Charles Bordenave

AND Pietro Caputo

Djalil Chafa

SPECTRUM OF NON-HERMITIAN HEAVY TAILED RANDOM MATRICES

Keywords: Mathematics Subject Classification. 47A10, 15A52, 05C80 Spectral theory, Objective method, Operator convergence, Logarithmic potential, Random matrices, Random Graphs, Heavy tailed distributions, α-stable laws

Let (X jk ) j,k 1 be i.i.d. complex random variables such that X jk is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X jk , we prove that there exist a deterministic sequence an ∼ n 1/α and a probability measure µα on C depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a -1 n X jk ) 1 j,k n converges weakly to µα as n → ∞. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of µα. In contrast with the Hermitian case, we find that µα is not heavy tailed.

Let (X ij ) i,j 1 be i.i.d. complex random variables with cumulative distribution function F . Consider the matrix X = (X ij ) 1 i,j n . Following Dozier and Silverstein [20,19], if F has finite positive variance σ 2 , then for every z ∈ C, there exists a probability measure Q σ,z on [0, ∞) depending only on σ and z, with explicit Cauchy-Stieltjes transform, such that a.s. (almost surely)

ν 1 √ n X-zI n→∞ Q σ,z (1.1)
where denotes the weak convergence of probability measures. The proof of (1.1) is based on a classical approach for Hermitian random matrices with bounded second moment: truncation, centralization, recursion on the resolvent, and cubic equation for the limiting Cauchy-Stieltjes transform. In the special case z = 0, the statement (1.1) reduces to the quarter-circular law theorem (square version of the Marchenko-Pastur theorem, see [START_REF] Ben Arous | The spectrum of heavy tailed random matrices[END_REF][START_REF] Wachter | The strong limits of random matrix spectra for sample matrices of independent elements[END_REF][START_REF] Yin | Limiting spectral distribution for a class of random matrices[END_REF]) and the probability measure Q σ,0 is the quarter-circular law with Lebesgue density

x → 1 πσ 2 4σ 2 -x 2 1 [0,2σ] (x).
(1.2)

Girko's famous circular law theorem [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF] states under the same assumptions that a.s.

µ 1 √ n X n→∞ U σ (1.3)
where U σ is the uniform law on the disc {z ∈ C; |z| σ}. This statement was established through a long sequence of partial results [START_REF] Mehta | Random matrices and the statistical theory of energy levels[END_REF][START_REF] Girko | The circular law[END_REF]26,[START_REF] Hwang | A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries, Random matrices and their applications[END_REF]21,[START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF][START_REF] Bai | Circular law[END_REF]27,5,40,[START_REF] Götze | The Circular Law for Random Matrices[END_REF][START_REF] Tao | Random matrices: the circular law[END_REF][START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF], the general case (1.3) being finally obtained by Tao and Vu [START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF] by using Girko's Hermitization with logarithmic potentials and uniform integrability, the convergence (1.1), and polynomial bounds on the extremal singular values.

1.1. Main results. The aim of this paper is to investigate what happens when F does not have a finite second moment. We shall consider the following hypothesis: (H1) there exists a slowly varying function L (i.e. lim t→∞ L(x t)/L(t) = 1 for any x > 0) and a real number α ∈ (0, 2) such that for every t 1 Assumption (H1) states a complex version of the classical criterion for the domain of attraction of a real α-stable law, see e.g. Feller [23,Theorem IX.8.1a]. For instance, if X 11 = V 1 + iV 2 with i = √ -1 and where V 1 and V 2 are independent real random variables both belonging to the domain of attraction of an α-stable law then (H1) holds. When (H1) holds, we define the sequence a n ) = lim n→∞ na -α n L(a n ) = 1. It follows then classically that a n = n 1/α ℓ(n) for every n 1, for some slowly varying function ℓ. The additional possible assumptions on F to be considered in the sequel are the following:

(H2) P(|X 11 | t) ∼ t→∞ c t -α for some c > 0 (this implies a n ∼ n→∞ c 1/α n 1/α ) (H3) X 11 has a bounded probability Lebesgue density on R or on C. One can check that (H1-H2-H3) hold e.g. when |X 11 | and X 11 /|X 11 | are independent with |X 11 | = |S| where S is real symmetric α-stable. Another basic example is given by X 11 = εW -1/α with ε and W independent such that ε takes values in S 1 and W is uniform on [0, 1].

For every n 1, let us define the i.i.d. n × n complex matrix A = A n by

A ij := a -1 n X ij (1.4)
for every 1 i, j n. Our first result concerns the singular values of A -zI, z ∈ C.

Theorem 1.1 (Singular values). If (H1) holds then for all z ∈ C, there exists a probability measure ν α,z on [0, ∞) depending only on α and z such that a.s.

ν A-zI n→∞ ν α,z .

The case z = 0 was already obtained by Belinschi, Dembo and Guionnet [START_REF] Belinschi | Spectral measure of heavy tailed band and covariance random matrices[END_REF]. Theorem 1.1 is a heavy tailed version of the Dozier and Silverstein theorem (1.1). Our main results below give a non-Hermitian version of Wigner's theorem for Lévy matrices [START_REF] Bouchaud | Theory of Lévy matrices[END_REF]7,[START_REF] Belinschi | Spectral measure of heavy tailed band and covariance random matrices[END_REF]11], as well as a heavy tailed version of Girko's circular law theorem (1.3).

Theorem 1.2 (Eigenvalues). If (H1-H2-H3) hold then there exists a probability measure µ α on C depending only on α such that a.s.

µ A n→∞ µ α .

Theorem 1.3 (Limiting law). The probability distribution µ α from theorem 1.2 is isotropic and has a continuous density. Its density at z = 0 equals

Γ(1 + 2/α) 2 Γ(1 + α/2) 2/α πΓ(1 -α/2) 2/α .
Furthermore, up to a multiplicative constant, the density of µ α is equivalent to

|z| 2(α-1) e -α 2 |z| α as |z| → ∞.
Recall that for a normal matrix (i.e. which commutes with its adjoint), the absolute value of the eigenvalues are equal to the singular values. Theorem 1.3 reveals a striking contrast between µ α and ν α,0 . The limiting law of the eigenvalues µ α has a stretched exponential tail while the limiting law ν α,0 of the singular values is heavy tailed with power exponent α, see e.g. [START_REF] Belinschi | Spectral measure of heavy tailed band and covariance random matrices[END_REF]. This does not contradict the identity in contrast with the density (1.2) of the quarter-circular law Q σ,0 , even the supports differ by a factor 2.

The proof of theorem 1.1 is given in section 2.8. It relies on an extension to non-Hermitian matrices of the "objective method" approach developed in [11]. More precisely, we build an explicit operator on Aldous' Poisson Weighted Infinite Tree (PWIT) and prove that it is the local limit of the matrices A n in an appropriate sense. While Poisson statistics arises naturally as in all heavy tailed phenomena, the fact that a tree structure appears in the limit is roughly explained by the observation that non vanishing entries of the rescaled matrix A n = a -1 n X can be viewed as the adjacency matrix of a sparse random graph which locally looks like a tree. In particular, the convergence to PWIT is a weighted-graph version of familiar results on the local structure of Erdős-Rényi random graphs.

The proof of theorem 1.2 is given in section 3. It relies on Girko's Hermitization method with logarithmic potentials, on theorem 1.1, and on polynomial bounds on the extremal singular values needed to establish a uniform integrability property. This extends the Hermitization method to more general settings, by successfully mixing various arguments already developed in [11,[START_REF]Circular Law Theorem for Random Markov Matrices[END_REF][START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF]. Following Tao and Vu, one of the key steps will be a lower bound on the distance of a row of the matrix A to a subspace of dimension at most n -n 1-γ , for some small γ > 0.

Girko's Hermitization method gives a characterization of µ α in terms of its logarithmic potential (see appendix A). In our settings, however, this is not convenient to derive properties of the measure µ α , and our proof of theorem 1.3 is based on an analysis of a self-adjoint operator on the PWIT and a recursive characterization of the spectral measure from the resolvent of this operator. This method is explained in section 2 while the actual computations on the PWIT are performed in section 4.

Let us conclude with some final remarks. Following [16], the derivation of a Markovian version of theorems 1.1 and 1.2 is an interesting problem, see [START_REF] Ch | Spectrum of large random reversible Markov chains: two examples, ALEA Lat[END_REF]11] for the symmetric case and [18,[START_REF]Circular Law Theorem for Random Markov Matrices[END_REF] for the light tailed non-symmetric case. In another direction, it is also tempting to seek for an interpretation of ν α,z and µ α in terms of a sort of graphical free probability theory. Indeed, our random operators are defined on trees and tree structures are closely related to freeness. Also, with a proper notion of trace, it is possible to define the spectral measure of an operator, see e.g. [15,[START_REF] Haagerup | Brown measures of unbounded operators affiliated with a finite von Neumann algebra[END_REF][START_REF] Lyons | Identities and Inequalities for Tree Entropy[END_REF]. However these notions are usually defined on algebras of bounded operators and we will not pursue this goal here. Note finally that theorems 1.1 and 1.2 remain available for additive perturbations of finite rank, by following the methodology used in [17,[START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF]47]. 1.2. Notation. Throughout the paper, the notation n ≫ 1 means large enough n. For any c ∈ [0, ∞] and any couple f, g of positive functions defined in a neighborhood of c , we say that f (t) ∼ g(t) as t goes to c, if lim t→c f (t)/g(t) = 1. We denote by D ′ (C) the set of Schwartz-Sobolev distributions endowed with its usual convergence with respect to all infinitely differentiable functions with bounded support C ∞ 0 (C). We will consider the differential operators on C ≃ R 2 , for z = x + iy (here i = √ -1)

∂ = 1 2 (∂ x -i∂ y ) and ∂ = 1 2 (∂ x + i∂ y ).
We have ∂ z = ∂z = 0, ∂z = ∂ z = 1 and the Laplace differential operator on C is given by

∆ = 4∂ ∂ = ∂ 2 x + ∂ 2 y .
We use sometimes the shortened notation A -z instead of A -zI.

Bipartized resolvent matrix

The aim of this section is to develop an efficient machinery to analyze the spectral measures of a non-hermitian matrix which avoids a direct use of the logarithmic potential and the singular values. Our approach builds upon similar methods in the physics literature [22,[START_REF] Gudowska-Nowak | Towards non-Hermitian random Lévy matrices[END_REF]44,[START_REF] Rogers | Universal sum and product rules for random matrices[END_REF].

2.1. Bipartization of a matrix. Let n be an integer, and A be a n × n complex matrix. We introduce the symmetrized version of ν A-z , νA-z = 1 2n

n k=1 δ σ k (A-z) + δ -σ k (A-z) .
Let C + = {z ∈ C : Im(z) > 0} and consider the quaternionic-type set

H + = U = η z z η , η ∈ C + , z ∈ C ⊂ M 2 (C).
For z ∈ C, η ∈ C + and 1 i, j n integers, we define the elements of H + and M 2 (C) respectively,

U (z, η) = η z z η and B ij = 0 A ij Āji 0 .
We define the matrix in

M n (M 2 (C)) ≃ M 2n (C), B = (B ij ) 1 i,j n . Since B * ji = B ij
, as an element of M 2n (C), B is an Hermitian matrix. Graphically, the matrix A can be identified with an oriented graph on the vertex set {1, • • • , n} with weight on the oriented edge (i, j) equal to A ij . Then, the matrix B can be thought of as the bipartization of the matrix A, that is a nonoriented graph on the vertex set {1, -1, • • • , -n, n}, for every integers 1 i, j n the weight on the non-oriented edge {i, -j} is A ij , and there is no edge between i and j or -i and -j.

For U ∈ H + , let U ⊗ I n ∈ M n (M 2 (C)) be the matrix given by (U ⊗ I n ) ij = δ ij U , 1 i, j n. The resolvent matrix is defined in M n (M 2 (C)) by R(U ) = (B -U ⊗ I n ) -1 , so that for all 1 i, j n, R(U ) ij ∈ M 2 (C). For 1 k n, we write, with U = U (z, η), R(U ) kk = a k (z, η) b k (z, η) b ′ k (z, η) c k (z, η) . (2.1)
The modulus of the entries of the matrix R(U ) kk are bounded by (Im(η)) -1 (see the forthcoming lemma 2.2). As an element of M 2n (C), R is the usual resolvent of the matrix

B(z) = B -U (z, 0) ⊗ I n . Indeed, with U = U (z, η), R(U ) = (B(z) -ηI 2n ) -1 . (2.2)
In the next proposition, we shall check that the eigenvalues of B(z) are ±σ k (A -z), 1 k n, and consequently µ B(z) = νA-z .

(2.

3)

It will follow that the spectral measures µ A and νA-z can be easily recovered from the resolvent matrix. Recall that the Cauchy-Stieltjes transform of a measure ν on R is defined, for η ∈ C + , as

m ν (η) = R 1 x -η ν(dx).
The Cauchy-Stieltjes transform characterizes the measure. For a probability measure on C, it is possible to define a Cauchy-Stieltjes-like transform on quaternions, by setting for

U ∈ H + , M µ (U ) = C 0 λ λ 0 -U -1 µ(dλ) ∈ H + .
This transform characterizes the measure : in

D ′ (C), lim t↓0 (∂M µ (U (z, it)) 12 = -πµ. If A is normal, i.e. if A * A = AA * , then it can be checked that R(U ) kk ∈ H + and 1 n n k=1 R(U ) kk = M µA (U ).
However, if A is not normal, the above formula fails to hold and the next proposition explains how to recover anyway µ A from the resolvent.

Theorem 2.1 (From resolvent to spectral measure).

Let U = U (z, η) ∈ H + , and a k , b k , b ′ k , c k be as in (2.1). Then (2.3) holds, m νA-z (η) = 1 2n n k=1 (a k (z, η) + c k (z, η)) ,
and, in D ′ (C),

µ A = - 1 πn n k=1 ∂b k (•, 0) = lim t↓0 - 1 πn n k=1 ∂b k (•, it).
In particular, if A is a random matrix with exchangeable entries, then by linearity we get

m EνA-z (η) = Ea 1 (z, η),
and, in D ′ (C),

Eµ A = - 1 π ∂Eb 1 (•, 0) = lim t↓0 - 1 π ∂Eb 1 (•, it).
Proof of theorem 2.1. Through a permutation of the entries, the matrix B(z) is similar to

0 (A -z) (A -z) * 0 ,
whose eigenvalues are easily seen to be ±σ

k (A -z), 1 k n. We get trR = n k=1 (a k + c k ) = n k=1 (σ k (A -z) -η) -1 + n k=1 (-σ k (A -z) -η) -1 .
And the first statement and (2.3) follow. Also, from (A.3), in Appendix, for z / ∈ supp(µ A ),

U µA (z) = ln |x|µ B(z) (dx) = 1 2n ln | det B(z)|, (2.4)
where U µ is the logarithmic potential of a measure µ on C, see (A.1). Recall that the differential of X → det(X) at point X (invertible) in the direction Y is tr(X -1 Y ) det(X) (this is sometimes referred as the Jacobi formula). The sign of det B(z) is (-1) n . We deduce that in D ′ (C),

∂ ln | det B(z)| = ∂ det B(z) det B(z) = tr B(z) -1 ∂ 0 -z -z 0 ⊗ I n . With R kk = R(U (z, 0)) kk = (B(z) -1 ) kk , we get from ∂z = 0, ∂ z = 1, ∂ ln | det B(z)| = n k=1 tr R kk 0 0 -1 0 = - n k=1 b k (z, 0). Now from Equation (A.2), in D ′ (C), using ∆ = 4∂ ∂, 2πµ A = ∆U µA = ∆ 1 2n ln | det B(z)| = - 2 n n k=1
∂b k .

To get the limit as t ↓ 0, we note that for real t > 0,

ln |x -it|µ B(z) (dx) = 1 2n ln | det(B(z) -it)|.
Note that det(B(z) -it) is real and its sign is (-1) n . As t ↓ 0, the left hand side of the above identity converges in D ′ (C), to U µA . Taking the Laplacian, and arguing as above, we get

∆ ln |x -it|µ B(z) (dx) = - 2 n n k=1 ∂b k (z, it).
(2.5)

The conclusion follows.

Note that even ifk ∂b k is a measure on C, for each 1 k n, -∂b k is not in general a measure on C (default of positivity, this can be checked on 2 × 2 matrices).

2.2.

Bipartization of an operator. We shall generalize the above finite dimensional construction. Let V be a countable set and let ℓ 2 (V ) denote the Hilbert space defined by the scalar product φ, ψ :=

u∈V φu ψ u , φ u = δ u , φ ,
where δ u is the unit vector supported on u ∈ V . Let D(V ) denote the dense subset of ℓ 2 (V ) of vectors with finite support. Let (w uv ) u,v∈V be a collection of complex numbers such that for all u ∈ V ,

v∈V |w uv | 2 + |w vu | 2 < ∞
We may then define a linear operator A on D(V ), by the formula,

δ u , Aδ v = w uv . (2.6)
Let V be a set in bijection with V , the image of v ∈ V being denoted by v ∈ V . We set V b = V ∪ V and define the symmetric operator B on D(V b ), by the formulas,

δ u , Bδ v = δ v , Bδ u = w uv δ u , Bδ v = δ û, Bδ v = 0.
(2.7)

In other words, if Π u : ℓ 2 (V b ) → C 2 denotes the orthogonal projection on (u, û),

Π u BΠ * v = 0 w uv wvu 0 .
For z ∈ C, we also define on D(V b ), the symmetric operator B(z): for all u, v in V ,

δ u , B(z)δ v = δ v , B(z)δ u = w uv -z1(u = v) δ u , B(z)δ v = δ û, B(z)δ v = 0.
Hence, if we identify V b with {1, 2} × V , we have

B(z) = B -U (z, 0) ⊗ I V . (2.8)
The operator B(z) is symmetric and it has a closure on a domain D(B) ⊂ ℓ 2 (V b ). We also denote by B(z) the closure of B(z). If B is self-adjoint then B(z) is also self-adjoint (recall that the sum of a bounded self-adjoint operator and a self-adjoint operator is also a self-adjoint operator). Recall also that the spectrum of a self-adjoint operator is real. For all U = U (z, η) ∈ H + , B(z) -ηI V b = B -U (z, η) ⊗ I V is invertible with bounded inverse and the resolvent operator is then well defined by

R(U ) = (B(z) -ηI V b ) -1 .
We may then define

R(U ) vv = Π v R(U )Π * v = a v (z, η) b v (z, η) b ′ v (z, η) c v (z, η)
.

In the sequel, we shall use some properties of resolvent operators.

Lemma 2.2 (Properties of resolvent). Let B be the above bipartized operator. Assume that B is self-adjoint and let

U = U (z, η) ∈ H + , v ∈ V . Then, a v , c v ∈ C + , for each z ∈ C, the functions a v (z, •), b v (z, •), b ′ v (z, •), c v (z,
•) are analytic on C + , and

|a v | (Im(η)) -1 , |c v | (Im(η)) -1 , |b v | (2Im(η)) -1 and |b ′ v | (2Im(η)) -1 . Moreover, if η ∈ iR + ,
then a v and c v are pure imaginary and b ′ v = bv . Proof. For a proof of the first statements refer e.g. to Reed and Simon [START_REF] Reed | Methods of modern mathematical physics[END_REF]. For the last statement concerning η ∈ iR + , we define the skeleton of B(z) as the graph on V b obtained by putting an edge between two vertices u, v in V b , if δ u , B(z)δ v = 0. Then since there is no edge between two vertices of V or V , the skeleton of B(z) is a bipartite graph.

Assume first that B(z) is bounded: for all u ∈ V b , B(z)δ u C. Then for |η| > C, the series expansion of the resolvent gives

R(U ) = - ∞ n=0 B(z) n η n+1 .
However since the skeleton is a bipartite graph, all cycles have an even length. It implies that for n odd, δ u , B(z) n δ u = 0. Applied first to v ∈ V , we deduce that for |η| > C, a(z, -η) = -ā(z, η) and then applied to v, we get c(z, -η) = -c(z, η). We may then extend to C + this last identity by analyticity. For η = it ∈ iR + , we deduce that a v and c v are pure imaginary. Similarly, since the skeleton is a bipartite graph, a path from a vertex v ∈ V to a vertex û ∈ V must of be of odd length. We get for |η| > C

b′ v (z, -η) = δ v , R(U (z, -η))δ v = - ∞ n=0 δ v , B(z) 2n+1 δ v η 2n+2 = δ v , R(U )δ v = b v (z, η),
where we have used the symmetry of B(z). It follows that b ′ v (z, -η) = bv (z, η). If B(z) is not bounded, then B(z) is limit of a sequence of bounded operators and we conclude by invoking Theorem VIII.25(a) in [START_REF] Reed | Methods of modern mathematical physics[END_REF].

2.3.

Operator on a tree. We keep the setting of the above paragraph and consider a (nonoriented) tree T = (V, E) on the vertices V with edge set E (recall that a tree is a connected graph without cycles). For ease of notation, we note u ∼ v if {u, v} ∈ E. We assume that if {u, v} / ∈ E then w uv = w vu = 0. In particular w vv = 0 for all v ∈ V . We continue to consider the operator A defined by (2.6).

In the special case when w uv = w vu for all u, v in V , the operator A is symmetric and we first look for sufficient conditions for A to be essentially self-adjoint. Lemma 2.3 (Criterion of self-adjointness). Let κ > 0 and T = (V, E) be a tree. Assume that for all u, v ∈ V , w uv = w vu and that if {u, v} / ∈ E then w uv = w vu = 0. Assume also that there exists a sequence of connected finite subsets When the operator B is self-adjoint, the resolvent operator has a nice recursive expression due to the tree structure. Let ∅ ∈ V be a distinguished vertex of V (in graph language, we root the tree T at ∅). For each v ∈ V \{∅}, we define V v ⊂ V as the set of vertices whose unique path to the root ∅ contains v. We define

(S n ) n 1 in V , such that S n ⊂ S n+1 , ∪ n S n = V ,
T v = (V v , E v ) as the subtree of T spanned by V v . We finally consider A v , the projection of A on V v , and B v the bipartized operator of A v . The skeleton of A v is contained in T v . Finally, we note that if B is self-adjoint then so is B v (z) for every z ∈ C.
The next lemma can be interpreted as a Schur complement formula on trees.

Lemma 2.5 (Resolvent on a tree). Assume that B is self-adjoint and let

U = U (z, η) ∈ H + . Then R(U ) ∅∅ = -U + v∼∅ 0 w ∅v w v∅ 0 R(U ) vv 0 w v∅ w ∅v 0 -1 , where R(U ) vv = Π v R Bv (U )Π * v and R Bv (U ) = (B v (z) -η) -1
is the resolvent operator of B v . Proof. Define the operator C on D(V b ) by its matrix elements

C ∅ := Π ∅ CΠ * ∅ = -U (z, 0) , C v := Π ∅ CΠ * v = Π v CΠ * ∅ = 0 w ∅v w v∅ 0
for all v ∈ V such that v ∼ ∅, and Π u CΠ * v = 0 otherwise. The operator C is symmetric and bounded. Its extension to ℓ 2 (V b ) is thus self-adjoint (also denoted by C). In this way, we have from

V = {∅} v∼∅ V v , B(z) = C + B with B = v∼∅ B v (z).
We shall write R(U ) = ( B -ηI) -1 for the associated resolvent of B. From the resolvent identity, these operators satisfy

R(U )CR(U ) = R(U ) -R(U ) . (2.9) Set R uv = Π u R(U )Π * v and R uv = Π u R(U )Π * v . Observe that R ∅∅ = -η -1 I 2 . Also the direct sum decomposition V = {∅} v∼∅ V v implies R vv = Π v R Bv (U )Π * v and R uv = 0 for every u = v with u ∼ ∅, v ∼ ∅. Similarly we have that R ∅v = 0 = R v∅ for every v ∈ V \{∅}. Using the identity u∈V Π * u Π u = I, we get Π ∅ R(U )CR(U )Π * ∅ = R ∅∅ C ∅ R ∅∅ + v∼∅ R ∅∅ C v R v∅ = η -1 U (z, 0)R ∅∅ -η -1 v∼∅ C v R v∅ .
We compose the identity (2.9) on the left by Π v and on the right by Π * ∅ , we obtain, for v ∼ ∅,

R vv C * v R ∅∅ = -R v∅ .
We finally compose (2.9) on the left by Π ∅ and on the right by Π

* ∅ , η -1 U (z, 0)R ∅∅ + η -1 v∼∅ C v R vv C * v R ∅∅ = -η -1 I 2 -R ∅∅ , or equivalently (U (z, η) + v∼∅ C v R vv C * v )R ∅∅ = -I 2 . 2.4.
Local operator convergence. In the next paragraphs, we are going to prove that the sequence of random matrices (A n ) converges to a limit random operator on an infinite tree. Let us recall a notion of convergence that we have already used in [11].

Definition 2.6 (Local convergence). Suppose (A n ) is a sequence of bounded operators on ℓ 2 (V )
and A is a linear operator on ℓ 2 (V ) with domain D(A) ⊃ D(V ). For any u, v ∈ V we say that (A n , u) converges locally to (A, v), and write

(A n , u) → (A, v) , if there exists a sequence of bijections σ n : V → V such that σ n (v) = u and, for all φ ∈ D(V ), σ -1 n A n σ n φ → Aφ , in ℓ 2 (V ), as n → ∞.
Assume in addition that A is closed and D(V ) is a core for A (i.e. the closure of A restricted to D(V ) equals A). Then, the local convergence is the standard strong convergence of operators up to a re-indexing of V which preserves a distinguished element. With a slight abuse of notation we have used the same symbol σ n for the linear isometry σ n : ℓ 2 (V ) → ℓ 2 (V ) induced in the obvious way. As pointed out in [11], the point for using Definition 2.6 lies in the following theorem on strong resolvent convergence.

Theorem 2.7 (From local convergence to resolvents). Assume that (A n ) and A satisfy the conditions of Definition 2.6 and

(A n , u) → (A, v) for some u, v ∈ V . Let B n be the self-adjoint bipartized operator of A n . If the bipartized operator B of A is self-adjoint and D(V b ) is a core for B, then, for all U ∈ H + , R Bn (U ) uu → R B (U ) vv . (2.10) where R B (U ) vv = Π v R B (U )Π * v and R B (U ) = (B(z) -η) -1
is the resolvent of B(z). Proof of theorem 2.7. It is a special case of Reed and Simon [START_REF] Reed | Methods of modern mathematical physics[END_REF]Theorem VIII.25(a)]. Indeed, we first fix z ∈ C and extend the bijection σ n to V b by the formula, for all w ∈ V , σ n ( ŵ) = σn (w). Then we define B n (z) = σ -1 n B n (z)σ n , so that B n (z)φ → B(z)φ for all φ in a common core of the self-adjoint operators B n (z), B(z). This implies the strong resolvent convergence, i.e. ( B n (z)-ηI) -1 ψ → (B(z)-ηI) -1 ψ for any η ∈ C + , ψ ∈ ℓ 2 (V ). We conclude by using the identities

: Π v ( B n (z) -ηI) -1 δ v = Π u (B n (z) -ηI) -1 δ u and Π v ( B n (z) -ηI) -1 δ v = Π u (B n (z) -ηI) -1 δ û.
We shall apply the above theorem in cases where the operators A n and A are random operators on ℓ 2 (V ), which satisfy with probability one the conditions of theorem 2.7. In this case we say that (A n , u) → (A, v) in distribution if there exists a random bijection σ n as in Definition 2.6 such that σ -1 n A n σ n φ converges in distribution to Aφ, for all φ ∈ D(V ) (where a random vector

ψ n ∈ ℓ 2 (V ) converges in distribution to ψ if lim n→∞ Ef (ψ n ) = Ef (ψ) for all bounded continuous functions f : ℓ 2 (V ) → R).
Under these assumptions then (2.10) becomes convergence in distribution of (bounded) complex random variables. Note that in order to prove theorems 1.1, 1.2, we will also need almost-sure convergence statements.

Poisson Weighted Infinite Tree (PWIT).

We now define an operator on an infinite rooted tree with random edge-weights, the Poisson weighted infinite tree (PWIT) introduced by Aldous [START_REF] Aldous | Asymptotics in the random assignment problem[END_REF], see also [START_REF] Aldous | The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures[END_REF].

Let ρ be a positive Radon measure on R such that ρ(R) = ∞. PWIT(ρ) is the random weighted rooted tree defined as follows. The vertex set of the tree is identified with N f := ∪ k∈N N k by indexing the root as N 0 = ∅, the offsprings of the root as N and, more generally, the offsprings of some v ∈ N k as (v1), (v2), • • • ∈ N k+1 (for short notation, we write (v1) in place of (v, 1)). In this way the set of v ∈ N n identifies the n th generation. We then define T as the tree on N f with (non-oriented) edges between the offsprings and their parents.

We denote by Be(1/2) the Bernoulli probability distribution 1 2 δ 0 + 1 2 δ 1 . Now assign marks to the edges of the tree T according to a collection {Ξ v } v∈N f of independent realizations of the Poisson point process with intensity measure ρ ⊗ Be(1/2) on R × {0, 1}. Namely, starting from the root ∅, let Ξ ∅ = {(y 1 , ε 1 ), (y 2 , ε 2 ), . . . } be ordered in such a way that |y 1 | |y 2 | • • • , and assign the mark (y i , ε i ) to the offspring of the root labeled i. Now, recursively, at each vertex v of generation k, assign the mark (y vi , ε vi ) to the offspring labeled vi, where

Ξ v = {(y v1 , ε v1 ), (y v2 , ε v2 ), . . . } satisfy |y v1 | |y v2 | • • • .
The Bernoulli mark ε vi should be understood as an orientation of the edge {v, vi} : if ε vi = 1, the edge is oriented from vi to v and from v to vi otherwise.

For a probability measure θ on S 1 , we introduce the measure on C, for all Borel D:

ℓ θ (D) = ∞ 0 S 1 1 {ω -α r∈D} θ(dω)dr (2.11)
Consider a realization of PWIT(2ℓ θ ). We now define a random operator A on D(N f ) by the formula, for all v ∈ N f and k ∈ N,

δ v , Aδ vk = ε vk y -1/α vk and δ vk , Aδ v = (1 -ε vk )y -1/α vk (2.12)
and δ v , Aδ u = 0 otherwise. It is an operator as in §2.3. Indeed, if u = vk is an offspring of v, we set

w vu = ε vk y -1/α vk and w uv = (1 -ε vk )y -1/α vk , (2.13)
otherwise, we set w uv = 0. We may thus consider the bipartized operator B of A.

Proposition 2.8 (Self-adjointness of bipartized operator on PWIT). Let A be the random operator defined by (2.12). With probability one, for all z ∈ C, B(z) is self-adjoint.

We shall use Corollary 2.4. We start with a technical lemma proved in [11, Lemma A.4].

Lemma 2.9.

Let κ > 0, 0 < α < 2 and let 0 < x 1 < x 2 < • • • be a Poisson process of intensity 1 on R + . Define τ = inf{t ∈ N : ∞ k=t+1 x -2/α k κ}.
Then Eτ is finite and goes to 0 as κ goes to infinity.

Proof of proposition 2.8. For κ > 0 and v ∈ N f , we define

τ v = inf{t 0 : ∞ k=t+1 |y vk | -2/α κ}.
The variables (τ v ) are i.i.d. and by lemma 2.9, there exists κ > 0 such that Eτ v < 1. We fix such κ. Now, we put a green color to all vertices v such that τ v 1 and a red color otherwise. We consider an exploration procedure starting from the root which stops at red vertices and goes on at green vertices. More formally, define the sub-forest T g of T where we put an edge between v and vk if v is a green vertex and 1 k τ v . Then, if the root ∅ is red, we set

S 1 = C g (T ) = {∅}.
Otherwise, the root is green, and we consider T g ∅ = (V g ∅ , E g ∅ ) the subtree of T g that contains the root. It is a Galton-Watson tree with offspring distribution τ ∅ . Thanks to our choice of κ, T g ∅ is almost surely finite. Consider L g ∅ the leaves of this tree (i.e. the set of vertices v in V g ∅ such that for all 1 k τ v , vk is red). We set

S 1 = V g ∅ v∈L g ∅ {1 k τ v : vk}.
Clearly, the set S 1 satisfies the condition of Lemma 2.3. Now, we define the outer boundary of {∅} as

∂ τ {∅} = {1, • • • , τ ∅ } and for v = (i 1 , • • • i k ) ∈ N f \{∅} we set ∂ τ {v} = {(i 1 , • • • , i k-1 , i k + 1)} ∪ {(i 1 , • • • , i k , 1), • • • , (i 1 , • • • , i k , τ v )}.
For a connected set S, its outer boundary is

∂ τ S = v∈S ∂ τ {v} \S. Now, for each vertex u 1 , • • • , u k ∈ ∂ τ S 1 ,
we repeat the above procedure to the rooted subtrees

T u1 , • • • , T u k . We set S 2 = S 1 ∪ 1 i k C b (T ui )
. Iteratively, we may thus almost surely define an increasing connected sequence (S n ) of vertices with the properties required for Corollary 2.4.

2.6.

Local convergence to PWIT. We may now come back to the random matrix A n defined by (1.4). We extend it as an operator on D(N f ) by setting for 1 i, j n, δ i , Aδ j = A i,j and otherwise, if either i or j is in

N f \{1, • • • n}, δ i , Aδ j = 0.
The aim of this paragraph is to prove the following theorem.

Theorem 2.10 (Local convergence to PWIT). Assume (H1). Let A n be as above and A be the operator associated to PWIT(2ℓ θ ) defined by (2.12). Then in distribution (A n , 1) → (A, ∅).

Up to small differences, this theorem has already been proved in [11, Section 2]. We review here the method of proof and stress the differences. The method relies on the local weak convergence, a notion introduced by Benjamini and Schramm [8], Aldous and Steele [START_REF] Aldous | The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures[END_REF], see also Aldous and Lyons [START_REF] Aldous | Processes on unimodular random networks[END_REF].

We define a network as a graph with weights on its edges taking values in some metric space. Let G n be the complete network on {1, . . . , n} whose weight on edge {i, j} equals (ξ n i,j ), for some collection (ξ n i,j ) 1 i j n of i.i.d. complex random variables. We set ξ n j,i = ξ n i,j . We consider the rooted network (G n , 1) obtained by distinguishing the vertex labeled 1.

We follow Aldous [1, Section 3]. For every fixed realization of the marks (ξ n ij ), and for any B, H ∈ N, such that (B H+1 -1)/(B -1) n, we define a finite rooted subnetwork (G n , 1) B,H of (G n , 1), whose vertex set coincides with a B-ary tree of depth H with root at 1. To this end we partially index the vertices of (G n , 1) as elements in

J B,H = ∪ H ℓ=0 {1, • • • , B} ℓ ⊂ N f
, the indexing being given by an injective map σ n from J B,H to V n := {1, . . . , n}. We set

I ∅ = {1} and the index of the root 1 is σ -1 n (1) = ∅. The vertex v ∈ V n \I ∅ is given the index (k) = σ -1 n (v), 1 k B, if ξ n 1,v
has the k th smallest absolute value among {ξ n 1,j , j = 1}, the marks of edges emanating from the root 1. We break ties by using the lexicographic order. This defines the first generation. Now let I 1 be the union of I ∅ and the B vertices that have been selected. If H 2, we repeat the indexing procedure for the vertex indexed by (1) (the first child) on the set V n \I 1 . We obtain a new set {11, • • • , 1B} of vertices sorted by their weights as before (for short notation, we concatenate the vector (1, 1) into 11). Then we define I 2 as the union of I 1 and this new collection. We repeat the procedure for (2) on V n \I 2 and obtain a new set {21, • • • , 2B}, and so on. When we have constructed {B1, • • • , BB}, we have finished the second generation (depth 2) and we have indexed (B 3 -1)/(B -1) vertices. The indexing procedure is then repeated until depth H so that (B H+1 -1)/(B -1) vertices are sorted. Call this set of vertices V B,H n = σ n J B,H . The subnetwork of G n generated by V B,H n is denoted (G n , 1) B,H (it can be identified with the original network G n where any edge e touching the complement of V B,H n is given a mark x e = ∞). In (G n , 1) B,H , the set {u1, • • • , uB} is called the set of offsprings of the vertex u. Note that while the vertex set has been given a tree structure, (G n , 1) B,H is still a complete network on V B,H n . The next proposition shows that it nevertheless converges to a tree (i.e. extra marks diverge to ∞) if the ξ n i,j satisfy a suitable scaling assumption.

Let ρ be a Radon measure on C and let T be a realization of PWIT(ρ) defined in §2.5. For the moment, we remove the Bernoulli marks (ε v ) v∈N f and, for v ∈ N f and k ∈ N, we define the weight on edge {v, vk} to simply be y vk . Then (T, ∅) is a rooted network. We call (T, ∅) B,H the finite random network obtained by the same sorting procedure. Namely, (T, ∅) B,H consists of the subtree with vertices in J B,H , with the marks inherited from the infinite tree. If an edge is not present in (T, ∅) B,H , we assign to it the mark +∞.

We say that the sequence of random finite networks (G n , 1) B,H converges in distribution (as n → ∞) to the random finite network (T, ∅) B,H if the joint distributions of the marks converge weakly. To make this precise we have to add the points {±∞} as possible values for each mark, and continuous functions on the space of marks have to be understood as functions such that the limit as any one of the marks diverges to +∞ exists and coincides with the limit as the same mark diverges to -∞. We may define C = C ∪ {±∞}. The next proposition generalizes [1, Section 3], for a proof see [11, Proposition 2.6] (the proof there is stated for a measure ρ on R, the complex case extends verbatim). Let G n be the complete network on {1, . . . , n} whose mark on edge {i, j} equals ξ n ij , and T a realization of PWIT(ρ). Then, for all integers B, H,

(G n , 1) B,H n→∞ (T, ∅) B,H .
Now, we shall extend the above statement to directed networks. More precisely, let (ξ n i,j ) 1 i,j n be i.i.d. real random variables. We consider the complete graph Ḡn on V n whose weight on edge {i, j} equals, if i j, (ξ n i,j , ξ n j,i ) ∈ R 2 . As above, we partially index the vertices of ( Ḡn , 1) as elements in

J B,H = ∪ H ℓ=0 {1, • • • , B} ℓ ⊂ N f
, the indexing being given by an injective map σ n from J B,H to V n such that σ -1 n (1) = ∅. The difference with the above construction, is that the vertex

v ∈ V n \{1} is given the index (k) = σ -1 n (v), 1 k B, if min(|ξ n 1,v |, |ξ n v,1 |) has the k th smallest value among {min(|ξ n 1,j |, |ξ n j,1 |), j = 1}.
Similarly, let (T, ∅) be the infinite random rooted network with distribution PWIT(ρ). This time we do not remove the Bernoulli marks (ε v ) v∈N f and define the weight on edge {v, vk} as (y vk , ∞) if ε vk = 1 and (∞, y vk ) if ε vk = 0. Again, we call (T, ∅) B,H the finite random network obtained by the sorting procedure : (T, ∅) B,H consists of the subtree with vertices in J B,H , with the marks inherited from the infinite tree.

We apply proposition 2.11 to the complete network G n with mark on edge {i, j} equals, if i j, to min(|ξ n i,j |, |ξ n j,i |). This network satisfies (2.14) with 2ρ. We remark that if u, v ∈ J B,H then from (2.14), max(|ξ n σn(u),σn(v) |, |ξ n σn(v),σn(u) |) diverges weakly to infinity. We also notice that, given (G n , 1) B,H , with equal probability |ξ n σn(u),σn(v) | is larger or less than |ξ n σn(v),σn(u) |. We deduce the following.

Corollary 2.12 (Local weak convergence to a tree). Let ρ be a Radon measure on C with no mass at 0. Let (ξ n i,j ) 1 i,j n be a collection of i.i.d. random variables in C such that (2.14) holds. Let Ḡn be the complete network on {1, . . . , n} whose mark on edge {i, j} equals, if i j, (ξ n i,j , ξ n j,i ), and T a realization of PWIT(2ρ). Then, for all integers B, H, ( Ḡn , 1) B,H n→∞ (T, ∅) B,H .

We may now prove theorem 2.10.

Proof of theorem 2.10. We argue as in the proof of theorem 2.3(i) in [11, Section 2]. We first define the weights (ξ n i,j ) i,j∈N f as follows. For integers 1 i, j n, we set

ξ n i,j = A -α i,j = a α n X -α i,j
, with the convention that ξ n i,j = ∞ if X i,j = 0. For this choice, by assumption (H1), (2.14) holds with ρ = ℓ θ and ℓ θ in (2.11). If i or j is in N f \{1, • • • , n}, we set ξ n i,j = ∞. Let Ḡn denote the complete network on {1, • • • , n} with marks (ξ n i,j , ξ n j,i ) on edge {i, j}, if i j. From Corollary 2.12, for all B, H, ( Ḡn , 1) B,H converges weakly to (T, ∅) B,H , where T has distribution PWIT(2ℓ θ ). Let A be the random operator associated to T .

Let σ B,H n be the map σ n associated to the network ( Ḡn , 1) B,H . The maps σ n are arbitrarily extended to a bijection N f → N f . From the Skorokhod Representation Theorem we may assume that ( Ḡn , 1) B,H converges a.s. to (T, ∅) B,H for all B, H. Thus we may find sequences B n , H n tending to infinity and a sequence of bijections σ n := σ Bn,Hn n such that (B Hn+1 n -1)/(B n -1) n and such that for any pair u, v ∈ N f we have ξ n σn(u), σn(v) which converge a.s. to    y uk if for some integer k, v = uk and ε uk = 1 y vk if for some integer k, u = vk and ε vk = 0 ∞ otherwise It follows that a.s.

δ u , σ -1 n A n σ n δ v = ξ n σn(u), σn(v)
For any v, set ψ v n := σ -1 n A n σ n δ v . To prove theorem 2.10, it is sufficient to show that for any v ∈ N f , ψ v n → Aδ v in ℓ 2 (N f ) almost surely as n goes to infinity, i.e.,

u ( δ u , ψ v n -δ u , Aδ v ) 2 → 0 .
From what precedes, we know that δ u , ψ v n → δ u , Aδ v for every u. The claim follows if we have (almost surely) uniform (in n) square-integrability of ( δ u , ψ v n ) u . This in turn follows from Lemma 2.4(i) and Lemma 2.7 in [11].

2.7. Convergence of the resolvent matrix. Let A n and A be as in theorem 2.10. From proposition 2.8, we may almost surely define the resolvent R of the bipartized random operator of

A. For U = U (z, η) ∈ H + , we set R(U ) ∅∅ = Π ∅ R(U )Π * ∅ = a(z, η) b(z, η) b ′ (z, η) c(z, η) . (2.15) We define similarly, R n (U ) = (B n (z) -η) -1 , the resolvent of B n , the bipartized operator of A n . We set R n (U ) 11 = Π 1 R n (U )Π * 1 . Theorem 2.
m να,z (η) = Ea(z, η),
where a(z, η) was defined in (2.15).

Proof. For every z ∈ C, by proposition 2.8, the operator B(z) is a.s. self-adjoint. It implies that there exists a.s. a measure on R, ν ∅,z , called the spectral measure with vector δ ∅ , such that for all η ∈ C + ,

a(z, η) = δ ∅ , R(U )δ ∅ = ν ∅,z (dx) x -η = m ν∅,z (η).
We define R n as the resolvent matrix of B n , the bipartized matrix of

A n . For U = U (z, η) ∈ H + , we write R n (U ) kk = a k b k b ′ k c k . By theorem 2.1, m EνA n-z (η) = Ea 1 (z, η).
By lemma 2.2, for U ∈ H + , the entries of the matrix R n (U ) 11 are bounded. It follows from theorem 2.13 that for all U ∈ H + ,

lim n→∞ ER n (U ) 11 = E a b b ′ c ,
where the limit matrix was defined in (2.15). Hence, for all z ∈ C + ,

lim n→∞ m EνA n -z (η) = Ea(z, η).
We deduce that Eν An-z converges to the measure ν α,z = Eν ∅,z . This convergence can be improved to almost sure by showing that the random measure νAn-z concentrates around its mean. This is done by applying Borel-Cantelli Lemma and lemma C.2 to the matrix B n (z) whose spectral measure equals νAn-z , see (2.3).

Theorem 1.1 is a corollary of the above theorem up to the fact that Ea(z, η) does not depend on the measure θ which appears in (H1). The latter will be a consequence of the forthcoming theorem 4.1.

3.

Convergence of the spectral measure 3.1. Tightness. In this paragraph, we prove that the counting probability measures of the eigenvalues and singular values of the random matrices (A n ) defined by (1.4) are a.s. tight. For ease of notation, we will often write A in place of A n . Lemma 3.1 (Tightness). If (H1) holds, there exists r > 0 such that for all z ∈ C, a.s. Proof. In both cases, the a.s. tightness follows from the moment bound and the Markov inequality. The moment bound on µ A follows from the statement on ν A (take z = 0) by using the Weyl inequality (B.6). It is therefore enough to establish the moment bound on ν A-zI for every C. Let us fix z ∈ C and r > 0. By definition of ν A-zI we have

∞ 0 t r ν A-zI (dt) = 1 n n k=1 s k (A -zI) r .
From (B.2) we have s k (A -zI) s k (A) + |z| for every 1 k n, and one can then safely assume that z = 0 for the proof. By using (B.7) we get for any 0 r 2,

∞ 0 t r ν A (dt) Z n := 1 n n i=1 Y n,i where Y n,i :=   n j=1 a -2 n |X ij | 2   r/2
.

We need to show that (Z n ) n 1 is a.s. bounded. Assume for the moment that sup

n 1 E(Y 4 n,1 ) < ∞ (3.1)
for some choice of r. Since Y n,1 , . . . , Y n,n are i.i.d. for every n 1, we get from (3.1) that

E((Z n -EZ n ) 4 ) = n -4 E   1 i,j n (Y n,i -EY n,i ) 2 (Y n,j -EY n,j ) 2   = O(n -2 ).
Therefore, by the monotone convergence theorem, we get

E( n 1 (Z n -EZ n ) 4 ) < ∞, which gives n 1 (Z n -EZ n ) 4 < ∞ a.s. and thus Z n -EZ n → 0 a.s. Now the sequence (EZ n ) n 1 = (EY n,1
) n 1 is bounded by (3.1) and it follows that (Z n ) n 1 is a.s. bounded.

It remains to show that (3.1) holds, say if 0 < 4r < α. To this end, let us define

S n,a,b := n j=1 a -2 n |X 1j | 2 1 {a -2 n |X1j | 2 ∈[a,b)} for every a < b. Now Y 4 n,1 = (S n,0,∞ ) 2r = (S n,0,1 + S n,1,∞ ) 2r and thus, E(Y 4 n,1 ) 2 2r-1 E(S 2r n,0,1 ) + E(S 2r n,1,∞ ) . (3.2)
We have sup n 1 E(S 2r n,0,1 ) < ∞. Indeed, since 2r < 1, by the Jensen inequality, E(S 2r n,0,1 ) (ES n,0,1 ) 2r and by lemma C.1, ES n,0,1 ∼ n α/(2 -α). For the second term of the right hand side of (3.2), we set

M n := max 1 j n a -1 n |X 1j |1 {a -1 n |X1j |>1} and N n := #{1 j n s.t. a -1 n |X 1j | > 1}.
From Hölder inequality, if 1/p + 1/q = 1, we have

E(S 2r n,1,∞ ) E N 2r n M 4r n EN 2rp n 1/p EM 4rq n 1/q . (3.3) Recall that P(|X 12 | > a n ) = (1 + o(1)
)/n 2/n for n ≫ 1. By the union bound, for n ≫ 1,

P(N n k) n k P(|X 12 | > a n ) k n k k! 2 k n k = 2 k k! .
In particular, we have sup n 1 EN η n < ∞ for any η > 0. Similarly, since the function L is slowly varying, for n ≫ 1 and all t 1, we have

P(M n t) nP(|X 12 | > ta n ) = na -α n t -α L(a n t) 2t -α . It follows that if γ < α, sup n 1 EM γ n < ∞.
Taking p and q so that 4rq < α, we thus conclude from (3.3) that sup n 1 E(S 2r n,1,∞ ) < ∞. 3.2. Invertibility. In this paragraph, we find a lower bound for the smallest singular value of the random matrix A -zI where A is defined by (1.4), in other words an upper bound on the operator norm of the resolvent of A. Such lower bounds on the smallest singular value of random matrices were developed in the recent years by using Littlewood-Offord type problems, as in [START_REF] Tao | Random matrices: the circular law[END_REF][START_REF]Smooth analysis of the condition number and the least singular value[END_REF] and [45]. The available results require moments assumptions which are not satisfied when the entries have heavy tails. Here we circumvent the problem by requiring the bounded density hypothesis (H3). The removal of this hypothesis can be done by adapting the Rudelson and Vershynin approach already used by Götze and Tikhomirov [START_REF] Götze | The Circular Law for Random Matrices[END_REF]. Lemma 3.2 (Invertibility). If (H3) holds then for some r > 0, every z ∈ C, a.s.

lim n→∞ n r s n (A -zI) = +∞.
Proof. For every x, y ∈ C n and S ⊂ C n , we set x • y := x 1 y 1 + • • • + x n y n and x 2 := √ x • x and dist(x, S) := min y∈S x -y 2 . Let R 1 , . . . , R n be the rows of A -zI and set R -i := span{R j ; j = i} for every 1 i n. From lemma B.2 we have min

1 i n dist(R i , R -i ) √ n s n (A -zI)
and consequently, by the union bound, for any u 0,

P( √ n s n (A -zI) u) n max 1 i n P(dist(R i , R -i ) u).
Let us fix 1 i n. Let Y i be a unit vector orthogonal to R -i . Such a vector is not unique. We just pick one. This defines a random variable on the unit sphere S n-1 = {x ∈ C n : x 2 = 1}. By the Cauchy-Schwarz inequality,

|R i • Y i | π i (R i ) 2 Y i 2 = dist(R i , R -i )
where π i (•) is the orthogonal projection on the orthogonal complement of R -i . Let ν i be the distribution of Y i on S n-1 . Since Y i and R i are independent, for any u 0,

P(dist(R i , R -i ) u) P(|R i • Y i | u) = S n-1 P(|R i • y| u) dν i (y)
Let us first consider the case where X 11 has a bounded density ϕ on C. Since y 2 = 1 there exists an index j 0 ∈ {1, . . . , n} such that y j0 = 0 with |y j0 | -1

√ n. The complex random variable R i • y is a sum of independent complex random variables and one of them is a -1 n X ij0 y j0 , which is absolutely continuous with a density bounded above by a n √ n ϕ ∞ . Consequently, by a basic property of convolutions of probability measures, the complex random variable R i • y is also absolutely continuous with a density ϕ i bounded above by a n √ n ϕ ∞ , and thus

P(|R i • y| u) = C 1 |s| u ϕ i (s) ds πu 2 a n √ n ϕ ∞ .
Therefore, for every b > 0,

P(s n (A -zI) n -b-1/2 ) = O(n 3/2-2b a n )
where the O does not depend on z. By taking b large enough, the first Borel-Cantelli lemma implies that there exists r > 0 such that a.s. for every z ∈ C and n ≫ 1,

s n (A -zI) n -r .
It remains to consider the case where X 11 has a bounded density ϕ on R. As for the complex case, let us fix y ∈ S n-1 . Since y 2 = 1 there exists an index j 0 ∈ {1, . . . , n} such that |y j0 | -1 √ n. Also, either |Re(y j0 )| -1 √ 2n or |Im(y j0 )| -1 √ 2n. Assume for instance that |Re(y j0 )| -1 √ 2n. We observe that for every u 0,

P(|R i • y| u) P(|Re(R i • y)| u).
The real random variable Re(R i • y) is a sum of independent real random variables and one of them is a -1 n X ij0 Re(y j0 ), which is absolutely continuous with a density bounded above by a n √ 2n ϕ ∞ . Consequently, by a basic property of convolutions of probability measures, the real random variable Re(R i • y) is also absolutely continuous with a density ϕ i bounded above by a n √ 2n ϕ ∞ . Therefore, we have for every u 0,

P(|Re(R i • y)| u) = [-u,u] ϕ i (s) ds 2 3/2 a n √ n u ϕ ∞ .
We skip the rest of the proof, which is identical to the complex case.

3.3. Distance from a row to a vector space. In this paragraph, we give two lower bounds on the distance of a row of the random matrix A -z defined by (1.4) to a vector space of not too large dimension. The first ingredient is an adaptation of Proposition 5.1 in Tao and Vu [START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF].

Proposition 3.3 (Distance of a row to a subspace). Assume that (H1) holds. Let 0 < γ < 1/2, and let R be a row of a n (A -z). There exists δ > 0 depending on α, γ such that for all d-dimensional subspaces W of C n with n -d n 1-γ , one has

P dist(R, W ) n (1-2γ)/α e -n δ .
The proof of proposition 3.3 is based on a concentration estimate for the truncated variables X 1i 1 {|X1i| bn} for suitable sequences b n . We first recall a concentration inequality of Talagrand.

Theorem 3.4 (Talagrand concentration inequality [START_REF] Talagrand | Concentration of measure and isoperimetric inequalities in product spaces[END_REF] and [34, Corollary 4.10]). Let us denote by D := {z ∈ C; |z| 1} the complex unit disc and let P be a product probability measure on the product space D n . Let F : D n → R be a Lipschitz convex function on D n with F Lip 1. If M (F ) is a median of F under P then for every r 0,

P (|F -M (F )| r) 4e -r 2 /4 .
Proof of proposition 3.3. We first perform some pre-processing of the vector R as in Tao-Vu [START_REF]Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur[END_REF].

To fix ideas, we may assume that R is the first row of a n (A -z). Then R = X 1 -za n e 1 where X 1 is the first row of X = a n A. We then have dist(R, W ) dist(X 1 -za n e 1 , span(W, e 1 )) = dist(X 1 , W 1 ).

where we have set W 1 = span(W, e 1 ). Note that d dim W 1 d + 1.

For any sequence b n , from the Markov inequality,

P n i=1 1 {|X1i| bn} √ n e -√ n Ee 1 |X 11 | bn n e - √ n 1 + eL(b n )b -α n n e - √ n+enL(bn)b -α n . (3.4) Choose b n = a n n -2γ/α . Clearly, b n /n (1-2γ)/α ∈ [n -ε , n ε ]
eventually for all ε > 0. Let J denote the set of indexes i such that |X 1i | b n . From (3.4) we see that, for some δ > 0:

P(|J | < n - √ n) e -n δ .
It follows that it is sufficient to prove the statement conditioned on the event {|J | n -√ n}. In particular, we shall prove that for any fixed I ⊂ {1, . . . , n}, such that |I| n -√ n,

P dist(X 1 , W 1 ) n (1-2γ)/α | J = I e -n δ . (3.5)
Without loss of generality, we assume that

I = {1, • • • , n ′ } with n ′ n - √ n. Let π I be the orthogonal projection on span(e i : i ∈ I). If W 2 = π I (W 1 ), we find d - √ n dim(W 2 ) dim(W 1 ) d + 1 and dist(X 1 , W 1 ) dist(π I (X 1 ), W 2
). Note that π I (X 1 ) is simply the vector X 1i , i = 1, . . . , n ′ . We set

W ′ = span(W 2 , E[π I (X 1 ) | J = I]) , Y = π I (X 1 ) -E[π I (X 1 ) | J = I] , so that d - √ n dim(W ′ ) d + 2 and dist(π I (X 1 ), W 2 ) dist(Y, W ′ ).
Let P denote the orthogonal projection matrix to the orthogonal complement of

W ′ in C n ′ . We have dist 2 (Y, W ′ ) = i,j Y i P ij Ȳj , and, since Y = (Y i ) 1 i n ′ is a mean zero vector under P(• | I = I), E[dist 2 (Y, W ′ ) | J = I] = E i,j Y i P ij Ȳj | J = I = n ′ i=1 P ii E[|Y i | 2 | J = I] = E[|Y 1 | 2 | J = I] trP.
We have for any ε > 0 and for n ≫ 1:

E[|Y 1 | 2 | J = I] = E[|X 11 | 2 | J = I] -(E[|X 11 | | J = I]) 2 b 2-α n n -ε ,
where the last bound follows from lemma C.1, since by independence one has

E[|X 11 | 2 | J = I] = E[|X 11 | 2 | |X 11 | b n ] ,
and

|E[X 11 | J = I]| 2 = |E[X 11 | |X 11 | b n ]| 2 is O(1) if α > 1, while (by lemma C.1) it is O(b 2-2α+ε n ) for any ε > 0, if α ∈ (0, 1]. Using trP = n ′ -dim(W ′ ) 1 2 (n -d), it follows that, for any ε > 0, for n ≫ 1: E[dist 2 (Y, W ′ ) | J = I] cL(b n )b 2-α n (n -d) n q(ε) , (3.6)
where

q := (1 -2γ) 2 α + γ -ε. Under P(• | J = I), the vector (Y 1 /b n , • • • , Y n ′ /b n ) is
a vector of independent variables on D n ′ , where D be the unit complex ball. We consider the function F : x → dist(x, W ′ ). The mapping F is 1-Lipschitz and convex. From theorem 3.4, we deduce that

P(|dist(Y, W ′ ) -M (dist(Y, W ′ ))| r | J = I) 4e -r 2 8b 2 n (3.7)
where

M (dist(Y, W ′ )) is a median of dist(Y, W ′ ) under P(• | J = I).
It follows that, for e.g. δ = γ/2, taking ε = γ/4 in (3.6), we obtain q(ε) = (1 -2γ) 2 α + δ + ε, and therefore there exists c > 0 such that n ≫ 1,

b -2 n E[dist 2 (Y, W ′ ) | J = I] c n q(ε) b 2 n c n δ . (3.8) From (3.7) it follows that E |M (dist(Y, W ′ )) -dist(Y, W ′ )| 2 | J = I = O b 2 n
From the Cauchy-Schwarz inequality we then have

M (dist(Y, W ′ )) -E[dist 2 (Y, W ′ ) | J = I] 2 E |M (dist(Y, W ′ )) -dist(Y, W ′ )| 2 | J = I = O b 2 n .
The above estimates, with (3.6) and (3.8), imply that M (dist(Y, W ′ ))

1 2 n q(ε)/2 for n ≫ 1. Therefore, for n ≫ 1, P dist(Y, W ′ ) n (1-2γ)/α | J = I P |M (dist(Y, W ′ )) -dist(Y, W ′ )| 1 4 n q(ε)/2 | J = I .
The desired conclusion (3.5) now follows from (3.7) and (3.8).

So far we have shown that under assumption (H1), the distance of a row to a space with codimension n -d n 1-γ is at least n (1-2γ)/α with large probability. We want a sharper estimate, namely at the order n 1/α . We will obtain such a bound in a weak sense in the forthcoming proposition 3.7. Furthermore, we shall require assumption (H2) to do so. We start with some preliminary facts.

Below we write Z = Z (β) , β ∈ (0, 1), for the one-sided β-stable distribution such that for all s 0, E exp(-sZ i ) = exp(-s β ) .

From the standard inversion formula, for m > 0

y -m = Γ(m) -1 ∞ 0 x m-1 e -x y dx ,
we see that all moments

E[Z -m ] = Γ(m) -1 ∞ 0 x m-1 e -x β dx (3.9)
are finite for m > 0. Also, recall that if (Z i ) 1 i n is an i.i.d. vector with distribution Z then, for every (w i )

1 i n ∈ R n + , in distribution n i=1 w i Z i d = n i=1 w β i 1/β Z 1 . (3.10)
Indeed, (3.10) follows from E exp(-s w i Z i ) = exp(-s β w β i ) and a change of variables.

Lemma 3.5. Assume (H2).

There exists ε > 0 and p ∈ (0, 1) such that the random variable |X 11 | 2 dominates stochastically the random variable ε D Z, where P(D = 1) = 1 -P(D = 0) = p is a random variable with law Be(p) , Z = Z (β) with β = α 2 , and D and Z are independent. Proof. From our assumptions, there exist δ > 0 and x 0 > 0 such that

P(|X 11 | 2 > x) δ x -β P(δ 2 Z > x) , x > x 0 .
Let p be the probability that |X 11 | 2 > x 0 . If x > x 0 then P(|X 11 | 2 > x) p P(δ 2 Z > x) = P(δ 2 D Z > x). On the other hand, if x x 0 then P(|X 11 | 2 > x) p P(δ 2 D Z > x). In any case, setting ε = δ 2 we have

P(|X 11 | 2 > x) P(ε D Z > x) , x > 0.
This implies the lemma.

Lemma 3.6. Assume (H2). Let ω

i ∈ [0, 1] be numbers such that ω(n) := n i=1 ω i n 1 2 +ε for some ε > 0. Let X 1 = (X 1i ) 1 i n be i.i.d. random variables distributed as X 11 , and let Z = Z (β) with β = α 2 .
There exist δ > 0 and a coupling of X 1 and Z such that

P n i=1 ω i |X 1i | 2 δ ω(n) 1 β Z e -n δ . (3.11)
Proof. Let D = (D i ) 1 i n denote an i.i.d. vector of Bernoulli variables with parameter p given by lemma 3.5. From this latter lemma and (3.10) we know that there exist ε > 0 and a coupling of X 1 , D and Z such that

P n i=1 ω i |X 1i | 2 ε n i=1 ω β i D i 1 β Z = 1 .
It remains to show that for some ε ′ > 0:

P n i=1 ω β i D i ε ′ ω(n) e -n ε ′ . Observe that ω β i ω i , so that E n i=1 ω β i D i p ω(n). Therefore, for 0 < ε ′ < p, P n i=1 ω β i D i ε ′ ω(n) P n i=1 ω β i D i -Eω β i D i (p -ε ′ ) ω(n) 2e -2(p-ε ′ ) 2 ω(n) 2 /n ,
where we have used the Hoeffding inequality in the last bound. Since ω(n) n 1 2 +ε , this implies the lemma. Proposition 3.7. Assume (H2) and take 0 < γ α/4. Let R be the first row of the matrix a n (A -z). There exists a constant c > 0 and an event E such that for any d-dimensional subspace W of C n with codimension n -d n 1-γ , we have

E[dist -2 (R, W ) ; E] c (n -d) -2 α and P(E c ) c n -(1-2γ)/α .
Proof. As in the proof of proposition 3.3, we have

dist(R, W ) dist(X 1 , W 1 ) ,
where W 1 = span(W, e 1 ), d dim W 1 d + 1, and X 1 = (X 1i ) 1 i n is the first row of X = a n A. Let I denote the set of indexes i such that |X 1i | a n . From (3.4) we know that

P(|I| < n - √ n) < e -n δ ,
for some δ > 0. It is thus sufficient to prove that for any set I ⊂ {1, . . . , n} such that |I| n -√ n,

E[dist -2 (R, W ) ; E I | I = I] c (n -d) -2
α , for some event E I satisfying P((E I ) c | I = I) n -(1-2γ)/α . We will then simply set

E = E I ∩ {|I| n - √ n}.
Without loss of generality, we assume that

I = {1, • • • , n ′ } with n ′ n -n 1/2
. Let π I be the orthogonal projection on span(e i : i ∈ I). If W 2 = π I (W 1 ), set

W ′ = span (W 2 , E(π I (X 1 ) | I = I)) . Note that d - √ n dim(W ′ ) dim(W 1 ) + 1 d + 2. Defining Y = π I (X 1 ) -E(π I (X 1 ) | I = I), we have dist(R, W ) dist(X 1 , W 1 ) dist(Y, W ′ ). Thus, Y = (Y i ) 1 i n ′
is an i.i.d. mean zero vector under P(• | I = I). Let P denote the orthogonal projection matrix to the orthogonal of W ′ in C n ′ . By construction, we have

E dist 2 (Y, W ′ ) | I = I = E n ′ i,j=1 Y i P ij Ȳj | I = I = E |Y 1 | 2 | I = I trP .
Here trP = n ′ i=1 P ii , where

P ii = (e i , P e i ) ∈ [0, 1] and trP = n ′ -dim(W ′ ) satisfies 2(n -d) trP 1 2 (n -d) . (3.12) Let S = n ′ i=1 P ii |Y i | 2 .
We have

E (dist 2 (Y, W ′ ) -S) 2 | I = I = E   i =j Y i P ij Ȳj 2 | I = I   = (i1 =j1),( i2 =j2)
P i1j1 P i2j2 E Y i1 Ȳj1 Y i2 Ȳj2 | I = I = 2 i1 =j1 P 2 i1j1 E[|Y 1 | 2 | I = I] 2E[|Y 1 | 2 | I = I] trP 2 .
Note that,

E[|Y 1 | 2 | I = I] E[|X 11 | 2 | I = I] = E[|X 11 | 2 | |X 11 | a n ] E[|X 11 | 2 ; |X 11 | a n ] P(|X 11 | a n ) = O(a 2 n /n) ,
where the last bound follows from lemma C.1. Since P 2 = P , we deduce that

E (dist 2 (Y, W ′ ) -S) 2 | I = I = O a 2 n n -d n . (3.13)
Next, let Z = Z (β) with β = α 2 , as in lemma 3.6. Set ω i = P ii , i = 1, . . . n ′ , and for ε > 0, consider the event

Γ I =    n ′ i=1 ω i |X 1i | 2 ε (n -d) 1 β Z    .
From lemma 3.6 (with n replaced by n ′ n -n 1/2 ) and using (3.12) there exists a coupling of the vector X 1i , i = 1, . . . , n ′ and Z such that

P(Γ c I ) e -n δ , (3.14)
for some δ > 0 and some choice of ε > 0. Also, since (a -b) 2 a 2 /2 -b 2 for all a, b ∈ R, we have S 1 2 S a -S b , where

S a = n ′ i=1 ω i |X 1i | 2 , S b = n ′ i=1 ω i E [|X 1i | | |X 1i | a n ] 2 .
From Lemma C.1 and (3.12) we have

S b = E [|X 11 | | |X 11 | a n ] 2 trP = h (α) (n, d) (3.15)
where

h (α) (n, d) ∼ (n -d)a 2 n /n 2 if α ∈ (0, 1] and h (α) (n, d) ∼ (n -d) if α ∈ (1, 2).
Let G 1 I be the event that S a 3 S b . From (3.15) and the definition of Γ I we have, for some c 0 > 0

P((G 1 I ) c ∩ Γ I | I = I) P(Z c 0 (n -d) -1/β h (α) (n, d) | I = I) .
Note that, thanks to the assumptions n -d n 1-γ , γ α/4, we have (n -d) -1/β h (α) (n, d) n -ε0 for some ε 0 = ε 0 (α) > 0 for all α ∈ (0, 2), for n ≫ 1. Therefore, for n ≫ 1,

P((G 1 I ) c ∩ Γ I | I = I) P(Z c 0 n -ε0 | I = I) = P(Z c 0 n -ε0 ; |X 1i | a n , ∀i = 1, . . . , n ′ ) P(|X 1i | a n , ∀i = 1, . . . , n ′ ) ,
where the last identity follows from the independence of the X 1i . Observing that the probability for the event {|X 1i | a n , ∀i = 1, . . . , n ′ } is lower bounded by 1/c > 0 uniformly in n, we obtain

P((G 1 I ) c ∩ Γ I | I = I) c P(Z c 0 n -ε0
) . The latter probability can be estimated using Markov's inequality and the fact that E[Z -m ] = u m is finite (cf. (3.9)). Indeed, for every m > 0, P(Z t) u m t -m . Thus, we have shown that for every p > 0 there exists a constant κ p such that

P((G 1 I ) c ∩ Γ I | I = I) κ p n -p . (3.16)
Next, we set Γ I = G 1 I ∩ Γ I and we claim that

E S -2 ; Γ I | I = I = O (n -d) -4/α , (3.17)
Indeed, on Γ I we have S

E S -2 ; Γ I | I = I c 1 (n -d) -4/α E Z -2 | I = I .
Using independence as before, and recalling that the event {|X 1i | a n , ∀i = 1, . . . , n ′ } has uniformly positive probability we have

E Z -2 | I = I c E[Z -2 ] = c u 2 .
This proves (3.17). Now, for the event Markov's and Cauchy-Schwarz' inequalities lead to

P dist 2 (Y, W ′ ) S/2 ; Γ I | I = I P |dist 2 (Y, W ′ ) -S| S 1/2 ; Γ I | I = I 2E |dist 2 (Y, W ′ ) -S| S ; Γ I | I = I 2 E |dist 2 (Y, W ′ ) -S| 2 | I = I E S -2 ; Γ I | I = I .
Hence, if G 2 I denotes the event {dist 2 (Y, W ′ ) S/2}, we deduce from (3.13) and (3.17)

P (G 2 I ) c ∩ Γ I | I = I = O a n n -1 2 (n -d) 1 2 -2 α . (3.18)
Note that, using n -d n 1-γ , the last expression is certainly O(n -1 α (1-2γ) ). On the other hand, by (3.17) and Cauchy-Schwarz' inequality

E dist -2 (X, W ) ; G 2 I ∩ Γ I | I = I 2 E S -1 ; Γ I | I = I = O (n -d) -2/α . (3.19)
To conclude the proof we take

E I = G 2 I ∩ Γ I = G 1 I ∩ G 2 I ∩ Γ I . We have P((E I ) c | I = I) P ((Γ I ) c | I = I) + P (G 1 I ) c ∩ Γ I | I = I + P (G 2 I ) c ∩ G 1 I ∩ Γ I | I = I . From (3.16
) and (3.18) we see that,

P (G 1 I ) c ∩ Γ I | I = I + P (G 2 I ) c ∩ G 1 I ∩ Γ I | I = I = O n -1 α (1-2γ) ,
and all it remains to prove is an upper bound on P ((Γ I ) c | I = I). By independence, as before It follows from Markov inequality that for all t 1, ∞ t ln(x)ν An-z (dx) < c 0 / ln t. The upper part (δ -1 , ∞) of (3.20) is thus not an issue. For the lower part (0, δ), it is sufficient to prove that

P ((Γ I ) c | I = I) c P ((Γ I ) c ; |X 1i | a n , ∀i = 1, . . . , n ′ ) .
1 n n-1 i=0 1 {σn-i δn} ln σ -2 n-i
converges in probability to 0 for any sequence (δ n ) n converging to 0. From lemma 3.2, we may a.s. lower bound σ n-i by cn -r for some constant c and all integer n 1. Take 0 < γ < α/4 to be fixed later. Using this latter bound for every 1 i n 1-γ , it follows that it is sufficient to prove that

1 n n-1 i=⌊n 1-γ ⌋ 1 {σn-i δn} ln σ -2 n-i
converges in probability to 0. We are going to prove that there exists an event F n such that, for some δ > 0 and c > 0, P((F n ) c ) c exp(-n δ ), (3.21) and

E σ -2 n-i | F n c n i 2 α +1 . (3.22)
We first conclude the proof before proving (3.21)- (3.22). From Markov inequality, and (3.22), we deduce that

P(σ n-i δ n ) P((F n ) c ) + c δ 2 n n i 2 α +1
.

If follows that there exists a sequence

ε n = δ 1/( 2 α +1) n
tending to 0 such that the probability that P(σ n-⌊nεn⌋ δ n ) converges to 0. We obtain that it is sufficient to prove that

1 n ⌊εnn⌋ i=⌊n 1-γ ⌋ ln σ -2 n-i
given F n converges in probability to 0. However, using the concavity of the logarithm and (3.22) we have

E   1 n ⌊εnn⌋ i=⌊n 1-γ ⌋ ln σ -2 n-i F n   1 n ⌊εnn⌋ i=⌊n 1-γ ⌋ ln E[σ -2 n-i |F n ] c 1 n ⌊εnn⌋ i=1 ln n i = c 1 -ε n ln ε n + ε n + O(n -1
σ n-i σ ′ n-i a n .
By the Tao-Vu negative second moment lemma B.3, we have

σ ′ -2 1 + • • • + σ ′ -2 n-⌈i/2⌉ = dist -2 1 + • • • + dist -2 n-⌈i/2⌉
, where dist j is the distance from the j-th row of B n to the subspace spanned by the other rows of B n . In particular,

i 2 σ -2 n-i a 2 n n-⌊i/2⌋ j=1 dist -2 j .
Let F n be the event that for all 1 j n -⌊i/2⌋, dist j n (1-2γ)/α . Since the dimension of the span of all but one rows of B n is at most d n -i/2, we can use proposition 3.3, to obtain

P((F n ) c ) exp(-n δ ) ,
for some δ > 0. Then we write

i 2 σ -2 n-i 1 Fn a 2 n n-⌊i/2⌋ j=1 dist -2 j 1 Fn ,
Taking expectation, we get

E iσ -2 n-i ; F n 2 a 2 n nE dist -2 1 ; F n , (3.23)
Since we are on F n we can always estimate dist 1 n (1-2γ)/α . By introducing a further decomposition we can strengthen this as follows. Recall that from proposition 3.7, there exists an event E independent from the rows j = 1 such that P((E) c ) n -(1-2γ)/α and for any W ⊂ C n with dimension d < n -n 1-γ one has

E[dist(R, W ) -2 ; E] c (n -d) -2/α .
Here R is the first row of the matrix B n . By first conditioning on the value of the other rows of B n and recalling that the dimension d of the span of these is at most n -i/2 n -2n 1-γ , we see that

E[dist -2 1 ; E] = O i -2/α . Therefore E dist -2 1 ; F n E(dist -2 1 ; E) + P((E) c ) n -2(1-2γ)/α c 2 i -2/α + n -3(1-2γ)/α . (3.24)
Now, if γ < 1/6 we have 3(1 -2γ)/α > 2/α and therefore n -3(1-2γ)/α i -2/α . Thus, (3.24) Let us upgrade this convergence to an a.s. convergence. By lemmas 3.1 and A.1, it is sufficient to prove that for every z ∈ C, a.s.

implies E dist -2 1 ; F n 2 c 2 i -2/α . ( 3
lim n→∞ U µA n (z) = U µa (z).
Let us fix z ∈ C from now on. Since L = U µa (z) is deterministic, it actually suffices to show that there exists a deterministic sequence L n such that a.s. L n := E f n (s)dν An-zI (s), then we have, for every ε > 0,

P f n (s)dν An-zI (s) -L n ε 2 exp -2 nε 2 (c log n) 2 .
In particular, from the first Borel-Cantelli lemma, a.s., lim n→∞ f n (s) dν An-zI (s) -L n = 0. Finally, using (3.27), we deduce that (3.26) holds almost surely, as required.

Limiting spectral measure

In this section, we give a close look to the resolvent of the random operator on the PWIT and we deduce some properties of the limiting spectral measure µ α . For ease of notation we set β = α 2 and define the measure on R + , By lemma 2.2, for t ∈ R + , a(z, it) is pure imaginary and we set

Λ α = α 2 x -α 2 -1 dx.
h(z, t) = Im(a(z, it)) = -ia(z, it) ∈ [0, t -1 ].
The random variables a(z, η) and h(z, t) solve a nice recursive distribution equation. (i) L U solves the equation in distribution

a d = η + k∈N ξ k a k |z| 2 -η + k∈N ξ k a k η + k∈N ξ ′ k a ′ k , ( 4
h d = t + k∈N ξ k h k |z| 2 + t + k∈N ξ k h k t + k∈N ξ ′ k h ′ k (4.2)
where h, (h k ) k∈N and (h ′ k ) k∈N are i.i.d. with law L z,t , independent of {ξ k } k∈N , {ξ ′ k } k∈N two independent Poisson point processes on R + with intensity Λ α . (iii) For t = 0 there are two probability distributions on [0, ∞) solving (4.2) such that Eh α/2 < ∞: δ 0 and another denoted by L z,0 . Moreover, for the topology of weak convergence, L z,t converges to L z,0 as t goes to 0.

We start with an important lemma.

Lemma 4.2. For every Proof of lemma 4.2. Consider a realization of PWIT(2ℓ θ ) on the tree T . For k ∈ N, we define T k as the subtree of T spanned by kN f . With the notation of lemma 2.5, for k

U = U (z, η) ∈ H + , a b b ′ c is equal in distribution to 1 |z| 2 -η + k∈N ξ k a k η + k∈N ξ ′ k a ′ k η + k∈N ξ k a k -z -z η + k∈N ξ ′ k a ′ k , (4.3
∈ N, R B k (U ) = (B k (z) -η) -1 is the resolvent operator of B k and set R(U ) kk = Π k R B k (U )Π * k = a k b k b ′ k c k .
Then, by lemma 2.5 and (2.13), we get The next well-known and beautiful lemma will be crucial in the computations that will follow. It is a consequence of the LePage-Woodroofe-Zinn representation of stable laws [START_REF] Lepage | Convergence to a stable distribution via order statistics[END_REF], see also Panchenko

R(U ) ∅∅ = -U + k∈N 0 ε k y -1/α k (1 -ε k )y -1/α k 0 a k b k b ′ k c k 0 (1 -ε k )y -1/α k ε k y -1/α k 0 -1 = -U + k∈N (1 -ε k )|y k | -2/α c k 0 0 k∈N ε k |y k | -2/α a k -1 = D -1 η + k∈N ε k |y k | -2/α a k -z -z η + k∈N (1 -ε k )|y k | -2/α c k , with D = |z| 2 -η + k∈N ε k |y k | -2/α a k η + k∈N (1 -ε k )|y k | -2/α c k .
} k∈N , such that E[Y β 1 ] < ∞ then k∈N ξ k Y k d = E[Y β 1 ] 1 β k∈N ξ k d = E[Y β 1 ] 1 β S,
where S is the positive β-stable random variable with Laplace transform for all x 0,

E exp(-xS) = exp -Γ(1 -β)x β . (4.4)
Proof of lemma 4.3. Recall the formulas, for y 0, η > 0 and 0 < η < 1 respectively,

y -η = Γ(η) -1 ∞ 0 x η-1 e -xy dx and y η = Γ(1 -η) -1 η ∞ 0 x -η-1 (1 -e -xy )dx. (4.5)
From the Lévy-Khinchin formula we deduce that, with s 0,

E exp -s k ξ k Y k = exp E ∞ 0 (e -xsY1 -1)βx -β-1 dx = exp -Γ(1 -β)s β E[Y β 1 ] .
Proof of theorem 4.1. Statement (i) is contained in lemma 4.2. For (ii), let t > 0 and h a solution of (4.2). Then h is positive and is upper bounded by 1/t. By lemma 4.3, we may rewrite (4.2) as

h d = t + E[h β ] 1/β S |z| 2 + t + E[h β ] 1/β S t + E[h β ] 1/β S ′ (4.6)
where S and S ′ are i.i.d. variables with common Laplace transform (4.4). In particular, E[h β ] 1/β is solution of the equation in y:

y β = E t + yS |z| 2 + (t + yS) (t + yS ′ ) β Since t > 0, E[h β ] > 0, it follows that E[h β ] 1/β is solution of the equation in y: 1 = E ty -1 + S |z| 2 + (t + yS) (t + yS ′ ) β . (4.7)
For every S, S ′ > 0, the function y →

ty -1 +S |z| 2 +(t+yS)(t+yS ′ ) is decreasing in y. It follows that y → E ty -1 + S |z| 2 + (t + yS) (t + yS ′ ) β
is decreasing in y. As y goes to 0 it converges to ∞ and as y goes to infinity, it converges to 0. In particular, there is a unique point, y * (|z| 2 , t) of such that (4.7) holds. This proves (ii) since from (4.6), the law of h is determined by E

[h β ] 1/β = y * (|z| 2 , t).
For Statement (iii) and t = 0, then h = 0 is a particular solution of (4.2). If h is not a.s. equal to 0, then E[h β ] 1/β > 0 and the argument above still works since, for every s, s ′ > 0, the function y → s |z| 2 +y 2 ss ′ is decreasing in y. We deduce the existence of a unique positive solution y * (|z| 2 , 0) of (4.7). We also have the continuity of the function t → y * (|z| where S and S ′ are i.i.d. variables with common Laplace transform (4.4) and y * := y * (|z| 2 , t) is the unique solution in (0, ∞) of (4.7) (uniqueness is proved in theorem 4.1). We extend continuously the function y * (r, t) for t = 0 by defining y * (|z| 2 , 0) as the unique solution in (0, ∞):

1 = E S |z| 2 + y 2 SS ′ β . (4.8) Lemma 4.4. The function y * : [0, ∞) 2 → (0, ∞) is C 1 .
For every t 0, the mapping r → y * (r, t) is decreasing to 0.

Proof. For every t 0, the derivative in y > 0 of the function E

ty -1 +S |z| 2 +(t+yS)(t+yS ′ ) β is -βty -2 E (ty -1 + S) β-1 (|z| 2 + (t + yS) (t + yS ′ )) β -βE (ty -1 + S) β (S(t + yS ′ ) + S ′ (t + yS)) (|z| 2 + (t + yS) (t + yS ′ )) β+1 . (4.9)
The last computation is justified since all terms are integrable, indeed we have

(ty -1 + S) β-1 (|z| 2 + (t + yS) (t + yS ′ )) β y -β+1 (t + yS) (t + yS ′ ) β y -2β SS ′ β
and from (4.5), for all η > 0,

ES -η = Γ(η) -1 x η-1 e -Γ(1-β)x β dx < ∞. (4.10)
Similarly, for the second term of (4.9), we write

(ty -1 + S) β (S(t + yS ′ ) + S ′ (t + yS)) (|z| 2 + (t + yS) (t + yS ′ )) β+1 y -1 S(t + yS ′ ) + S ′ (t + yS) (t + yS) (t + yS ′ ) β+1 y -1 S (t + yS) (t + yS ′ ) β + y -1 S ′ (t + yS ′ ) β+1 y -β-2 S ′ -β + y -β-2 S ′ -β
The expression (4.9) is finite and strictly negative for all y > 0. The statement follows from the implicit function theorem.

From (4.3), for all t > 0, b(z, it) d = - z |z| 2 + (t + y * (|z| 2 , it)S) (t + y * (|z| 2 , it)S ′ )
.

By lemma 4.4, we may also define

b(z, 0) = lim t↓0 b(z, it) d = - z |z| 2 + y 2 * (|z| 2 , 0)SS ′ .
For ease of notation, we set y * (r) = y * (r, 0). Since ∂z = 1, ∂|z| 2 = z, we deduce that

-E∂b(z, 0) = E∂ z |z| 2 + y 2 * (|z| 2 )SS ′ = E |z| 2 + y 2 * (|z| 2 )SS ′ -1 -|z| 2 E |z| 2 + y 2 * (|z| 2 )SS ′ -2 -2|z| 2 y * (|z| 2 )y ′ * (|z| 2 )ESS ′ |z| 2 + y 2 * (|z| 2 )SS ′ -2 = y 2 * (|z| 2 ) -2|z| 2 y * (|z| 2 )y ′ * (|z| 2 ) E SS ′ (|z| 2 + y 2 * (|z| 2 )SS ′ ) 2 . (4.11)
The latter is justified since

SS ′ |z| 2 + y 2 SS ′ -2 y -4 (SS ′ ) -1 .
is integrable from (4.10). The next lemma is an important consequence of Theorems 2.13 and 1.2.

Lemma 4.5. The following identity holds in D ′ (C):

µ α = - 1 π ∂Eb(•, 0).
Therefore the measure µ α is isotropic and has a continuous density given by 1/π times the right hand side of (4.11).

Proof. Let R n be the resolvent matrix of B n , the bipartized matrix of A n defined by (1.4) The conclusion follows from (4.12), (4.13) and (4.14).

It is possible to compute explicitly the expression (4.11) at z = 0. We also have

ES -1 = e -Γ(1-β)x β dx = 1 βΓ(1 -β) 1/β
x 1/β-1 e -x dx = Γ(1 + 1/β) Γ(1 -β) 1/β , where we have used again the identity zΓ(z) = Γ(1 + z). Then the right hand side of (4.11) at z = 0 is equal to y 2 * (0)y -4 * (0)E(SS ′ ) -1 = y -2 * (0) ES -1 2 . In particular, the distribution of Y t given {Y t ε} converges weakly as t goes to infinity to the distribution with density βx -β-1 ε β dx. Since f is bounded and L slowly varying, we get

E f Y t 1 {Y εt} = P(Y t ε)E f (Y t ) Y t ε ∼ L(εt)ε -β t -β ∞ ε f (y)βy -β-1 ε β dy ∼ βL(t)t -β ∞ ε f (y)y -β-1 dy.
Finally, by assumption, for some constant, c > 0,

E f Y t 1 {Y εt} ct -β-δ E[Y β+δ 1 {Y εt} ].
Thus by lemma C.1, for some new constant c > 0 and all t 1/ε,

E f Y t 1 {Y εt} ct -β-δ L(εt)(εt) δ = ct -β L(t)ε δ L(εt) L(t) .
We may thus conclude by letting t tend to infinity and then ε to 0.

Lemma 4.8. Let S be a random variable with Laplace transform (4.4). There exists a constant c 0 > 0 such that as t goes to infinity, ES β 1 {S t} = ln t + c 0 + o(1).

Proof. Let g β be the density function of S. From Equation (2.4.8) in Zolotarev [START_REF] Zolotarev | One-dimensional stable distributions[END_REF], g β has a convergent power series representation

g β (x) = 1 π ∞ n=1
(-1) n-1 Γ(nβ + 1) Γ(n + 1)Γ(1 -β) n sin(πnβ)x -nβ-1 .

The Stirling formula Γ(x) ∼ x→∞ 2π x x e

x implies that the convergence radius of the series is +∞. Recall that Γ(β + 1) = βΓ(β), and the Euler reflection formula, Γ(1 -β) sin(πβ)/π = Γ(β). Thus, as x goes to infinity, g β (x) = βx -β-1 + O(x -2β-1 ).

The next lemma is a consequence of the Karamata Tauberian theorem.

Lemma 4.9. As t goes to infinity, P(SS ′ t) ∼ βt -β ln t, and, with c 1 = β 2 ∞ 0 (x + 1) -2 x -β dx, E SS ′ (t + SS ′ ) 2 ∼ c 1 t -1-β ln t. Once again, Laplace method implies that, as t goes infinity,

x 1/β e -x e -x 1/β t Γ(1-β) 1/β dx ∼ Γ(β + 1)Γ(1 -β) 1/β+1 t -β-1 .

In particular, for all ε > 0 there exists t 0 such that

(1 -ε)t -β-1 ES β 1 {S t/t0} 1 Γ(1 -β) 1/β+1 Γ(1 + β) x 1/β e -x Ee - x 1/β t SΓ(1-β) 1/β S -1 1 {S t/t0} dx (1 + ε)t -β-1 ES β 1 {S t/t0} .
By lemma 4.8, ES β 1 {S t/t0} ∼ ln t. It follows that for some t 1 > t 0 and all t t 1 , (1-2ε)t -β-1 ln t 1 Γ(1 -β) 1/β+1 Γ(1 + β)

x 1/β e -x Ee -x 1/β t SΓ(1-β) 1/β S -1 1 {S t/t0} dx (1+2ε)t -β-1 ln t.

On the other hand, for some constant c > 0 and all t 1,

x 1/β e -x Ee -x 1/β t SΓ(1-β) 1/β S -1 1 {S t/t0} dx x 1/β e -x dxP(S t/t 0 ) ct -β-1 t β+1 0 .

We thus have proved that 1 Γ(1 -β) 1/β+1 Γ(1 + β)

x 1/β e -x Ee -x 1/β t SΓ(1-β) 1/β S -1 dx ∼ t -β-1 ln t, and ∂ r G(y * (r), r) ∼ -r -β-1 ln(ry -2 ) ∼ -r -1 .

The statement follows.

Proof of theorem 1.3. From Equation (4.11) and lemma 4.9, the density at r = |z| 2 is equivalent to 1/π times Lemma B.6 (Schatten bound [55, proof of Theorem 3.32]). Let A be an n × n complex matrix with rows R 1 , . . . , R n . Then for every 0 < r 2, the n × n Hermitian matrix given by H(x) ij := x i,j for 1 j i n. We have µ H = µ H(H1,...,Hn) . For all x ∈ X and x ′ i ∈ C i-1 × R, the matrix H(x 1 , . . . , x i-1 , x i , x i+1 , . . . , x n ) -H(x 1 , . . . , x i-1 , x ′ i , x i+1 , . . . , x n ) has only the i-th row and column possibly different from 0, and thus rank(H(x 1 , . . . , x i-1 , x i , x i+1 , . . . , x n ) -H(x 1 , . . . , x i-1 , x ′ i , x i+1 , . . . , x n )) 2. Therefore from C.1, we obtain, for every f : R → R with f TV 1,

  λ k ( √ M M * ) where M * = M ⊤ is the conjugate transpose of M . We define the empirical spectral measure and the empirical singular values measure as

P

  (|X 11 | t) = {z∈C;|z| t} dF (z) = L(t)t -α , and there exists a probability measure θ on the unit circle S 1 := {z ∈ C; |z| = 1} of the complex plane such that for every Borel set D ⊂ S 1 , lim t→∞ P X 11 |X 11 | ∈ D |X 11 | t = θ(D).

  a n := inf{a > 0 s.t. nP(|X 11 | a) 1} and (H1) implies that lim n→∞ nP(|X 11 |

n

  k=1 |λ k (A)| = n k=1 s k (A), but it does indicate that A is typically far from being a normal matrix. A similar shrinking phenomenon appears already in the finite second moment case (1.1)-(1.3): the law of the absolute value under the circular law U σ has density r → 2σ -2 r1 [0,σ] (r)

lim n→∞ ∞ 0 t

 0 r ν A-zI (dt) < ∞, and thus (ν A-zI ) n 1 is tight. Moreover, a.s. lim n→∞ C |z| r µ A (dz) < ∞, and thus (µ A ) n 1 is tight.

1 6 S a ε 6

 66 (n -d) 2/α Z and therefore, for some constant c 1 ,

From ( 3 . 14 ) 3 . 4 . 1 ln 2

 3143412 we obtain P ((Γ I ) c | I = I) c e -n δ . This ends the proof. Uniform integrability. Let z ∈ C and σ n • • • σ 1 be the singular values of A n -z with A n defined by (1.4). For 0 < δ < 1, we define K δ = [δ, δ -1 ]. In this paragraph, we prove the uniform integrability in probability, meaning that for all ε > 0, there exists δ > 0 such thatP K c δ | ln(x)|ν An-z (dx) > ε → 0.(3.20)From lemma 3.1, with probability 1 there exists c 0 > 0, such that for all n, ∞ (x)ν An-z (dx) < c 0 .

. 25 )From ( 3 .

 253 23) we obtainE iσ -2 n-i ; F n 2 c 2 a 2 n n i -2/α. From (H2) it follows that(3.22) holds. This concludes the proof of (3.21)-(3.22).3.5. Proof of theorem 1.2. We may now invoke theorem 1.1 and (3.20). From lemma A.2, µ An converges in probability to µ α , where for almost all z ∈ C, U µα (z) = ln(x)ν α,z (dx).

U

  lim n→∞ µA n (z) -L n = 0. (3.26) Now, by lemmas 3.1 and 3.2, there exists b > 0 such that a.s. for n ≫ 1,supp(ν An-zI ) ⊂ [s n (A n -zI), s 1 (A n -zI)] ⊂ [n -b , n b ]. Denoting f n : x ∈ R + → f n (x) = 1 [n -b ,n b ] (x) log(x), we get that a.s. for n ≫ 1,U µA n (z) = -∞ 0 log(s) dν An-zI (s) = -∞ 0 f n (s) dν An-zI (s). (3.27)The total variation of f n is bounded by c log n for some c > 0. Hence by lemma C.2, if

4. 1 .

 1 Resolvent operator on the Poisson Weighted Infinite Tree. In this paragraph, we analyze the random variable R(U ) ∅∅ = a(z, η) b(z, η) b ′ (z, η) c(z, η) .

Theorem 4 . 1 (

 41 Recursive Distributional Equation). Let U = U (z, η) ∈ H + , t ∈ R + . Let L U be the distribution on C + of a(z, η) and L z,t the distribution of h(z, t).

  Now the structure of the PWIT implies that (i) a k and c k have common distribution L U ; and (ii) the variables (a k , c k ) k∈N are i.i.d.. Also the thinning property of Poisson processes implies that (iii) {ε k |y k | -2/α } k∈N and {(1 -ε k )|y k | -2/α } k∈N are independent Poisson point process with common intensity Λ α .

(4. 13 ) 2 ln(x 2 + 2 + t 2 )

 132222 Moreover from theorem 1.2, Equation (3.20), lemma A.2, in D ′ (C): ∆ ln(x)ν α,z (dx) = 2πµ α .(4.14)On the other hand, 1 t 2 )ν An-z (dx) = 1 2n ln | det(B(z) -itI 2n ))|, and from (2Eν An-z , (dx) = -2∂Eb 1 (z, it).

Lemma 4 . 6 .

 46 The density of µ α at z = 0 is1 π Γ(1 + 1/β) 2 Γ(1 + β) 1/β Γ(1 -β) 1/β .Proof. By definition, the real y * (0) solves the equation1 = E S y 2 SS ′ β = y -2β ES -β = y -2β Γ(β) x β-1 e -Γ(1-β)x β dx.With the change of variable x → x β and the identity zΓ(z) = Γ(1 + z), we find easily,ES -β = (Γ(1 -β)Γ(1 + β)) -1 and y * (0) = (Γ(1 -β)Γ(1 + β))

Lemma 4 .

 4 11. As r goes to infinity,y ′ (r) ∼ -c -1 3 y * (r)r β-1 , β) 1/β βs -β-1 dxds/ (Γ(1 + β)Γ(1 -β)).Proof. We defineG(y, r) = E S r + y 2 SS ′ β = 1 Γ(β) x β-1 e -x β y 2β Γ(1-β) Ee -xr S dx.From the implicit function theoremy ′ * (r) = -∂ r G(y * , r) ∂ y G(y * , r) .We have∂ y G(y, r) = -2βΓ(1 -β)y 2β-1 Γ(β) x 2β-1 e -x β y 2β Γ(1-β) Ee -xr S dx = -2 y 2β+1 Γ(1 + β)Γ(1 -β) xe -x Ee -x 1/β ry -2 SΓ(1-β) 1/β dxThe Laplace method implies that, as t goes infinity,xe -x e -x 1/β t Γ(1-β) 1/β dx ∼ Γ(2β)Γ(1 -β) 2 t -2β .Thus by lemma 4.7, we deduce thatxe -x Ee -x 1/β t SΓ(1-β) 1/β dx ∼ t -β xe -x e -x 1/β sΓ(1-β) 1/β βs -β-1 dxds ∼ ct -β .Applying the above to t = ry -2 * (r) we deduce, withc 3 = 2c/(Γ(1 + β)Γ(1 -β)), ∂ y G(y * , r) ∼ -c 3 r -β y -1 * (r). Similarly, the derivative of G with respect to r is ∂ r G(y, r) = -1 y 2β+2 Γ(1 -β) 1/β+1 Γ(1 + β)x 1/β e -x Ee -x 1/β ry -2 SΓ(1-β) 1/β S -1 dx.

  f dµ H(x1,...,xi-1,xi,xi+1,...,xn) -f dµ H(x1,...,xi-1,x ′ i ,xi+1,...,xn) 2 n . The desired result follows now from the Azuma-Hoeffding inequality, see e.g. [38, Lemma 1.2].

  Corollary 2.4 (Criterion of self-adjointness of bipartized operator). Let κ > 0 and T = (V, E) be a tree. Assume that if {u, v} / ∈ E then w uv = w vu = 0. Assume also that there exists a sequence of connected finite subsets (S n ) n 1 in V , such that S n ⊂ S n+1 , ∪ n S n = V , and for every n and v ∈ S n , as follows. The trees G ∅ and Ĝ∅ are the unique trees such that ∅ ∈ V ∅ , ∅ ∈ V∅ and that satisfy the following properties (i) if {u, v} ∈ E and u in V ∅ (or V∅ ) then v ∈ V ∅ (or V∅ ) and {u, v} ∈ E ∅ (or Ê∅ ), (ii) if {u, v} ∈ E and û in V ∅ (or V∅ ) then v ∈ V ∅ (or V∅ ) and {û, v} ∈ E ∅ (or Ê∅ ).We note that by construction if u ∈ V ∅ and v ∈ V∅ then δ u , Bδ v = 0. If follows that the operator B decomposes orthogonally into two operators B ∅ and B∅ on domains in ℓ 2 (V ∅ ) and ℓ 2 ( V∅ ) respectively: B = B ∅ ⊕ B∅ . We may then safely apply lemma 2.3 to B ∅ and B∅ .

	u / ∈Sn:u∼v Then for all z ∈ C, B(z) is self-adjoint.	|w uv | 2 + |w vu | 2	κ.
	Proof. From (2.8), it is sufficient to check that B is self-adjoint. Let ∅ ∈ V be a distinguished vertex, we define two disjoint trees G ∅ = (V ∅ , E ∅ ) and Ĝ∅ = ( V∅ , Ê∅ ) on a partition (V ∅ , V∅ )
	of V b
				and for
	every n and v ∈ S n , Then A is essentially self-adjoint.	u / ∈Sn:u∼v	|w uv | 2 κ.
	For a proof, see [11, Lemma A.3]. The above lemma has an interesting corollary for the bipartized
	operator B of A defined by (2.7)-(2.8).

  13 (Convergence of the Resolvent matrix). Let A n and A be as in theorem 2.10. For all U = U (z, η) ∈ H + , R n (U ) 11 n→∞ R(U ) ∅∅ .

	Proof of theorem 2.13. We apply proposition 2.8, theorem 2.10 and the"in distribution" version of
	theorem 2.7.

2.8. Proof of theorem 1.1. Again, we consider the sequence of random n × n matrices (A n ) defined in introduction by

(1.4)

.

Theorem 2.14. For all z ∈ C + , almost surely the measure νAn-z (dx) converges weakly to a measure να,z (dx) whose Cauchy-Stieltjes transform is given, for η ∈ C + , by

  It thus remain to prove (3.21)-(3.22). Let B n be the matrix formed by the first n -⌊i/2⌋ rows of a n (A n -zI). If σ ′

	Lemma B.4,	1	n-⌊i/2⌋ are the singular values of B n , then by the Cauchy interlacing • • • σ ′

) .

  .1) where a, (a k ) k∈N and (a ′ k ) k∈N are i.i.d. with law L U independent of {ξ k } k∈N , {ξ ′ k } k∈N two independent Poisson point processes on R + with intensity Λ α . (ii) L z,t is the unique probability distribution on [0, ∞) such that

  and Talagrand [41, Lemma 2.1]. Lemma 4.3. Let {ξ k } k∈N be a Poisson process with intensity Λ α . If (Y k ) is an i.i.d. sequence of non-negative random variables, independent of {ξ k

  (|z| 2 , 0)S/(|z| 2 + y 2 * (|z| 2 , 0)SS ′ ), and from (4.6), it implies the weak convergence of L z,t to L z,0 . 4.2. Density of the limiting measure. In this paragraph, we analyze the RDE (4.3). For all t > 0, let L z,t be as in theorem 4.1. From Equation (4.6), h may be expressed as h S |z| 2 + (t + y * S) (t + y * S ′ )

2

, t) on [0, ∞). Finally

h d = y * d = t + y *

  From theorem 2.14, Eν An-z converge weakly to ν α,z and, by lemma 3.1, for all t > 0, + t2 )Eν An-z dx) = 1 2 ln(x 2 + t 2 )ν α,z (dx).From Equation (3.20), ln(x)ν α,z (dx) is integrable. We deduce that for all z 0 ∈ C, there exists an open neighborhood of z 0 and a sequence (t n ) n 1 converging to 0 such that for all z in the neighborhood,

						. By
	theorem 2.13 and lemma 2.2, for all t > 0 and z ∈ C,
		lim n→∞	ER n (U (z, it)) 11 =	iEh(z, t) Eb(z, it) E b(z, it) iEh(z, t)	,
	lim n→∞	1 2 ln(x 2 lim n→∞ ER n (U (z, it n )) 11 =	iEh(z, 0) Eb(z, 0) E b(z, 0) iEh(z, 0)	,	(4.12)
	and
	lim n→∞	1 2	ln(x 2 + t 2

n )Eν An-z (dx) = ln(x)ν α,z (dx).

  4.3.Proof of theorem 1.3. In this subsection, we prove the last statement of theorem 1.3 (the first part of the theorem being contained in lemmas 4.5, 4.6). We start with a first technical lemma. Lemma 4.7. Let 0 < β < 1, δ > 0, and f be a bounded measurable R + → R function such that f (y) = O(y β+δ ) as y ↓ 0. Let Y be a random variable such that P(Y t) = L(t)t -β for some slowly varying function L. Then as t goes to infinity Proof. Define Y t = Y /t. We fix ε > 0 and consider the distribution P(Y t ∈ •|Y t ε). By assumption, for s > ε, P(Y t s|Y t ε) ∼ (s/ε) -β .

	Ef	Y t	∼ βL(t)t -β	0	∞	f (y)y -β-1 dy.

Appendix C. Additional lemmas References

Introduction

The eigenvalues of an n×n complex matrix M are the roots in C of its characteristic polynomial. We label them λ 1 (M ), . . . , λ n (M ) so that |λ From Corollary 8.1.7 in [START_REF] Bingham | Regular variation[END_REF], we have as t goes to infinity, P(S > t) ∼ t -β . In particular, we have P(S β > t) ∼ t -1 and a new application of Corollary 8.1.7 in [START_REF] Bingham | Regular variation[END_REF] gives as x ↓ 0,

1 -E exp(-xS β ) ∼ x ln x -1 .

We obtain

1 -E exp(-xSS ′ ) ∼ Γ(1 -β)x β ln(Γ(1 -β)x -β ) ∼ βΓ(1 -β)x β ln(x -1 ).

We then conclude by a third application of Corollary 8.1.7 in [START_REF] Bingham | Regular variation[END_REF]. The second statement is a consequence of lemma 4.7.

The next lemma gives the asymptotic behavior of y * (r) as r goes to infinity.

Lemma 4.10. There exists a constant c 2 > 0 such that as r goes to infinity,

Proof. From Equations (4.5), (4.8), we have with y * = y * (r),

By lemma 4.4, lim r→∞ y * (r) = 0. Hence, from the above expression, we deduce that the term ry -2 * goes to infinity as r goes to infinity. Define

with I 0 (y) = I(y)1 {y 1} ,

The function I is increasing and lim y→∞ I(y) < ∞. Also, the function I 0 is equal to 0 in a neighborhood of 0. By lemma 4.7, we get as t goes to infinity,

for some positive constant

Also, from Laplace method, I 2 (y) ∼ -Γ(2β)Γ(1 -β) 2 y 2β as y goes to 0. By lemma 4.7,

* , we get from (4.15)

). In other words, r β = ln(ry -2 * ) + (c 0 + a 0 + a 2 ) + o(1). We conclude by setting c 2 = exp((c 0 + a 0 + a 2 )/2).

Appendix A. Logarithmic potentials and Hermitization

Let P(C) be the set of probability measures on C which integrate ln |•| in a neighborhood of infinity. For every µ ∈ P(C), the logarithmic potential U µ of µ on C is the function

Note that in classical potential theory, the definition is opposite in sign, but ours turns out to be more convenient (lightweight) for our purposes. Since ln |•| is Lebesgue locally integrable on C, one can check by using the Fubini theorem that U µ is Lebesgue locally integrable on C. In particular, U µ < ∞ a.e. (Lebesgue almost everywhere) and

and thus µ = ν as measures since µ and ν are Radon measures.

If A is an n × n complex matrix and P A (z) := det(A -zI) is its characteristic polynomial,

for every z ∈ C \ {λ 1 (A), . . . , λ n (A)}. We have also the alternative expression

The identity above bridges the eigenvalues with the singular values, and is at the heart of the following lemma, which allows to deduce the convergence of µ A from the one of ν A-zI . The strength of this Hermitization lies in the fact that contrary to the eigenvalues, one can control the singular values with the entries of the matrix. The price paid here is the introduction of the auxiliary variable z and the uniform integrability. We recall that on a Borel measurable space (E, E), we say that a Borel function f :

We will use this property as follows: if η n η and f is continuous and uniformly integrable for (η n ) n 1 then f is η-integrable and lim n→∞ f dη n = f η. Similarly for a sequence random probability measures (η n ) n 1 we will say that f is uniformly integrable for (η n ) n 1 in probability, if for all ε > 0 lim

A proof of lemma A.2 below can be found in [START_REF]Circular Law Theorem for Random Markov Matrices[END_REF] which covers the "a.s." case, the "in probability" case being similar. It relies only on the unicity lemma A.1, the classical Prohorov theorem, and the Weyl inequalities of Lemma B.5 linking eigenvalues and singular values.

Lemma A.2 (Girko's Hermitization method). Let (A n ) n 1 be a sequence of complex random matrices where A n is n × n for every n 1. Suppose that for Lebesgue almost all z ∈ C, there exists a probability measure ν z on [0, ∞) such that (i) a.s. (ν An-zI ) n 1 tends weakly to ν z (ii) a.s. (resp. in probability) ln(•) is uniformly integrable for (ν An-zI ) n 1

Then there exists a probability measure µ ∈ P(C) such that (j) a.s. (resp. in probability) (µ An ) n 1 converges weakly to µ (jj) for a.a. z ∈ C,

Appendix B. General spectral estimates

Lemma B.1 (Basic inequalities [32]

). If A and B are n × n complex matrices then

and max

Lemma B.2 (Rudelson-Vershynin row bound [45,[START_REF]Circular Law Theorem for Random Markov Matrices[END_REF]). Let A be a complex n × n matrix with rows R 1 , . . . , R n . Define the vector space R -i := span{R j ; j = i}. We have then

Recall that the singular values s 1 (A), . . . , s n ′ (A) of a rectangular n ′ × n complex matrix A with n ′ n are defined by s i (A) := λ i ( √ AA * ) for every 1 i n ′ .

Lemma B.3 (Tao-Vu negative second moment [50, Lemma A4]).

If A is a full rank n ′ × n complex matrix (n ′ n) with rows R 1 , . . . , R n ′ , and R -i := span{R j ; j = i}, then

Lemma B.4 (Cauchy interlacing by rows deletion [32]

). Let A be an n × n complex matrix. If B is n ′ × n, obtained from A by deleting n -n ′ rows, then for every

Lemma B.5 (Weyl inequalities [53]

). For every n × n complex matrix A, we have

for all 1 k n. In particular, by viewing |det(A)| as a volume,

where R 1 , . . . , R n are the rows of A. Moreover, for every increasing function ϕ from (0, ∞) to (0, ∞) such that t → ϕ(e t ) is convex on (0, ∞) and ϕ(0) := lim t→0 + ϕ(t) = 0, we have

for every 1 k n. In particular, with ϕ(t) = t r/2 , r > 0, and k = n, we obtain

We begin with a lemma on truncated moments. We skip the proof since it follows from an adaptation of the proof in the real case given by e.g. Feller [23, Theorem VIII.9.2].

Lemma C.1 (Truncated moments). If (H1) holds then for every p > α,

where c(p) := α/(p -α). In particular, we have

We end up this section by a result on the concentration of the spectral measure of Hermitian or Hermitized random matrices, mentioned in [START_REF] Ch | The rank of diluted random graphs[END_REF]. The total variation norm of f : R → R is

where the supremum runs over all sequences (x k ) k∈Z such that x k+1

x k for any k ∈ Z. If f = 1 (-∞,s] for some real s then f TV = 1, while if f has a derivative in L 1 (R), we get

The following lemma comes with remarkably weak assumptions, and allows to deduce the almost sure weak convergence of empirical spectral measures of random matrices without any moment assumptions on the entries. We discovered that this lemma was obtained independently by Guntuboyina and Leeb in [30], where they discuss the relationships with more classical results.

Lemma C.2 (Concentration for spectral measures). Let H be an n × n random Hermitian matrix. Let us assume that the vectors (H i ) 1 i n , where H i := (H ij ) 1 j i ∈ C i , are independent. Then for any f : R → R going to 0 at ±∞ and such that f TV 1 and every t 0,

Similarly, if M is an n × n complex random matrix with independent rows (or with independent columns) then for any f : R + → R going to 0 at +∞ with f TV 1 and every t 0,

Proof. We prove only the Hermitian version, the non-Hermitian version being entirely similar. Let us start by showing that for every n × n deterministic Hermitian matrices A and B and any measurable function f with f TV = 1,

Indeed, it is well known (follows from interlacing, see e.g. [START_REF] Thompson | The behavior of eigenvalues and singular values under perturbations of restricted rank[END_REF] or [5, Theorem 11.42]) that F A -F B ∞ rank(A -B) n where F A and F B are the cumulative distribution functions of µ A and µ B respectively. Now if f is smooth, we get, by integrating by parts,

and since the left hand side depends on at most 2n points, we get (C.1) by approximating f by smooth functions. Next, for any x = (x 1 , . . . , x n ) ∈ X := {(x i ) 1 i n : x i ∈ C i-1 × R}, let H(x) be