{'fraction_non_alphanumeric': 0.06724659929214107, 'fraction_numerical': 0.028442283911133855, 'mean_word_length': 3.5326633165829144, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 30, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Local elliptic law

The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this convergence on all mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum with an optimal convergence rate. As a corollary we obtain complete delocalisation for the corresponding eigenvectors in any basis.

eigenvalue spacing. Such strong local law is a signature of the logarithmic repulsion between eigenvalues and is in stark contrast to the behaviour of the Poisson point process, for which the independence between point positions leads to much larger fluctuations. The first local laws on the optimal spectral scale in the bulk appeared in the context of Wigner matrices [28,30] (see [29] for a historical account). Since then they have been established for a wide range of self-adjoint models, including random matrices with deterministic deformations [37,42], variance profiles [3], general decaying correlations [4,27], heavy tailed entries [1,16], band matrices [19,22], sparse random graphs [12,13,23], invariant ensembles [17] as well as specific polynomials [9,25] and rational functions [24] in several random matrices.
In the non-Hermitian setup local laws were first established for matrices with i.i.d. entries [18,59]. They were extended to ensembles with independent non-identically distributed entries [5,6], ensembles with decaying correlations [8], products of independent matrices [36,48] and to matrices that are either multiplied by independent Haar unitaries on the left and right [11] or by a matrix with i.i.d. entries on one side [58]. In all these cases the spectral density becomes radially symmetric in the limit of large dimension. Control on the fluctuation of the ESD down to local scales is also a vital ingredient in spectral universality proofs [20,55].
In this work we establish the local law in the spectral bulk of elliptic n × n -random matrices X. The spectral resolution and convergence rates are optimal, up to factors n o (1) . As a corollary we show complete delocalisation of the eigenvectors of X in any basis. For matrices with independent entries similar results have been proved in [43,44,51], for matrices with decaying correlations in [8] and for a model with elliptic type correlations and non-random imaginary part in [52].
Our proof relies on Girko's Hermitization trick, i.e. on expressing the spectral distribution of X in terms of the eigenvalues of the ζ-dependent family of Hermitian matrices at the origin. Optimal control on the resolvent G(ζ, η) := (H ζ − iη) −1 for η ≫ n −1 in combination with a bound on the smallest singular value of X − ζ, which we import from [49], allows to infer the local law for X. The Hermitization H ζ belongs to a general class of self-adjoint random matrices with correlated entries, for which it was shown in [27] that G = G(ζ, η) satisfies the non-linear matrix Dyson equation (cf. (5.7)) with solution M = M (ζ, η) in the limit n → ∞. In contrast to [27], however, the equation associated to H ζ has a structural instability on local scales η ≪ 1. We treat this instability by projecting the Dyson equation onto a codimension one subspace to conclude G − M → 0. A similar strategy has been used in [8]. In contrast to previous works we establish the convergence x , (G − M )y → 0 isotropically for all deterministic vectors x, y without performing an elaborate separate step (see e.g. [15,40]) and without tracking a family of high probability sets associated to a large number of vectors for which the convergence holds (see e.g. [27,37]). Compared to the general settings in the latter two references, here this simplification is achievable because of the simple structure within both, the correlations and expectation values of H ζ entries. We reduce the stability analysis of the matrix Dyson equation to a finite dimensional problem by taking partial traces of G − M and at the same time treat all directions simultaneously by working with isotropic L p -norms directly on the space of random matrices. This strategy streamlines the proof of the local law for H ζ and is generalisable to random matrices with block correlation structures. More precisely, it is applicable to Kronecker random matrices (see [7] for a definition) with constant variance and expectation profiles, i.e. for a i = a and s µ ij = s µ , t µ ij = t µ in [7, eq. (2.1) and eq. (2. 3)], respectively.

Main results
In this section, we state our assumptions and the main results.

Local elliptic law and delocalisation
We introduce σ ̺ : C → [0, ∞), the uniform probability density on the ellipse E ̺ defined through The next theorem, the main result of the present work, will provide detailed information about the eigenvalue density of X on all mesoscopic scales. Such scale is probed by the observables f ζ 0 ,α defined through where f : C → C is an arbitrary function, ζ 0 ∈ C and α > 0.
Theorem 2.1 (Local elliptic law). Let δ ∈ (0, 1) and α ∈ (0, 1/2). Then, for any ε > 0 and ν ∈ N, there is C > 0 such that uniformly for all n ∈ N, ζ 0 ∈ E ̺,δ , and every f ∈ C 2 0 (C) satisfying f (ζ) = 0 for all ζ ∈ C with |ζ| ≥ ϕ and ∆f L 2+a ≤ n D ∆f L 1 with some constants ϕ, a > 0 and D ∈ N. Theorem 2.1 will be proved at the end of Section 3 below. We also obtain complete delocalisation of eigenvectors of X in any basis. Corollary 2.2 (Isotropic eigenvector delocalisation). Let δ ∈ (0, 1) and denote the set of eigenvectors of X with eigenvalue in E ̺,δ . Then, for any ε > 0 and ν > 0, there is C > 0 such that for all w ∈ C n and all n ∈ N. Here, w , u denotes the Euclidean scalar product of w and u.
Corollary 2.2 is a consequence of the local law for the Hermitization H ζ of X from (1.1) which is the main input in the proof of Theorem 2.1. Its proof will be given in Section 5.1 below. • The random variables have mean zero and variance 1 n , i.e. Ex ij = 0 and E|x ij | 2 = 1 n for all i, j ∈ n , and satisfy E[ For simplicity of the presentation, we stated Theorem 2.1 and Corollary 2.2 under the same assumptions. The stronger assumptions are solely needed to control the smallest singular value via a result by Nguyen-O'Rourke (see Theorem 3.4

below).
(ii) A sufficiently strong local law also implies that there are no eigenvalues away from the support of the limiting eigenvalue density. This is also correct in the present setup. However, for elliptic random matrices the absence of eigenvalues away from E ̺ has been shown in [50,Theorem 2.2] for real matrices. Therefore, we refrain from reproving such result here.
Moreover, for real and complex matrices, exclusion of eigenvalues can also be derived from [26] by combining [26,Lemma 4.8] and the displayed equation after [26, eq. (7.2)].
We now give an overview of the remainder of this paper. In the next subsection, we explain a few notations and conventions used throughout this work. Section 3 summarises the main ingredients and contains the proof of Theorem 2.1 given these ingredients. In Section 4, we analyse the Dyson equation and its solution which is a deterministic approximation of the resolvent of the Hermitization of X. In Section 5, this analysis is used to prove a local law for the Hermitization, a key input for the proof of the local law for X.

Notations
Here, we introduce and collect a few notations used throughout the paper. We first recall the definition n . . = {1, . . . , n} for n ∈ N. We write #Σ for the number of elements of a set Σ. For r > 0, we denote the disk of radius r in the complex plane centred at the origin by D r . . = {z ∈ C : |z| < r}. For functions of ζ ∈ C, we write ∂ and∂ for their derivatives with respect to ζ andζ, respectively, i.e. ∂ = 1 2 (∂ Re ζ − i∂ Im ζ ) and∂ = 1 2 (∂ Re ζ + i∂ Im ζ ). We remark that, in matrix equations, scalars are identified with the corresponding multiple of the identity matrix, e.g. H ζ − iη. For a matrix A ∈ C l×l , we denote its normalised trace by A = 1 l Tr A. Throughout the paper, we use the convention that all generic constants are denoted by c and C and may always depend on the distribution of ξ := (ξ 0 , ξ 1 , ξ 2 ). These constants are uniform in all other parameters, e.g. n, ζ, etc., within specified parameter sets.
If f and g are two real scalars then we write f g and g f if there is a constant C > 0 such that f ≤ Cg. If f g and f g then we write f ∼ g. In case, the constant C depends on a parameter δ we write δ , δ and ∼ δ , respectively. The same notation is also used for Hermitian matrices f and g, where f ≤ Cg is interpreted in a quadratic form sense. If f is complex and g ≥ 0 then we write We omit the subscripts from and O for the parameters 1 − |̺|, α from (2.2) as well as ϕ, D or a from Theorem 2.1.

Proof of local elliptic law
In this section, we prove Theorem 2.1. To that end, we first collect the ingredients of its proof and then use them to conclude the local elliptic law. The novel ingredients will be proved in the following sections.
The first ingredient is a basic formula due to Girko [32] expressing the averaged linear statistics of X in a more tractable way. Indeed, since log is the fundamental solution of the Laplace equation in where we introduced the Hermitization The log-determinant of H ζ can be easily obtained from the resolvent G(ζ, η) := (H ζ − iη) −1 through the well-known identity for any T > 0 (see [55] for the use of (3.3) in a similar context). Recall that R denotes the normalised trace of a matrix R ∈ C 2n×2n . When n becomes large, G is well approximated by a deterministic function which we introduce next. For each ζ ∈ C and η > 0, let M ≡ M (ζ, η) ∈ C 2×2 be the unique solution of The existence and uniqueness of M is shown e.g. in [38]. In (3.4), a 11 , a 12 , a 21 , a 22 ∈ C. The relation in (3.4) is called Dyson equation and the linear map S : C 2×2 → C 2×2 is the self-energy operator or self-energy. It is easy to see that the unique solution M of (3.4) satisfies The following proposition states that G is well approximated by iv on all scales η ∈ [n −1+γ , n 100 ] slightly above the typical eigenvalue spacing of H ζ around zero, which is of order n −1 (in the bulk, i.e. for ζ ∈ E ̺,δ ). This proposition will be proved in Section 5.1 below.

Proposition 3.1 (Local law for H ζ , averaged version).
Let v be as in (3.5), γ > 0 and δ ∈ (0, 1). Then, for any ε > 0 and ν > 0, there is C ε,ν > 0 such that The previous proposition directly implies the bound on the number of small singular values of X − ζ in the next lemma. The singular values of X − ζ coincide with the moduli of the eigenvalues of H ζ . The latter are denoted by {λ 1 (ζ), . . . , λ 2n (ζ)} = Spec H ζ in the following.

Lemma 3.2 (Number of small singular values of
uniformly for all η ∈ [n −1+γ , n 100 ], ζ ∈ E ̺,δ and n ∈ N. As we have seen in the formulation of Proposition 3.1 and Lemma 3.2 the following notion of high probability events is useful. It will be used extensively in the proofs of Lemma 3.2 and Theorem 2.1 below. Definition 3.3 (With very high probability). The (sequence of) events (A n ) n∈N occur with very high probability if for every ν > 0 there is C ν > 0 such that, for all n ∈ N, Proof of Lemma 3.2. From the first bound in (3.7) in Lemma 3.5 below, we conclude that the trace of G is bounded by n, i.e. |Tr G(ζ, η)| n, with very high probability due to Proposition 3.1. Since The smallest singular value of X − ζ, which coincides with min j∈ 2n |λ j (ζ)|, is controlled in the following result from [49,Theorem 1.9]. See also [35] for a related result.
uniformly for all n ∈ N and ζ ∈ C with |ζ| ≤ n a .
The next lemma relates v from (3.5) with the elliptic law σ ̺ defined in (2.1) and will be proved in Section 4.3 below. The relation is expected due to the identities in (3.1) and (3.3).
where the integral in the definition of L exists in Lebesgue sense for all ζ ∈ C. Moreover, uniformly for all η ∈ (0, ∞), ζ ∈ D 10 and In order to use Proposition 3.1 when G is integrated with respect to ζ, we will approximate such integrals by an average of evaluations of the integrand at uniformly distributed points. This approximation is controlled by the next lemma which is a simplified version of [55,Lemma 36] used in a similar context. Then, for any δ > 0, we have Proof. The i.i.d. random variables F (ξ 1 ), . . . , F (ξ m ) have mean Ω F dµ and variance Ω |F − Ω F dµ| 2 dµ. Thus, Markov's inequality implies Lemma 3.6.
We now combine the results introduced above in order to prove Theorem 2.1. The argument is very similar to the proof of [8,Theorem 2.7]. However, we detail it here for the convenience of the reader.
Since Ω ⊂ D 10 , combining the first step in (3.1), (3.6) and the last estimate in (3.7) yields Here, we denoted by µ the normalised Lebesgue measure on Ω and by F the function defined through The rest of this proof is devoted to estimating Ω F dµ. We now apply Lemma 3.6. The function Hence, the second bound in (3.7) implies that, for any For any ν > 0, we apply Lemma 3.6 with δ = n −ν and m = n ν+2D+20 to get with probability at least 1 − n −ν . Here, ξ 1 , . . . , ξ m are uniformly distributed on Ω.
From (3.10), we conclude that with probability at least 1 − 3n −ν . Here, we used Lemma 3.6 in the second step. Finally, we use (3.9), (3.11) and the assumption ∆f L 2+a ≤ n D ∆f L 1 to conclude that the first term on the right-hand side of (3.8) is bounded by n −1+ε+2α ∆f L 1 with very high probability. Since T = n −100 , this completes the proof of Theorem 2.1.

Solution and stability of Dyson equation
In this section, we study the solution M ∈ C 2×2 of the Dyson equation, (3.4), and its stability. This will allow us to prove Lemma 3.5 (see Section 4.3 below) and be an important ingredient in the proof of Theorem 5.1 below.

Properties of M
We start this section with some bounds on M and an asymptotic expansion for M when η is large.
For the proof of (ii), let r > 0. Then there is C r > 0 such that η −1 (|ζ| + M ) ≤ 1/2 for all η ≥ C r and ζ ∈ D r . If η ∈ (0, C r ] then (4.2a) holds trivially. If η ≥ C r then we take the inverse of (3.4), expand the right-hand side of the result around (iη) −1 and obtain Here, we used in the last step that S The next lemma shows that, for ζ ∈ E ̺,δ , the origin is asymptotically in the bulk spectrum of H ζ . Moreover, for any ζ ∈ int(E ̺ ), we have Employing (4.3) again, we arrive at Proof. We now show that there is c ∼ δ 1 depending only on δ such that if v(ζ, η) ≤ c for some z ∈ D 10 and η ∈ (0, 1] then ζ / ∈ E ̺,δ . The previous statement implies (4.4). Suppose that v(z, η) ≤ c. First, we choose c ∼ 1 sufficiently small such that |b| 1. This is possible since v 2 + |b| 2 1 by (4.2b) for z ∈ D 10 and η ∈ (0, 1]. Note that |b| ≤ 1 by (4.1a) and (4.3).
Taking the real and imaginary part of (4.6), solving them for Re b and Im b, respectively, and combining the results to a relation for τ = |b| yield where we also used τ = |b| ∼ 1.

Stability
For the following analysis of the stability operator of the Dyson equation, (3.4), we introduce the matrix We will work on the linear subspace E ⊥ − ⊂ C 2×2 , where the orthogonality is understood with respect to the Hilbert-Schmidt scalar product on C 2×2 . The next proposition proves a precise bound on the inverse of the stability operator L : C 2×2 → C 2×2 of the Dyson equation, (3.4). The operator L is defined through for any R ∈ C 2×2 . This proposition will be a crucial ingredient in the proof of Proposition 3.1 and its generalisation, Theorem 5.1 below.
uniformly for all ζ ∈ D 10 and η ∈ (0, 1]. In (4.11) and the following, we write L −1 | E ⊥ − for the inverse of the restriction of L to E ⊥ − , i.e. the operator is first restricted to E ⊥ − and then inverted. In the next lemma and for the rest of this proof, · 2 is the operator norm on C k×k induced by the Euclidean norm on C k .

Lemma 4.5 (Rotation-Inversion).
Let T be a self-adjoint and U a unitary operator on C k . Suppose that Gap(T ) > 0 and T 2 ≤ 1. Then there is universal constant C > 0 such that where h is the normalised eigenvector of T corresponding to the nondegenerate eigenvalue T 2 .
We now explain how L can be represented as U − T for some U and T introduced next. From (3.5), we conclude that M is normal. Let M = |M |U = U |M | be its polar decomposition. Owing to (4.3), we have Consequently, L = U * (U − T ) with the definitions T . . = M 2 S and U R . . = U * RU * for any R ∈ C 2×2 . Note that S , U and U * , and, thus, L leave E ⊥ − invariant. Moreover, S is self-adjoint and U is unitary on E ⊥ − . In particular, M 2 S is self-adjoint on E ⊥ − . We now apply Lemma 4.5 by identifying E ⊥ − with C 3 and the choices of T and U made above. It is easily seen that Spec(S | E ⊥ − ) = {1, ̺, −̺} with the identity matrix, the first and second Pauli matrix being the respective eigenvectors. Hence, T 2 = M 2 S 2 = M 2 = v η+v ≤ 1 and Gap(T ) = M 2 (1 − |̺|). Moreover, the eigenvector h from Lemma 4.5 is the identity matrix. Thus, (3.5) and (4.3). Therefore, Lemma 4.5 implies where we identified E ⊥ − with C 3 and · 2 denotes the operator norm induced by the Hilbert-Schmidt scalar product on E ⊥ − ⊂ C 2×2 . Since M 1 and v = Im M η for η ∈ (0, 1] and ζ ∈ D 10 by (4.2b) and (4.2c), respectively, and the norms on the space of operators on C 2×2 are equivalent (with constants ∼ 1), we conclude (4.11) from (4.12). Corollary 4.6 (Bounded derivatives). The function M (ζ, η) is continuously differentiable with respect to ζ,ζ and η at any η > 0 and ζ ∈ C. Moreover, for any δ ∈ (0, 1), we have uniformly for η ∈ (0, 1] and ζ ∈ E ̺,δ . Proof. Since (3.4) is an equation on E ⊥ − and the linear stability operator L is invertible on E ⊥ − for any η > 0 and ζ ∈ C, the implicit function theorem implies the continuous differentiability with respect to ζ,ζ and η. The bound on the derivatives follows from differentiating (3.4) and using Corollary 4.4.

Proof of Lemma 3.5
Proof of Lemma 3.5. Note that the integral in the definition of L exists in the Lebesgue sense since, for every uniformly for all η > 0 and ζ ∈ D r by (4.1a) and (4.2a). The first inequality in (3.7) follows directly from (4.1a). The second and the third bound in (3.7) are immediate consequences of (4.13) with r = 10.

Local law for Hermitization and eigenvector delocalisation
In this section we prove Proposition 3.1 and Corollary 2.2, see Section 5.1 below. The main tool is the local law for H = H ζ from (3.2), Theorem 5.1 below, which states that as n tends to infinity the where every entry in this 2 × 2-block structure is a multiple of the identity matrix in C n×n , i.e. M = M ⊗ 1 ∈ C 2×2 ⊗ C n×n . We recall that M and v as well as b were defined in (3.4) and (3.5), respectively.
To express the convergence of G to M , we introduce appropriate norms. For any random matrix A ∈ C l×l in dimension l ∈ N we define the p-norms The proof of Theorem 5.1 will be presented in Section 5.2 below. For η ∈ [n −ε , n 100 ] with a very small ε > 0, it will directly follow from [27, Theorem 2.1]. As it stands, equation (3.4) is unstable in the local regime η ≤ n ε . Thus, the results from [27] do not directly extend to such small η. We will use the refined stability from Proposition 4.3, orthogonal to the unstable direction E − , to show (5.4) in the local regime.

Proofs of Proposition 3.1 and Corollary 2.2
We now conclude Proposition 3.1 and Corollary 2.2 from Theorem 5.1.
Proof of Proposition 3.1. We first note that M = iv by the definition of M in (5.1). Thus, Proposition 3.1 follows directly from Theorem 5.1 with suitably chosen γ and p as We use a standard argument, adjusted to our setting, to obtain eigenvector delocalisation from the resolvent control in the local law, see e.g. [28] for an early version of this argument.
The proof of Theorem 5.1 is based on the realisation that G approximately satisfies the matrix Dyson equation where Z = −EH ∈ C 2n×2n and S is the natural extension of S from (3.4) to C 2n×2n , i.e.
The matrix M defined in (5.1) solves (5.7). For consistency with the presentation in [27] we introduce the self-energy operator

SA := E(H + Z)A(H + Z) ,
which is a slight modification of S, Indeed, one easily verifies that where P = (p ij ) n i,j=1 and Q = (q ij ) n i,j=1 are the n × n-matrices with entries Then there is C * > 0 such that, uniformly for η ∈ [n −1 , 1] and ζ ∈ D 10 , we have the following bounds on the error matrix with q = C * p 4 /ε (choose µ = 1/4).
In the remainder of this section we will infer the bounds ( Note that M also satisfies (5.7) with S replaced by S since SM = SM due to the block diagonal structure of M in (5.1). The purpose of writing (5.14) in this fashion will become apparent when we take partial traces inside the 2 × 2 block structure underlying (5.14). In contrast to the setup in [27], equation (5.14) is unstable because the linear stability operator acting on ∆ on its left-hand side has a non-trivial kernel. This instability also prevents us from using [27,Theorem 5.2] directly to show stability of (5.14). The bounded invertibility of the stability operator in [27] was ensured by [27, Assumption (E)] which is violated by S as well as S.
We will now show how stability of (5.14) is nevertheless achieved when the equation is restricted to a codimension one subspace. First we reduce (5.14) to a matrix equation on C 2×2 through the partial trace operation C 2n×2n → C 2×2 , A → A defined via with A ij ∈ C n×n for all i, j = 1, 2. We apply this operation to (5.14) and find The first inequality in (5.23) holds because for any x, y ∈ C 2n we have Here, in the last step we used (5.9), (5.20) and for any random matrix A ∈ C 2n×2n and deterministic R ∈ C 2n×2n with x i := (r ij x j ) j and e i = (δ ij ) j . The second inequality in (5.23) follows from the Ward identity, (5.21). Finally, for any random matrix A ∈ C l×l we define a norm that tracks several p-norms simultaneously through for any fixed K ∈ N and # = av, iso. We use the convention A ⋆,K := A iso ⋆,K . This norm is convenient in the following to express inequalities such as (5.22), that estimate the p-norm of ∆ in terms of its 2p-norm, in a fixed norm. The ⋆-norm and p-norm are bounded in terms of each other through With these preparations we now formalise the bootstrapping step in which a rough bound on ∆ is transported from the almost global regime A δ,γ with γ close to 1 to the local regime A δ,γ with γ ≪ 1.
Proof. For transparency of the argument we write δ * := c * γ, where we will choose the constant c * > 0 sufficiently small later in the proof. Furthermore, we will omit the dependence on δ in the notation for the remainder of this proof. In particular, all constants implicit in the comparison relation may depend on δ and we write A γ = A δ,γ .
We assume that ∆ p p,γ n −γ/6 holds for all p on A γ . Since M = M 1 by (4.1a) we see that the resolvent is bounded on A γ , i.e. that G p p,γ 1. The function η → η G(ζ, η) p is monotonously increasing which is seen by checking positivity of its derivative. In particular, G(ζ, n −δ * η) p ≤ n δ * G(ζ, η) p .
By the isotropic global law from Proposition 5.3 we see that the bound ∆ p p,δ,γ n −γ/6 is satisfied on A δ,γ for some γ close to 1. Thus, repeated application of Lemma 5.4 implies the following corollary.
Armed with this rough bound that holds down to all mesoscopic scales, i.e. on A γ,γ for all γ > 0 we proof the local law for H ζ .