}B_n(z)\), \(n\varepsilon _n^{-1}C_n(z)\) and \(n\varepsilon _n^{-1}R_n^\varepsilon (z)\) converge to 0 in probability, we can conclude, by Slutsky’s theorem, that for any \(z\in \mathbb {C}\setminus \mathbb {R}\):

    $$\begin{aligned} n\varepsilon _n^{-1}\Delta \mathrm {G}_n(z) \xrightarrow [n\rightarrow \infty ]{\text {dist}} Z(z) \end{aligned}$$

  • if \(\varepsilon _n \sim \frac{c}{n}\), then, as Claims 2, 3 and 4 imply that the processes \(n\varepsilon _n^{-1}B_n(z)\), \(n\varepsilon _n^{-1}C_n(z)\) and \(n\varepsilon _n^{-1}R_n^\varepsilon (z)\) converge, respectively, to cB(z), 0 and 0 in probability, we can conclude, by Slutsky’s theorem, that for any \(z\in \mathbb {C}\setminus \mathbb {R}\):

    $$\begin{aligned} n\varepsilon _n^{-1}\Delta \mathrm {G}_n(z) \xrightarrow [n\rightarrow \infty ]{\text {dist}} Z(z) + cB(z) \end{aligned}$$

  • if \(n^{-1}\ll \varepsilon _n\ll n^{-1/3}\), then, as Claims 2, 3 and 4 imply that the three processes \(n\varepsilon _n^{-1}(B_n(z)-\varepsilon _n^2B(z))\), \(n\varepsilon _n^{-1}C_n(z)\) and \(n\varepsilon _n^{-1}R_n^\varepsilon (z)\) converge to 0 in probability, we can conclude, by Slutsky’s theorem, that for any \(z\in \mathbb {C}\setminus \mathbb {R}\):

    $$\begin{aligned} n\varepsilon _n^{-1} \left( \Delta \mathrm {G}_n(z)-\varepsilon _n^2B(z)\right) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z(z) \end{aligned}$$

Regarding the convergence in probability (22), in the case \(n^{-1} \ll \varepsilon _n\ll 1\), Claims 1, 2, 3 and 4 imply that the processes \(\varepsilon _n^{-2} A_n(z)\), \(\varepsilon _n^{-2} B_n(z) - B(z)\), \(\varepsilon _n^{-2} C_n(z)\) and \(\varepsilon _n^{-2} R_n^\varepsilon (z)\) converge to 0.

This finishes the proof of the convergences of Proposition 3. \(\square \)

6.3 Proof of Proposition 4

Recall that

$$\begin{aligned} B(z)=\int _{(s,t)\in [0,1]^2}\frac{\sigma ^2(s,t)}{(z-f(s))^2(z-f(t))}\mathrm {d}s\mathrm {d}t. \end{aligned}$$

Recall that \(\rho \) is the density of the push-forward of the uniform measure on [0, 1] by the map f.

Let \(\tau \) be as in Hypothesis (d). We have

$$\begin{aligned} B(z) = \ \int _{\mathbb {R}^2} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{(z-s)^2 \ (z-t)} \mathrm {d}s \mathrm {d}t. \end{aligned}$$

By a partial fraction decomposition, we have for all \(a \ne b\)

$$\begin{aligned} \frac{1}{(z-a)^2 (z-b)} = \frac{1}{(b-a)^2} \left( \frac{1}{z-b} - \frac{1}{z-a} - \frac{b-a}{(z-a)^2} \right) . \end{aligned}$$

Thus, as the Lebesgue measure of the set \(\left\{ (y_1,y_2) \in [0,1]^2 \ \,;\, \ y_1 = y_2 \right\} \) is null, we have

$$\begin{aligned} B(z) = \ \int _{\mathbb {R}^2} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{(t-s)^2} \left( \frac{1}{z-t} - \frac{1}{z-s} - \frac{t-s}{(z-s)^2} \right) \mathrm {d}s \mathrm {d}t. \end{aligned}$$

Moreover, for \(\varphi _z\) the function \(\varphi _z: x \longmapsto \frac{1}{z-x}\), we obtain

$$\begin{aligned} B(z) = \ \int _{\mathbb {R}^2} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{(t-s)^2} \left( \varphi _z(t) - \varphi _z(s) - (t-s) \varphi _z'(s) \right) \mathrm {d}s \mathrm {d}t. \end{aligned}$$

Now, we want to prove that \(\displaystyle B(z) = -\int _{\mathbb {R}^2} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{t-s} \ \varphi _z'(s) \ \mathrm {d}s \mathrm {d}t\).

To do this, we will use a symmetry argument: in fact both terms in \(\varphi _z(t)\) and \(\varphi _z(s)\) neutralize each other, and it remains only to prove that we did not remove \(\infty \) to \(\infty \) and that the remaining term has the desired form.

Let us define

$$\begin{aligned} B^\eta (z):= \ \int _{\begin{array}{c} |s-t|>\eta \end{array}} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{(t-s)^2} \left( \varphi _z(t) - \varphi _z(s) - (t-s) \varphi _z'(s) \right) \mathrm {d}s \mathrm {d}t. \end{aligned}$$

By the Taylor–Lagrange inequality we obtain:

$$\begin{aligned} \left| \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{(t-s)^2} \left( \varphi _z(t) - \varphi _z(s) - (t-s) \varphi _z'(s) \right) \right| \le \frac{ \rho (s) \ \rho (t) \ \Vert \tau (\cdot ,\cdot )\Vert _{L^\infty } \ \Vert \varphi _z''\Vert _{L^\infty }}{2}. \end{aligned}$$

So that, since \(\rho \) is a density, by dominated convergence, we have

$$\begin{aligned} \lim _{\eta \rightarrow 0} B^\eta (z) = B(z). \end{aligned}$$

Moreover, by symmetry, for any \(\eta \),

$$\begin{aligned} B^\eta (z) = \int _{\begin{array}{c} |s-t|>\eta \end{array}} \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{t-s} (-\varphi _z'(s) ) \mathrm {d}s \mathrm {d}t. \end{aligned}$$

So

$$\begin{aligned} B(z)= & {} \lim _{\eta \rightarrow 0} \int _{|s-t|>\eta } \frac{\tau (s,t) \ \rho (s) \ \rho (t)}{t-s} (-\varphi _z'(s) ) \mathrm {d}t \mathrm {d}s\nonumber \\= & {} -\lim _{\eta \rightarrow 0}\int _{s\in \mathbb {R}}F_\eta (s) \varphi _z'(s) \mathrm {d}s \end{aligned}$$
(27)

where for \(\eta > 0\) and \(s \in \mathbb {R}\), we define

$$\begin{aligned} F_\eta (s): = \rho (s)\int _{t \in \mathbb {R}\backslash [s-\eta ,s+\eta ]} \frac{\tau (s,t) \ \rho (t)}{t-s} \mathrm {d}t. \end{aligned}$$

Note that that by definition of the function F given at (6), for any s, we have

$$\begin{aligned} F(s)=\lim _{\eta \rightarrow 0}F_\eta (s). \end{aligned}$$
(28)

Thus by (27) and (28), to conclude the proof of Proposition 4, by dominated convergence, one needs only to state that \(F_\eta \) is dominated, uniformly in \(\eta \), by an integrable function. This follows from the following computation.

Note first that by symmetry, we have

$$\begin{aligned} F_\eta (s)= & {} \rho (s) \int _{t \in \mathbb {R}\backslash [s-\eta ,s+\eta ]} \frac{\tau (s,t) \ \rho (t)-\tau (s,s) \ \rho (s)}{t-s} \mathrm {d}t. \end{aligned}$$
(29)

Let \(M>0\) such that the support of the function \(\rho \) is contained in \([-M,M]\). Then, for \(\eta _0,\alpha ,C\) as in Hypothesis (e), using the expression of \(F_\eta (s)\) given at (29), we have

$$\begin{aligned} \left| F_\eta (s) \right|&\le 2C\rho (s)\int _{t=s}^{s+\eta _0}|t-s|^{\alpha -1}\mathrm {d}t\\&\quad + \int _{t\in [s-2M,s-\eta _0]\cup [s+\eta _0,s+2M]} \left| \frac{ \tau (s,t)\rho (s)\rho (t)}{t-s} \right| \mathrm {d}t \\&\le \frac{2C\rho (s)}{\alpha }\eta _0^\alpha + \frac{1}{\eta _0}\int _{t\in \mathbb {R}} \left| \tau (s,t)\rho (s)\rho (t) \right| \mathrm {d}t \\&\le \frac{2C\rho (s)}{\alpha }\eta _0^\alpha + \frac{\Vert \tau (\cdot ,\cdot )\Vert _{L^\infty }}{\eta _0}\rho (s). \end{aligned}$$

\(\square \)

6.4 A Local Type Convergence Result

One can precise the convergence (22) by replacing the complex variable z by a complex sequence \((z_n)\) which converges slowly enough to the real axis. This convergence won’t be used in the sequel. As it is discussed in [7], this type of result is a first step toward a local result for the empirical distribution.

Proposition 5

Under Hypotheses (a), (b), (f), if \(n^{-1}\ll \varepsilon _n\ll 1\), then for any non-real complex sequence \((z_n)\), such that

$$\begin{aligned} \mathfrak {Im}(z_n) \gg \max \left\{ (n\varepsilon _n)^{-1/2} \ , \ \left( \frac{\eta _n}{n\varepsilon _n}\right) ^{1/4} \ , \ \varepsilon _n^{2/5} \right\} \end{aligned}$$
(30)

the following convergence holds

$$\begin{aligned} \varepsilon _n^{-2}\Delta \mathrm {G}_n(z_n) - B(z_n) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{P}}\quad 0 \,. \end{aligned}$$

Remark

In the classical case where \(\displaystyle {\dfrac{\eta _n}{n\varepsilon _n} = \sup _{i\ne j }( |\sigma _n^2(i,j)-\sigma ^2(i/n,j/n)|+|\lambda _n(i)}\)\(\displaystyle {-f(i/n)|)}\) is of order \(\dfrac{1}{n}\), the above assumption boils down to \(\mathfrak {Im}(z_n) \gg \max \left\{ (n\varepsilon _n)^{-1/2} \ , \ \varepsilon _n^{2/5} \right\} \).

Proof

Assume \(n^{-1} \ll \varepsilon _n\ll 1\). One can directly obtain, for all non-real complex sequences \((z_n)\), that

  • by Claim 1, if \({\text {dist}}(z_n,\widetilde{\mathcal {S}}) \gg (n\varepsilon _n)^{-1/2}\), then

    $$\begin{aligned}\mathbb {E}\left[ \left| \varepsilon _n^{-2}A_n(z_n)\right| ^2\right] \le \frac{C}{ (n\varepsilon _n)^2 \ {\text {dist}}(z_n, \widetilde{\mathcal {S}})^4} \underset{n\rightarrow \infty }{\longrightarrow } 0,\end{aligned}$$

  • by Claim 2, if \({\text {dist}}(z_n,\widetilde{\mathcal {S}}) \gg \max \left\{ n^{-1/3} \ , \ (\eta _n/(n\varepsilon _n ))^{1/4}\right\} \), then

    $$\begin{aligned}{\mathbb {E}}\left[ \left| \varepsilon _n^{-2}B_n(z_n) - B(z_n)\right| ^2\right] \le \frac{C}{n^{2}{\text {dist}}(z_n, \widetilde{\mathcal {S}})^6}+\frac{C \eta _n^2 }{(n\varepsilon _n)^{2} \ {\text {dist}}(z_n, \widetilde{\mathcal {S}})^8} \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0,\end{aligned}$$

  • by Claim 3, if \({\text {dist}}(z_n,\widetilde{\mathcal {S}}) \gg \left( \varepsilon _n/n\right) ^{1/4}\), then

    $$\begin{aligned} \mathbb {E}\left[ \left| \varepsilon _n^{-2}C_n(z_n)\right| ^2\right] \le \frac{C\varepsilon _n^2}{n^{2} \ {\text {dist}}(z_n,\widetilde{\mathcal {S}})^{8}} \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0, \end{aligned}$$

  • by Claim 4, if \( |\mathfrak {Im}(z_n)| {\text {dist}}(z_n,\widetilde{\mathcal {S}})^4 \gg \varepsilon _n^{2}\), then

    $$\begin{aligned} \mathbb {E}\left[ \left| \varepsilon _n^{-2}R_n^\varepsilon (z_n)\right| ^2\right] \le \frac{O(\varepsilon _n^4)}{|\mathfrak {Im}(z_n)|^2 {\text {dist}}(z_n,\widetilde{\mathcal {S}})^8} \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0. \end{aligned}$$

Therefore, when

$$\begin{aligned} {\text {dist}}(z_n,\widetilde{\mathcal {S}})\gg & {} \max \left\{ (n\varepsilon _n)^{-1/2} \ , \ n^{-1/3} \ , \ \left( \frac{\eta _n}{n\varepsilon _n}\right) ^{1/4} \ , \ \left( \frac{\varepsilon _n}{n}\right) ^{1/4} \right\} \\&\text { and } \ |\mathfrak {Im}(z_n)| {\text {dist}}(z_n,\widetilde{\mathcal {S}})^4 \gg \varepsilon _n^{2}, \end{aligned}$$

the four processes, \(\varepsilon _n^{-2}A_n(z_n)\), \(\varepsilon _n^{-2}B_n(z_n) - B(z_n)\), \(\varepsilon _n^{-2}C_n(z_n)\) and \(\varepsilon _n^{-2}R_n^\varepsilon (z_n)\) converge to 0 in probability. Since \({\text {dist}}(z_n,\widetilde{\mathcal {S}}) \ge \mathfrak {Im}(z_n)\), the above condition is implied by

$$\begin{aligned} \mathfrak {Im}(z_n) \gg \max \left\{ (n\varepsilon _n)^{-1/2} \ , \ n^{-1/3} \ , \ \left( \frac{\eta _n}{n\varepsilon _n}\right) ^{1/4} \ , \ \left( \frac{\varepsilon _n}{n}\right) ^{1/4}, \ \varepsilon _n^{2/5} \right\} . \end{aligned}$$

Observing finally that the two terms \(n^{-1/3}\) and \(\left( \frac{\varepsilon _n}{n}\right) ^{1/4}\) are dominated by the maximum of the three other ones, we conclude the proof. \(\square \)

6.5 Possible Extensions to Larger \(\varepsilon _n\)

The convergence in distribution result of Theorem 1 is valid for \(\varepsilon _n\ll n^{-1/3}\) but fails above \(n^{-1/3}\). Let us consider, for example, the case where \(n^{-1/3}\ll \varepsilon _n\ll n^{-1/5}\). In this case, the contribution of the first term \(A_n(z)\) in the expansion of \(\Delta \mathrm {G}_n(z)\) which yields the random limiting quantity is dominated not only by the term \(B_n(z)\) as it used to be previously. It is also dominated by a further and smaller term \(D_n(z)\) of the expansion

$$\begin{aligned} \Delta \mathrm {G}_n(z) = A_n(z)+B_n(z)+C_n(z)+D_n(z)+E_n(z)+R_n^\varepsilon , \end{aligned}$$

with:

$$\begin{aligned} A_n(z):= & {} \frac{\varepsilon _n}{n} {\text {Tr}}\frac{1}{z-D}X\frac{1}{z-D}\\&\vdots&\\ E_n(z):= & {} \frac{\varepsilon _n^5}{n}{\text {Tr}}\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}\\ R_n^\varepsilon (z):= & {} \frac{\varepsilon _n^6}{n}{\text {Tr}}\frac{1}{z-D}X \frac{1}{z-D}X \frac{1}{z-D}X \frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D}X\frac{1}{z-D^\varepsilon }. \end{aligned}$$

In this case, the random term Z(z) is still produced by \(A_n(z)\) and has an order of magnitude of \(\varepsilon _n/n\). Meanwhile, the term \(D_n(z)\) writes

$$\begin{aligned} D_n(z) := \frac{\varepsilon _n^4}{n^3} \sum _{i,j,k,l=1}^n \frac{x_{i,j} \ x_{j,k} \ x_{k,l} \ x_{l,i}}{ (z-\lambda _n(i))^2 \ (z-\lambda _n(j)) \ (z-\lambda _n(k)) \ (z-\lambda _n(l)) }. \end{aligned}$$

All the indices satisfying \(j=l\) contribute to the previous sum, since they produce a term in \(|x_{i,l}|^2 |x_{k,l}|^2\). Their cardinality is of order \(n^3\). Therefore, the term \(D_n(z)\) is of order \(\varepsilon _n^4\) which prevails on the order \(\varepsilon _n/n\) of \(A_n(z)\), as soon as \(\varepsilon _n\gg n^{-1/3}\). One can also observe that the odd terms \(C_n(z)\) and \(E_n(z)\) in the expansion are negligible with respect to \(A_n(z)\) due to the fact that the entries \(x_{i,j}\) are centered. One can then state an analogous result to Proposition 3, but the deterministic limiting term D(z) arising from \(D_n(z)\) does not find a nice expression as the image of \(\varphi _z\) by a linear form as it was the case for B(z) in Proposition 4. Therefore, we did not state an extension of Theorem 1.

More generally, for all positive integer p, when \(n^{-1/(2p-1)}\ll \varepsilon _n\ll n^{-1/(2p+1)}\), the expansion will contain p deterministic terms, produced by the even variables, \(B_n(z)\), \(D_n(z)\), \(F_n(z)\), \(H_n(z)\)\(\dots \) All the other odd terms, \(C_n(z)\), \(E_n(z)\), \(G_n(z)\)\(\dots \) being negligible due to the centering of the entries. The limits of the even terms \(B_n(z)\), \(D_n(z)\), \(F_n(z)\), \(H_n(z)\)\(\dots \) can be expressed thanks to operator-valued free probability theory, using the results of [22] (namely Th. 4.1), but expressing these limits as the images of \(\varphi _z\) by linear forms is a quite involved combinatorial problem that we did not solve yet.

7 Convergence in Probability in the Semi-Perturbative Regime

Our goal now is to extend the convergence in probability result (22) of Proposition 3, proved for test functions \(\varphi _z(x) := \frac{1}{z-x}\), to any \({\mathcal {C}}^6\) and compactly supported function on \(\mathbb {R}\). We do it in the following lemma by using the Helffer–Sjöstrand formula which is stated in Proposition 9 of Appendix.

Lemma 6

If \(n^{-1} \ll \varepsilon _n \ll 1\), then, for any compactly supported \({\mathcal {C}}^6\) function \(\phi \) on \(\mathbb {R}\),

$$\begin{aligned} \varepsilon _n^{-2}(\mu _n^\varepsilon - \mu _n)(\phi ) \xrightarrow [n\rightarrow \infty ]{P} -\int \phi '(s) F(s) \ \mathrm {d}s\, . \end{aligned}$$

Proof

Let us introduce the Banach space \(\mathcal {C}^1_{{\text {b}},{\text {b}}}\) of bounded \(\mathcal {C}^1\) functions on \(\mathbb {R}\) with bounded derivative, endowed with the norm \(\Vert \phi \Vert _{\mathcal {C}^1_{{\text {b}},{\text {b}}}}:=\Vert \phi \Vert _\infty +\Vert \phi '\Vert _\infty \).

On this space, let us define the random continuous linear form

$$\begin{aligned} \Pi _n(\phi ):= \varepsilon _n^{-2}(\mu _n^\varepsilon - \mu _n)(\phi ) + \int \phi '(s) F(s) \ \mathrm {d}s. \end{aligned}$$

Convergence (22) of Proposition 3 can now be formulated as

$$\begin{aligned} \forall z\in \mathbb {C}\setminus \mathbb {R}, \qquad \Pi _n(\varphi _z) \xrightarrow [n\rightarrow \infty ]{P} 0. \end{aligned}$$

Actually, we can be more precise by adding the upper bounds of Claims 1, 2, 3 and 4, and obtain, uniformly in z,

$$\begin{aligned} \mathbb {E}\left[ \left| \Pi _n(\varphi _z)\right| ^2\right]&= \mathbb {E}\left[ \left| \varepsilon _n^{-2}\Delta \mathrm {G}_n(z)-B(z)\right| ^2\right] \nonumber \\&\le \frac{(n\varepsilon _n)^{-2}}{\min \left( {\text {dist}}(z,\widetilde{\mathcal {S}})^4 \ , \ {\text {dist}}(z,\widetilde{\mathcal {S}})^8 \ , \ |\mathfrak {Im}(z)|^2 {\text {dist}}(z,\widetilde{\mathcal {S}})^8 \right) }. \end{aligned}$$
(31)

Now, let \(\phi \) be a compactly supported \(\mathcal {C}^{6}\) function on \(\mathbb {R}\) and let us introduce the almost analytic extension of degree 5 of \(\phi \) defined by

An elementary computation gives, by successive cancellations, that

$$\begin{aligned} \bar{\partial }\widetilde{\phi }_5(z) = \frac{1}{2}\left( \partial _x + \mathrm {i}\partial _y \right) \widetilde{\phi }_5(x+\mathrm {i}y) = \frac{1}{2\times 5!} (\mathrm {i}y)^5 \phi ^{(6)}(x). \end{aligned}$$
(32)

Furthermore, by Helffer–Sjöstrand formula (Proposition 9), for \(\chi \in {\mathcal {C}}^\infty _c(\mathbb {C};[0,1])\) a smooth cutoff function with value one on the support of \(\phi \),

$$\begin{aligned} \phi (\cdot ) = -\frac{1}{\pi } \int _{\mathbb {C}} \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5}y^5\varphi _z(\cdot ) \, \mathrm {d}^2 z\, \end{aligned}$$

where \(\mathrm {d}^2 z\) denotes the Lebesgue measure on \(\mathbb {C}\).

Note that by (32), \(z \mapsto \mathbb {1}_{y\ne 0} \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5}\) is a continuous compactly supported function and that \(z\in \mathbb {C}\mapsto \mathbb {1}_{y\ne 0} y^5\varphi _z\in \mathcal {C}^1_{{\text {b}},{\text {b}}}\) is continuous, hence,

$$\begin{aligned} \Pi _n(\phi ) = \frac{1}{\pi } \int _{\mathbb {C}} \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5} \ y^5\Pi _n(\varphi _z) \, \mathrm {d}^2 z. \end{aligned}$$

Therefore, using the Cauchy–Schwarz inequality and the fact that \(\chi \) has compact support at the second step, for a certain constant C, we have

$$\begin{aligned} \mathbb {E}\left( \left| \Pi _n(\phi ) \right| ^2 \right)&= \mathbb {E}\left( \left| \frac{1}{\pi } \int _{\mathbb {C}} \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5} \ y^5\Pi _n(\varphi _z) \, \mathrm {d}^2 z \right| ^2 \right) \\&\le C \mathbb {E}\left( \int _{\mathbb {C}} \left| \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5} \ y^5\Pi _n(\varphi _z) \right| ^2 \, \mathrm {d}^2 z \right) \\&= C\int _{\mathbb {C}} \left| \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5}\right| ^2 \ y^{10} \ \mathbb {E}\left( \left| \Pi _n(\varphi _z)\right| ^2\right) \, \mathrm {d}^2 z\, . \end{aligned}$$

Since the function \(\left| \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5}\right| ^2\) is continuous and compactly supported and that, by (31), for \(n^{-1} \ll \varepsilon _n \ll 1\), uniformly in z,

$$\begin{aligned} y^{10} \ \mathbb {E}\left( \left| \Pi _n(\varphi _z)\right| ^2\right) \le y^{10} \ \frac{o(1)}{\min (y^4, \ y^{10})} \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0. \end{aligned}$$

Thus, for any compactly supported \(\mathcal {C}^6\) function on \(\mathbb {R}\),

$$\begin{aligned} \mathbb {E}\left( \left| \Pi _n(\phi ) \right| ^2 \right) \le C \int _{\mathbb {C}} \left| \frac{\bar{\partial }(\widetilde{\phi }_5(z) \chi (z))}{y^5}\right| ^2 \ y^{10} \ \mathbb {E}\left( \left| \Pi _n(\varphi _z)\right| ^2\right) \, \mathrm {d}^2 z \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0 \end{aligned}$$

which implies that \(\Pi _n(\phi )\) converges to 0 in probability. \(\square \)

8 Convergence in Distribution Toward the Gaussian Variable \(Z_\phi \)

The purpose of this section is to extend the convergences in distribution of Proposition 3, from test functions of the type \(\varphi _z:= \frac{1}{z-x}\), to compactly supported \({\mathcal {C}}^6\) functions on \(\mathbb {R}\). To do so, we will use an extension lemma of Shcherbina and Tirozzi, stated in Lemma 10 of Appendix, which concerns the convergence of a sequence of centered random fields with uniformly bounded variance. Hence, we need to show first that our non-centered random sequence is not far from being centered, which is done in Sect. 8.1 by using again the Helffer–Sjöstrand formula (9). In Sect. 8.2, we dominate the variance of this centered random field thanks to another result of Shcherbina and Tirozzi stated in Proposition 11 of Appendix. Section 8.3 collects the preceding results to conclude the proof.

8.1 Coincidence of the Expectation of \(\mu _n^\varepsilon \) with Its Deterministic Approximation

The asymptotic coincidence of the expectation of \(\mu _n^\varepsilon \) with its deterministic approximation is the content of next lemma:

Lemma 7

Let us define, for \(\phi \) a \(\mathcal {C}^1\) function on \(\mathbb {R}\),

$$\begin{aligned} \Lambda _n(\phi ):={\left\{ \begin{array}{ll}n\varepsilon _n^{-1}\left( \mathbb {E}[\mu _n^\varepsilon (\phi )]-\mu _n(\phi )\right) &{}\text { if }\varepsilon _n\ll n^{-1},\\ \\ n\varepsilon _n^{-1}\left( \mathbb {E}[\mu _n^\varepsilon (\phi )]-\mu _n(\phi )+\varepsilon _n^2\int \phi '(s)F(s)\mathrm {d}s\right) &{}\text {if } \varepsilon _n\sim c/n \text { or } n^{-1} \ll \varepsilon _n \ll n^{-1/3}\, . \end{array}\right. } \end{aligned}$$

Then, as \(n\rightarrow \infty \), for any compactly supported \(\mathcal {C}^6\) function \(\phi \) or any \(\phi \) of the type \(\varphi _z(x)=\frac{1}{z-x}\), \(z\in \mathbb {C}\backslash \mathbb {R}\), we have

$$\begin{aligned} \Lambda _n(\phi )\underset{n\rightarrow \infty }{\longrightarrow }0. \end{aligned}$$

Proof

First note that, as the variables \(x_{i,j}\) are centered, \(\mathbb {E}[A_n(z)] = 0\). Moreover, by adding the renormalized upper bounds of Claims 2, 3 and 4 one can directly obtain the two following inequalities for any \(z\in \mathbb {C}\setminus \mathbb {R}\):

  • If \( \varepsilon _n\ll n^{-1}\), then

    $$\begin{aligned} |\Lambda _n(\varphi _z)|= & {} n\varepsilon _n^{-1}| \mathbb {E}[\Delta \mathrm {G}_n(z)]| \\ \\\le & {} n\varepsilon _n^{-1}\left( |\mathbb {E}[A_n(z)]| + \mathbb {E}[|B_n(z)|] + \mathbb {E}[|C_n(z)|] + \mathbb {E}[|R_n^\varepsilon (z)|] \right) \\ \\\le & {} \frac{ C(n\varepsilon _n + \eta _n) }{ \min \left\{ {\text {dist}}(z, \widetilde{\mathcal {S}})^3, {\text {dist}}(z, \widetilde{\mathcal {S}})^4,|\mathfrak {Im}(z)| {\text {dist}}(z, \widetilde{\mathcal {S}})^4\right\} } \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0\,. \end{aligned}$$

  • If \(\varepsilon _n\sim c/n\) or \(n^{-1}\ll \varepsilon _n\ll n^{-1/3}\), then

    $$\begin{aligned} |\Lambda _n(\varphi _z)|= & {} n\varepsilon _n^{-1}|\mathbb {E}[\Delta \mathrm {G}_n(z)-\varepsilon _n^{2}B(z)]| \\ \\\le & {} n\varepsilon _n^{-1}\left( |\mathbb {E}[A_n(z)]|\! +\! \mathbb {E}[|B_n(z)-\varepsilon _n^{2}B(z)|]\! +\! \mathbb {E}[|C_n(z)|] + \mathbb {E}[|R_n^\varepsilon (z)|] \right) \\ \\\le & {} \frac{ C(\varepsilon _n + \eta _n + n\varepsilon _n^3) }{ \min \left\{ {\text {dist}}(z, \widetilde{\mathcal {S}})^3, {\text {dist}}(z, \widetilde{\mathcal {S}})^4,|\mathfrak {Im}(z)| {\text {dist}}(z, \widetilde{\mathcal {S}})^4\right\} } \quad \underset{n\rightarrow \infty }{\longrightarrow }\quad 0 \, . \end{aligned}$$

Hence, in all cases, \( \Lambda _n(\varphi _z)\underset{n\rightarrow \infty }{\longrightarrow }0 \).

The extension of this result to compactly supported \(\mathcal {C}^6\) test functions on \(\mathbb {R}\) goes the same way as for \(\Pi _n\) in the proof of Lemma 6. \(\square \)

8.2 Domination of the Variance of \(\mu _n^\varepsilon \)

The second ingredient goes through a domination of the variance of \(\mu _n^\varepsilon (\phi )\):

Lemma 8

Let \(s>5\). There is a constant C such that for each n and each \(\phi \in \mathcal {H}_s\),

$$\begin{aligned} {\text {Var}}\left( n\varepsilon _n^{-1}\mu _n^\varepsilon ( \phi )\right) \le C\Vert \phi \Vert _{\mathcal {H}_s}^2\,. \end{aligned}$$

Proof

By Proposition 11, it suffices to prove that

$$\begin{aligned} \int _{y=0}^{\infty }y^{2s-1}\mathrm {e}^{-y}\int _{x\in \mathbb {R}}{\text {Var}}(\varepsilon _n^{-1} {\text {Tr}}((x+\mathrm {i}y-D_n^\varepsilon )^{-1}))\mathrm {d}x\mathrm {d}y \end{aligned}$$

are bounded independently of n.

Note that for \(\Delta \mathrm {G}_n(z)\) defined in (18),

$$\begin{aligned} {\text {Var}}\left( \varepsilon _n^{-1} {\text {Tr}}((z-D_n^\varepsilon )^{-1})\right) =n^2\varepsilon _n^{-2} {\text {Var}}(\Delta \mathrm {G}_n(z)). \end{aligned}$$

Moreover, the sum of the inequalities of Claims 1, 2, 3 and 4 yields

$$\begin{aligned} {\text {Var}}\left( n\varepsilon _n^{-1}\Delta \mathrm {G}_n(z)\right)\le & {} \frac{C}{{\text {dist}}(z, \widetilde{\mathcal {S}})^4} + \frac{C}{|\mathfrak {Im}(z)|^2 {\text {dist}}(z, \widetilde{\mathcal {S}})^8} . \end{aligned}$$

Let \(M>0\) such that \(\widetilde{\mathcal {S}}\subset [-M,M]\). Then

$$\begin{aligned} {\text {dist}}(z,\widetilde{\mathcal {S}})\ge {\left\{ \begin{array}{ll} y&{}\text { if }|x|\le M, \\ \sqrt{y^2+(|x|-M)^2}&{}\text { if |x|> M}. \end{array}\right. } \end{aligned}$$

Thus \( {{\text {dist}}(z,\widetilde{\mathcal {S}})}\ge y\) if \(|x|\le M\) and, for \(|x|> M\),

$$\begin{aligned} \frac{1}{{\text {dist}}(z,\widetilde{\mathcal {S}})} \le \frac{y^{-1}}{\sqrt{ 1+((|x|-M)/y)^2 }} \end{aligned}$$

and for any \(y>0\),

$$\begin{aligned} \int _{x\in \mathbb {R}}{\text {Var}}(n\varepsilon _n^{-1}\Delta \mathrm {G}_n(x+\mathrm {i}y))\mathrm {d}x\le & {} 2CM (y^{-10} + y^{-4})\\&+ 2C \int _0^{+\infty }\frac{y^{-4}}{(1+(\frac{x}{y})^2)^{2}} + \frac{y^{-10}}{(1+(\frac{x}{y})^2)^{4}} \mathrm {d}x \\\le & {} 2CM (y^{-10} + y^{-4}) + C \left( \frac{\pi }{2} y^{-3} + \frac{5\pi }{16} y^{-9}\right) \\\le & {} k \left( y^{-10}+ y^{-3}\right) , \end{aligned}$$

for a suitable constant k.

We deduce that, as soon as \(2s-10>0\), i.e. \(s>5\),

$$\begin{aligned}&\int _{y=0}^{\infty }y^{2s-1}\mathrm {e}^{-y}\int _{x\in \mathbb {R}}{\text {Var}}\left( \varepsilon _n^{-1} {\text {Tr}}((x+\mathrm {i}y-D_n^\varepsilon )^{-1})\right) \mathrm {d}x\mathrm {d}y \\&\quad \le \ k \int _{0}^{\infty }y^{2s-1}e^{-y} (y^{-10}+y^{-3}) \mathrm {d}y \ < \ \infty . \end{aligned}$$

\(\square \)

8.3 Proof of the Convergences in Distribution of Theorem 1

Since we have proved in Lemma 7 that for all compactly supported \(\mathcal {C}^6\) function \(\phi \), the deterministic term \(\mu _n(\phi )\) could be replaced by \(\mathbb {E}[\mu _n^\varepsilon (\phi )]\), we only have to prove, that for all \(\phi \in \mathcal {C}^6\),

$$\begin{aligned} n\varepsilon _n^{-1}\left( \mu _n^\varepsilon (\phi )-\mathbb {E}\left[ \mu _n^\varepsilon (\phi )\right] \right) \quad {\mathop {\underset{n\rightarrow \infty }{\longrightarrow }}\limits ^{\mathrm {dist.}}}\quad Z_\phi . \end{aligned}$$

For the time being, we know this result to be valid for functions \(\phi \) belonging to the space \(\mathcal {L}_1\), defined as the linear span of the family of functions \(\varphi _z(x):=\frac{1}{z-x}\), \(z\in \mathbb {C}\backslash \mathbb {R}\).

By applying Lemma 10 to the centered field \(\mu _n^\varepsilon -\mathbb {E}[\mu _n^\varepsilon ]\), we are going to extend the result from the space \(\mathcal {L}_1\) to the Sobolev space \((\mathcal {H}_s, \Vert \cdot \Vert _{\mathcal {H}_s})\) with \(s\in (5,6)\). Note that, since \(s<6\), this latter space contains the space of \(\mathcal {C}^6\) compactly supported functions (see [16, Sec. 7.9]).

It remains to check the two hypotheses of Lemma 10. First, the subspace \(\mathcal {L}_1\) is dense in every space \((\mathcal {H}_s, \Vert \cdot \Vert _{\mathcal {H}_s})\). This is the content of Lemma 13 of Appendix. Second, by Lemma 8, since \(s>5\), \({\text {Var}}(n\varepsilon _n^{-1}\mu _n^\varepsilon ( \phi ))\le C\Vert \phi \Vert _{\mathcal {H}_s}^2\) for a certain constant C.

This concludes the proof.