FLUCTUATION AROUND THE CIRCULAR LAW FOR RANDOM MATRICES WITH REAL ENTRIES

. We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identicallydistributed complex entries to the real symmetry class. We ﬁnd that the expectation and variance substantially diﬀer from their complex counterparts, reﬂecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the ﬁrst four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is theanalysisofseveralweaklydependentDysonBrownianmotions (DBMs). Theconceptualnovelty of the real case compared with [22] is that the correlation structure of the stochastic diﬀerentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.


Introduction
We consider an ensemble of n × n random matrices X with real i.i.d. entries of zero mean and variance 1/n; the corresponding model with complex entries has been studied in [22]. According to the circular law [6,58,38] (see also [11]), the density of the eigenvalues {σ i } n i=1 of X converges to the uniform distribution on the unit disk. Our main result is that the fluctuation of their linear statistics is Gaussian, i.e.
L n (f ) : converges, as n → ∞, to a centred normal distribution for regular test functions f with at least 2 + δ derivatives. We compute the variance V f and the next-order deviation of the expectation E n i=1 f (σ i ) from the value n π |z|≤1 f (z) given by the circular law.
As in the complex case, both quantities depend on the fourth cumulant of the single entry distribution of X, but in the real case they also incorporate the spectral symmetry of X onto the real axis. Moreover, the expectation carries additional terms, some of them are concentrated around the real axis; a by-product of the approximately √ n real eigenvalues of X. For the Ginibre (Gaussian) case they may be computed from the explicit density [27,26], but for general distributions they were not known before. As expected, the spectral symmetry essentially enhances V f by a factor of two compared with the complex case but this effect is modified by an additional term involving the fourth cumulant. Previous works considered either the case of analytic test functions f [48,49] or the (approximately) Gaussian case, i.e. when X is the real Ginibre ensemble or at least the first four moments of the matrix elements of X match the Ginibre ensemble [59,44].
In both cases some terms in the unified formulas for the expectation and the variance vanish and thus the combined effect of the spectral symmetry, the eigenvalues on the real axis, and the role of the fourth cumulant was not detectable in these works. We remark that a CLT for polynomial statistics of only the real eigenvalues for real Ginibre matrices was proven in [56].
In [53] the limiting random field L(f ) := lim n→∞ L n (f ) for complex Ginibre matrices has been identified as a projection of the Gaussian free field (GFF) [55]. We extended this interpretation [22] to general complex i.i.d. matrices with non-negative fourth cumulant and obtained a rank-one perturbation of the projected GFF. As a consequence of the CLT in the present paper, we find that in the real case the limiting random field is a version of the same GFF, symmetrised with respect to the real axis, reflecting the fact that complex eigenvalues of real matrices come in pairs of complex conjugates.
In general, proving CLTs for the real symmetry class is considerably harder than for the complex one. The techniques based upon the first four moment matching [59,44] are insensitive to the symmetry class, hence these results are obtained in parallel for both real and complex ensembles. Beyond this method, however, most results on CLT for non-Hermitian matrices were restricted to the complex case [25,33,47,51,52,54], see the introduction of [22] for a detailed history, as well as for references to the analogous CLT problem for Hermitian ensembles and log-gases. The special role that the real axis plays in the spectrum of the real case substantially complicates even the explicit formulas for the Ginibre ensemble both for the density [26] as well as for the k-point correlation functions [37,12,42]. Besides the complexity of the explicit formulas, there are several conceptual reasons why the real case is more involved. We now explain them since they directly motivated the new ideas in this paper compared with [22].
In [22] we started with Girko's formula [38] in the form given in [59] that relates the eigenvalues of X with resolvents of a family of 2n × 2n Hermitian matrices parametrized by z ∈ C. For any smooth, compactly supported test function f we have  (1.6) where (u z i , v z i ) ∈ C 2n is the (normalized) eigenvector of H z corresponding to the eigenvalue λ z i . Note that Θ z,z ij = δ i,j , and for j = i we have that Λ z ij ≈ 0. Moreover, if z is very close to the real axis, then the eigenvectors of H z are essentially real and Λ z ii = Θ z,z ii ≈ Θ z,z ii = 1. With z = z , this leads to (1.4) being essentially a real DBM with β = 1. (We recall that the parameter β = 1, 2, customarily indicating the real or complex symmetry class of a random matrix, also expresses the ratio of the coefficient of the repulsion to the strength of the diffusion in the DBM setup.) However, if z and z are far away, i.e. z is away from the real axis, then we can show that the overlap Λ z = Θ z,z is small, hence Λ z ij ≈ 0 for all i, j, including i = j. Thus the variance of the driving Brownian motions in (1.5) with z = z is reduced by a factor of two, rendering (1.4) essentially a complex DBM with β = 2.
The appearance of Λ z in (1.4) and the second term Θ z,z in (1.5) is specific to the real symmetry class; they were not present in the complex case [22]. They have three main effects for our analysis. for different z = z are nontrivial. Thus in the complex case [22], lacking the term Θ z,z in (1.5), the DBM (1.4) for any fixed z was the conventional DBM with independent Brownian motions and parameter β = 2 (c.f. [22,Eq. (7.15)]) and only the DBMs for different z's were mildly correlated. In the real case the correlations are already present within (1.4) for the same z due to Λ z = Θ z,z = 0.
We note that Dyson Brownian motions with nontrivial coefficients in the repulsion term have already been investigated in [17] (see also [19]) in the context of spectral universality of addition of random matrices twisted by Haar unitaries, however the driving Brownian motions were independent. The issue of well-posedness, nevertheless, has already emerged in [17] when the more critical orthogonal group (β = 1) was considered. The corresponding part of our analysis partly relies on techniques developed in [17]. We have already treated the dependence of Brownian motions for different z's in [22] for the complex case; but the more general dependence structure characteristic to the real case is a new challenge that the current work resolves. denote vectors by bold-faced lower case Roman letters x, y, . . . , ∈ C k , for some k ∈ N, and use the notation dx := dx 1 . . . dx k . Vector and matrix norms, x and A , indicate the usual Euclidean norm and the corresponding induced matrix norm. For any k × k matrix A we set A := k −1 Tr A to denote the normalized trace of A. Moreover, for vectors x, y ∈ C k and matrices A, B ∈ C k×k we define x, y := x i y i , A, B := A * B = 1 k Tr A * B.
We will use the concept of "event with very high probability" meaning that for any fixed D > 0 the probability of the event is bigger than 1 − n −D if n ≥ n 0 (D). Moreover, we use the convention that ξ > 0 denotes an arbitrary small exponent which is independent of n.

Main results
We consider real i.i.d. matrices X, i.e. n × n matrices whose entries are independent and identically distributed as x ab d = n −1/2 χ for some real random variable χ, satisfying the following: Assumption 2.1. We assume that E χ = 0 and E χ 2 = 1. In addition we assume the existence of high moments, i.e. that there exist constants C p > 0, for any p ∈ N, such that E|χ| p ≤ C p . (2.1) The circular law [6,7,11,39,10,36,38,50,58] asserts that the empirical distribution of eigenvalues {σ i } n i=1 of a complex i.i.d. matrix X converges to the uniform distribution on the unit disk D, i.e.
with very high probability for any continuous bounded function f . Our main result is a central limit theorem for the centred linear statistics for general real i.i.d. matrices and generic test functions f , complementing the recent central limit theorem [22] for the linear statistics of complex i.i.d. matrices. This CLT, formulated in Theorem 2.2, and its proof have two corollaries of independent interest that are formulated in Section 2.1 and Section 2.2.
In order to state the result we introduce some notations. For any function h defined on the boundary of the unit disk ∂D we define its Fourier transform as h(k) = 1 2π  where, in a slight abuse of notation, we identified f and g with their restrictions to ∂D. We use the convention that f is extended to C by setting it equal to zero on Ω c . Finally, we introduce the projection (P sym f )(z) := f (z) + f (z) 2 .
(2.6) EJP 26 (2021), paper 24. which maps functions on the complex plane to their symmetrisation with respect to the real axis. ∇P sym g, ∇P sym f L 2 (D) + P sym g, P sym f Ḣ1/2 (∂D) For the k-th moments we have an effective convergence rate of for some constant c(k + l) > 0. Moreover, the expectation in (2.3) is given by for some small constant c > 0.

Remark 2.3.
(i) Both expectation E(f ) and covariance C(g, f ) only depend on the symmetrised functions P sym f and P sym g. Indeed, E(f ) = E(P sym f ), and the coefficient of κ 4 in (2.7) can also be written as an integral over P sym f and P sym g.
(ii) By polarisation, a multivariate central limit theorem as in [ where we used that z → f (z) and z → f (z) are assumed to have disjoint support. Due to the moment matching assumption, κ 4 = 0 in the setting of [44].
(ii) The central limit theorem [ Here in the first step we used the analyticity of f (see [22,Eq. (2.11)]), and in the second step we used that (∇f )(z), (∇f (·))(z) = 0 and that f (k) = 0 for k < 0 while f (·)(k) = 0 for k > 0 by analyticity. We thus arrived at (2.7), since the coefficient of κ 4 in (2.7) vanishes also by analyticity of f in the setting of [49].

Remark 2.5 (Comparison with the complex case).
We remark that the limiting variance in the case of complex i.i.d. matrices, as studied in [22], is generally different from the real case. In the complex case L n (f ) converges to a complex Gaussian with variance where · D denotes the averaging over D as in (2.7). In contrast, in the real case the limiting variance is given by Thus the variances agree exactly in the case of analytic test functions by (2.9) and V (C,2) f = 0, while e.g. in the case of symmetric test functions, f = P sym f and vanishing fourth cumulant κ 4 = 0 the real variance is twice as big as the complex one, V in terms of the upper incomplete Gamma function Γ(s, x). Using the large n asymptotics uniform in z = x + iy for the incomplete Gamma function [61, Eq. (2. 2)] we obtain which, using asymptotics of the error function for any fixed |z| < 1, gives that in agreement with the second term in the rhs. of (2.8) accounting for the n −1 -correction to the circular law away from the real axis.
The situation very close to the real axis is much more subtle. The density of the real Ginibre eigenvalues is explicitly known [27,Corollary 4.3] and it is asymptotically uniform on [−1, 1], see [27,Corollary 4.5], giving a singular correction of mass of order n −1/2 to the circular law. However, the abundance of real eigenvalues is balanced by the sparsity of genuinely complex eigenvalues in a narrow strip around the real axisa consequence of the factor |y| in (2.10). Since these two effects of order n −1/2 cancel each other on the scale of our test functions f , they are not directly visible in (2.8).
Instead we obtain a smaller order correction of order n −1 specific to the real axis, in form of the second, the penultimate and the ultimate term in (2.8 for integers k ≥ 1, as n → ∞ (note that the trace is unnormalised). The asymptotics (2.11) are consistent with (2.8) since

Connection to the Gaussian free field
It has been observed in [53] that for complex Ginibre matrices the limiting random field L(f ) can be viewed as a projection of the Gaussian free field (GFF) [55]. In [ (2.12) The Sobolev space H 1 0 (C) = C ∞ 0 (C) · H 1 0 (C) can be orthogonally decomposed into i.e. the H 1 0 -closure of smooth functions which are compactly supported in D or D c , and their orthogonal complement H 1 0 ((∂D) c ) ⊥ , the closed subspace of functions analytic outside of ∂D (see e.g. [55,Thm. 2.17]). With the orthogonal projection P onto the first and third of these subspaces, we have (see [22,Eq. (2.13)]) If κ 4 ≥ 0, then L can be interpreted as as a consequence of (2.12) and (2.13).

Universality of the local singular value statistics of X − z close to zero
As a by-product of our analysis we obtain the universality of the small singular values of X − z, and prove that (up to a rescaling) their distribution asymptotically agrees with the singular value distribution of a complex Ginibre matrix X if z / ∈ R, even though X is a real i.i.d. matrix. In the following by {λ z i } i∈[n] we denote the singular values of X − z in increasing order.
It is natural to express universality in terms of the k-point correlation functions p (n) k,z which are defined implicitly by − p matrix X, even if the entries of X are real i.i.d. random variables. It is expected that the same result holds for all (possibly n-dependent) z as long as | z| n −1/2 , while in the opposite regime | z| n −1/2 the local statistics of the real Ginibre prevails with an interpolating family of new statistics which emerges for | z| ∼ n −1/2 .
Besides the universality of small singular values of X − z, our methods also allow us to conclude the asymptotic independence of the small singular values of X − z 1 and those of X − z 2 for generic z 1 , z 2 . More precisely, similarly to (2.15), we define the correlation function p (n) k1,z1;k2,z2 for the singular values of X − z 1 and X − z 2 implicitly by for any test function f , and any k 1 , k 2 ∈ N, where we used the notations λ z1 ).
Theorem 2.10 (Asymptotic independence of small singular values of X − z 1 , X − z 2 ). Let z 1 , z 2 ∈ C be as z in Theorem 2.8, and assume that |z 1 − z 2 |, |z 1 − z 2 | ∼ 1. Let X be an i.i.d. matrix with real entries satisfying Assumption 2.1, then for any k 1 , k 2 ∈ N, and for any compactly supported test function F ∈ C 1 c (R k ), with k = k 1 + k 2 , using the notation where ρ z l = ρ z l (0), and c(k) > 0 is a small constant only depending on k. The implicit constant in O(·) may depend on k, F C 1 , and C p from (2.1).
Remark 2.11. We stated Theorem 2.8 for two different z 1 , z 2 for notational simplicity. The analogous result holds for any finitely many z 1 , . . . , z q such that |z l − z m |, |z l − z m | ∼ 1, with l, m ∈ [q].

Proof strategy
The proof of Theorem 2.2 follows a similar strategy as the proof of [22,Thm. 2.2] with several major changes. We use Girko's formula to relate the eigenvalues of X to the resolvent of the 2n × 2n matrix the so called Hermitisation of X − z. We denote the eigenvalues of H z , which come in pairs symmetric with respect to zero, by {λ z ±i } i∈ [n] . The local law, see Theorem 3.1 below, asserts that the resolvent G(w) = G z (w) := (H z − w) −1 of H z with η = w = 0 becomes approximately deterministic, as n → ∞. Its limit is expressed via the unique solution of the scalar equation which is a special case of the matrix Dyson equation (MDE), see e.g. [1] and (5.1) later.
Note that on the imaginary axis m z (iη) = i m z (iη). We define the self-consistent density EJP 26 (2021), paper 24. The local law for G z (w) in its full averaged and isotropic form has been obtained for w ∈ iR in [2] for the bulk regime |1 − |z|| ≥ and in [3] for the edge regime |1 − |z|| < . In fact, in the companion paper [22] on the complex CLT the local law for w on the imaginary axis was sufficient. For the real CLT, however, we need its extension to general spectral parameters w in the bulk |1 − |z|| ≥ case that we state below. We remark that tracial and entry-wise form of the local law in Theorem 3.1 has already been established in [16,Theorem 3.4].
Theorem 3.1 (Optimal local law for G). For any > 0 and z ∈ C with |1 − |z|| ≥ the resolvent G z at w ∈ H with η = w is very well approximated by the deterministic matrix M z in the sense that with very high probability for some C ≤ −100 , uniformly for η ≥ n −100 , |1 − |z|| ≥ , and for any deterministic matrices A and vectors x, y, and ξ > 0. Since it is not necessary for the present work we refrain from adapting this technique for general w and rather present a conceptually simpler proof resulting in the -dependent bounds (3.6).
As in [22] we express the linear statistics (1.1) of eigenvalues σ i of X through the resolvent G z via Girko's Hermitisation formula (1.3) for η 0 = n −1−δ0 , η c = n −1+δ1 , and T = n 100 , where J T in (3.7) corresponds to the rhs. of the first line in (3.7) whilst I η0 0 , I ηc η0 , I T ηc correspond to the three different η-integrals in the second line of (3.7). Here we used that by spectral symmetry of H z it follows that G z (iη) ∈ iR and therefore G z (iη) = G z (iη) /i in order to obtain (3.7) from (1.3). The regime J T can be trivially estimated by [22,Lemma 4.3], while the regime I η0 0 can be controlled using [60,Thm. 3.2] as in [22,Lemma 4.4] (see [22,Remark 4.5] for an alternative proof). Both contributions are negligible. For the main term I T ηc we prove the following resolvent CLT .  3.3 (CLT for resolvents). Let > 0, η 1 , . . . , η p > 0, and z 1 , . . . , z p ∈ C be such that for any i = j, min{η i , η j } ≥ n −1 |z i − z j | −2 . Then for any ξ > 0 the traces of the Moreover, the expectation of the normalised trace of G = G i is given by (i) the V -term for the variance appears in a symmetrised form with z j and z j , (ii) the error term (3.9) deteriorates as z i ≈ 0, and (iii) the expectation (3.11) has an additional subleading term which is even present in case κ 4 = 0 (second term in (3.12)). Finally, in order to show that I ηc η0 in (3.7) is negligible, we prove that G z1 (iη 1 ) and G z2 (iη 2 ) are asymptotically independent if z 1 , z 2 and z 1 , z 2 are far enough from each other, they are far away from the real axis, they are well inside D, and η 0 ≤ η 1 , η 2 ≤ η c . These regimes of the parameters z 1 , z 2 represent the overwhelming part of the d 2 z 1 d 2 z 2 integration in the calculation of E|I ηc η0 | 2 . The following proposition is the direct analogue of [22,Prop. 3.5].
Proposition 3.4 (Independence of resolvents with small imaginary part). Fix p ∈ N.
For any sufficiently small ω h , ω d > 0 there exist ω * , δ 0 , δ 1 with ω h δ m ω * 1, for m = 0, 1, such that for any choice of z 1 , . . . , z p with n ω * , (3.13) for any η 1 , . . . , η p ∈ [n −1−δ0 , n −1+δ1 ]. Fluctuation around the circular law As in the complex case [22], one key ingredient for both Propositions 3.3 and 3.4 is a local law for products of resolvents G 1 , G 2 for G i = G zi (w i ). We remark that local laws for products of resolvents have also been derived for (generalized) Wigner matrices [30,46] and for sample covariance matrices [20], as well as for the addition of random matrices [8].
Note that the deterministic approximation to G 1 G 2 is not given simply by M 1 M 2 where M i := M zi (w i ) from (3.4). To describe the correct approximation, as in [22, Section 5], we define the stability operator acting on the space of 2n × 2n matrices. Here the linear covariance or self-energy operator S : C 2n×2n → C 2n×2n is defined as i.e. it averages the diagonal blocks and swaps them. Here E denotes the expectation with respect to X, A = n −1 TrA and Gin C stands for the standard complex Ginibre ensemble. The ultimate equality in (3.15) follows directly from E x 2 ab = 0, E| x ab | 2 = n −1 . Note that as a matter of choice we define the stability operator (3.14) with the covariance operator S corresponding to the complex rather than the real Ginibre ensemble. However, to leading order there is no difference between the two and the present choice is more consistent with the companion paper [22]. The effect of this discrepancy will be estimated in a new error term (see (6.4) later).
For any deterministic matrix B we define (3.16) which turns out to be the deterministic approximation to G 1 BG 2 . Indeed, from the local law for G 1 , G 2 , Theorem 3.1, and [22, Thm. 5.2] we immediately conclude the following theorem.
Theorem 3.5 (Local law for G z1 BG z2 ). Fix z 1 , z 2 ∈ C with |1 − |z i || ≥ , for some > 0 and w 1 , w 2 ∈ C with |η i | := | w i | ≥ n −1 such that for some small * > 0, where β * is the, in absolute value, smallest eigenvalue of B 12 defined in (3.14). Then, for any bounded deterministic matrix B, B 1, the product of for some C with very high probability for any deterministic A, x, y and ξ > 0. If w 1 , w 2 ∈ iR we may choose C = 1, otherwise we can choose C ≤ −100 .
An effective lower bound on β * , hence on | β * |, will be given in Lemma 6.1 later.
The paper is organised as follows: In Section 4 we prove Theorem 2.2 by combining imaginary axis, Theorem 3.1. In Section 6 we prove Proposition 3.3, the Central Limit Theorem for resolvents using Theorem 3.5. In Section 7 we prove Proposition 3.4 again using Theorem 3.5, and conclude Theorem 2.8.
Note that Theorem 3.5, the local law for G z1 BG z2 , is used in two different contexts. Traces of AG z1 BG z2 , for some deterministic matrices A, B ∈ C 2n×2n , naturally arise along the cumulant expansion for i G i − E G i in Proposition 3.3. The proof of Proposition 3.4 is an analysis of weakly correlated DBMs, where the correlations are given by eigenvector overlaps (1.6), whose estimate is reduced to an upper bound on G z1 G z2 .
. The identities (4.1)-(4.2) will be proven separately below. The other two modifications concern the error terms in (3.9) and (3.11). Namely, there is an additional factor includ- Proof of (4.1). With the short-hand notation z = x + iy, we compute Fluctuation around the circular law using that u = 1 + O(η) for |z| ≤ 1 and u = |z| −2 + O(η) for |z| > 1, so that for (4.1) we need to compute We may assume that f is symmetric with respect to the real axis, i.e. f = P sym f with P sym as in (2.6) since L n (f − P sym f ) = 0 by symmetry of the spectrum and therefore L n (f ) = L n (P sym f ). Since the functions in (4.4) are singular we introduce an -regularisation which enables us to perform integration by parts. In particular, the integral in (4.4) is equal to the → 0 limit of where |z ± 1| ≥ denotes that |z − 1| ≥ and |z + 1| ≥ , and we used that the contribution from the regimes |y| ≤ and |z ± 1| ≤ are negligible as → 0. In the following equalities should be understood in the → 0 limit.
for |z| = 1, when integrating by parts in (4.5), the terms where either 1(|z| ≤ 1) or 1(|z| > 1) are differentiated are equal to zero, using that We remark that (4.6) is understood in the sense of distributions, i.e. the equality holds when tested against compactly supported test functions f : Moreover, with a slightly abuse of notation in (4.6) by 1(|z| = 1) dz we denote the clockwise contour integral over the unit circle. This notation is used in the remainder of this section.
Then, performing integration by parts with respect to ∂ z , we conclude that (4.5) is (4.7) In order to get (4.7) we used that for some small fixed δ > 0, by f ∈ H 2+δ , and similarly all the other -boundary terms tend to zero. This implies that when the ∂ z derivative hits the -boundary terms then these give a negligible contribution as → 0. We now consider the two terms in (4.7) separately.
Since the integral of y −1 over D is zero we can rewrite the first term in (4.7) as EJP 26 (2021), paper 24.

Fluctuation around the circular law
Then performing integration by parts we conclude that the first term in (4.7) is equal to where we used that to show that the terms when the ∂ z derivative hits the -boundary terms go to zero as → 0. Note that the integrals in (4.8) are absolutely convergent since f is symmetric with respect to the real axis. For the second term in (4.8) we further compute where we used that the term with cos θ/ sin θ is zero by symmetry.
Proof of (4.2). By change of variables z 2 → z 2 we can then write such that [22,Lemma 4.8] is applicable and (4.2) follows.

Local law away from the imaginary axis: proof of Theorem 3.1
The goal of this section is to prove a local law for G = G z (w) for z in the bulk, as stated in Theorem 3.1. We do not follow the precise -dependence in the proof explicitly but it can be checked from the arguments below that C = −100 clearly suffices. We denote the unique solution to the deterministic matrix equation (see e.g. [1]) where we recall the definition of S from (3.15). The solution to (5.1) is given by (3.4). To keep notations compact, we first introduce a commonly used (see, e.g. [28]) notion of high-probability bound.
are families of non-negative random variables indexed by n, and possibly some parameter u in a set U (n) , then we say that X is stochastically dominated by Y , if for all , for large enough n ≥ n 0 ( , D). In this case we use the notation X ≺ Y . Moreover, if we have |X| ≺ Y for families of random variables X, Y , we also write X = O ≺ (Y ).
Let us assume that some a-priori bounds for some deterministic control functions Λ and ξ depending on w, z have already been established, uniformly in x, y, A under the constraint x , y , A ≤ 1. From the resolvent where we introduced the self-renormalisation, denoted by underlining, of a random variable of the form W f (W ) for some regular function f as with X independent of X. The choice of defining the self-renormalisation in terms of the complex rather than real Ginibre ensemble has the consequence that an additional error term needs to be estimated. For real Ginibre we have but the renormalisation comprises only the S[G] term, i.e.
thus the T -term needs to be estimated. By the Ward identity GG and therefore, together with the bound (5.5) on the T -term we obtain We now consider the stability operator B := 1 − M S[·]M which expresses the stability of (5.1) against small perturbations. Since S only depends on the four block traces of the input matrix, and M is a multiple of the identity matrix in each block, the operator B can be understood as an operator acting on 2 × 2 matrices after taking a partial trace. Henceforth for all practical purposes we may identify B with this four dimensional operator. Written as a 4 × 4 matrix, it is given by Here the rows and columns of B are ordered in such a way that 2 × 2 matrices are mapped to vectors as in We first record some spectral properties of B in the following lemma, the proof of which we defer to the end of the section. Note that B * refers to the adjoint of B with respect to the scalar product A, B = (2n) −1 TrA * B, for any deterministic matrices A, B ∈ C 2n×2n . Lemma 5.2. Let w ∈ H, z ∈ C be bounded spectral parameters, |w| + |z| 1. Then the operator B has the trivial eigenvalues 1 with multiplicity 2, and furthermore has two non-trivial eigenvalues, and left and right eigenvectors (5.8) Moreover, for the second non-trivial eigenvalue we have the lower bound Corresponding to the two non-trivial eigenvalues of B we define the spectral projec-  We now distinguish the two cases ρ ∼ 1 and ρ 1. In the former we obtain and thus x, where we used that the image of xy * under (Q * B −1 ) * is of rank at most 4, hence it can be written as In the complementary case ρ 1 we similarly decompose Now we apply B to both sides of (5.13) and take the inner product with V l to obtain from (5.10). For the spectral projection Q we find Thus it follows that by a simple calculation. By using (5.10) in (5.14) it follows that from (5.2), (5.6) since, due to ||z| − 1| , we have |1 − m 2 − u 2 |z| 2 | ≥ ρ according to (5.9).
to (5.12) that By using the bounds in (5.12) and (5.18) in the two complementary regimes we improve the input bound in (5.2). We can iterate this procedure and obtain In order to make sure the iteration yields an improvement one needs an priori bound on ξ of the form ξ 1 since otherwise ξ 2 is difficult to control. For large η such an a priori bound is trivially available which can then be iteratively bootstrapped by monotonicity down to the optimal η n −1 . For details on this standard argument the reader is referred to e.g. [4,Section 3.3]. Then the local law for any η > 0 readily follows by exactly the same argument as in [23,Appendix A]. This completes the proof of Theorem 3.1.
Proof of Lemma 5.2. The fact that B has the eigenvalue 1 with multiplicity 2, and the claimed form of the remaining two eigenvalues and corresponding eigenvectors can be checked by direct computations. Taking the imaginary part of (3.2) we have which implies as m and w have the same sign. Here supp ρ should be understood as the support of the self-consistent density of states, as defined in (3.3), restricted to the real axis. The second bound in (5.9) then follows from (5.21) and The bound (5.22) can be improved in the case ρ 1 if w is near a regular edge of ρ, i.e. where ρ locally vanishes as a square-root. According to [ By the explicit form of e ± it follows that e ± 1 whenever |1 − |z|| ≥ . In contrast, if |z| = 1, then ρ has a cusp singularity in 0 where it locally vanishes like a cubic root. Near a regular edge we have m √ w, and therefore from (5.20) proving also the first inequality in (5.9).

CLT for resolvents: proof of Proposition 3.3
The goal of this section is to prove the CLT for resolvents, as stated in Proposition 3.3. The proof is very similar to [22, Section 6] and we focus on the differences specific to the real case. Within this section we consider resolvents G 1 , . . . , G p with G i = G zi (iη i ) and η i ≥ n −1 . As a first step we recall the leading-order approximation of G = G i , where the stability operator B has been defined in (5.7). Here β is the eigenvalue of B with eigenvector (1, 1, 0, 0) and is bounded by (see [22, Eq. (6.8b)]) |β| |1 − |z|| + η 2/3 .

(6.2)
One important input for the proof of Proposition 3.3 is a lower bound on the eigenvalues of the stability operator B, defined in (3.14), the proof of which we defer to the end of the section. Note that the two-body stability operator B and its eigenvalues β, β * are consistently decorated by hats ( ·) to distinguish them from their one-body analogues B, β. We will consistently equip B, B and their eigenvalues, β, β, β * with indices when instead of M they are defined with the help of M i = M zi (w i ); e.g. β 1i * is the lowest eigenvalue of B 1i = B(z 1 , z i , w 1 , w i ) defined analogously to (3.14).
Proof of Proposition 3.3. The proof of Proposition 3.3 goes in two steps. First, we use (6.1) and a cumulant expansion in order to prove the asymptotic representation of the expectation in (3.11). In the second step we then turn to the computation of higher moments and establish an asymptotic Wick theorem in the form of (3.8).
We use the notation ∆ ab for the matrix (∆ ab ) cd = δ ac δ bd and decompose W = ab w ab ∆ ab . For each a, b we then perform a cumulant expansion and obtain which has an additional term compared to the complex case [ not the κ(ab, ab) = 1 cumulant into account. Here κ(ab, cd, ef, . . .) denotes the joint cumulant of the random variables w ab , w cd , w ef , . . ., and we denote partial derivatives by ∂ α := ∂ wα 1 · · · ∂ wα k for tuples α = (α 1 , . . . , α k ), with α i ∈ [n] × [n]. In (6.4) we introduced the notation ab := a≤n b>n + a>n b≤n .
We note that by Assumption 2.1 the cumulants κ(α 1 , . . . , α k ) satisfy the scaling For the second term in (6.4) we find exactly as in [22, Eqs. (6.13)-(6.10)] that (6.6) For the first term in (6.4), which is new compared to [22, Eq. (6.11)], we rewrite where we used that (G z ) t = G z , and the convention that formulas containing (E, E ) are understood so that the matrices E, E are summed over the assignments (E, where z = x + iy, and the second step follows by explicitly computing the inverse in terms of the entries of M , noting that m z = m z and u z = u z . Then, using the definition v := −im > 0 and that Now (3.11) follows from combining (6.1) and (6.4)-(6.8).
We now turn to the computation of higher moments for which we recall from (6.1) and (3.11 with A i as in (6.1) and E i as in (3.12), and with the bound on β i from (6.2). We begin with the cumulant expansion of W G 1 to obtain where, compared to [22,Eq. (6.17)], the first line on the rhs. has an additional term specific to the real case, and W , as opposed to W in (5.4), is the Hermitisation of an independent real Ginibre matrix X with expectation E. The expansion of the third line on the rhs. of (6.11) is completely analogous to [22] since for cumulants of degree at least three nothing specific to the complex case was used. Therefore we obtain, from combining 2 [22, Eqs. (6.26), (6.29)], that Recall the definition of E i in (3.12), then using (6.7)-(6.8) and (6.12) in (6.11) we thus Fluctuation around the circular law It remains to consider the variance term in (6.13) for which we use the identity in order to compute for which we use the local law from Theorem 3.5 to conclude that the main terms in (6.15) are with V being exactly as in the complex case, i.e. as in (3.10). For the error term in (6. 16) we claim that The CLT for resolvents, as stated in (3.8) follows from inserting (6.15)-(6.19) into (6.13), and iteration of (6.13) for the remaining product.
In order to conclude the proof of Proposition 3.3 it remains to prove (6.19). Introduce the shorthand notation G i1i for generic finite sums of products of appearing in the first term in (6.19). We will prove the more general claim The proof is similar to [22,Eq. (6.32)]. Therefore we focus on the differences. In the cumulant expansion of (6.20) there is an additional term compared to [22, Eq. (6.33)] given by where we combined two terms of type G 1iii into one since in our convention G 1iii is a short-hand notation for generic sums of products. We now perform another cumulant expansion of (6.21) to obtain (6.22) where the first line on the rhs. corresponds to the term where the remaining W acts on G i1i within its own trace as in (6.21), and in the last line we used the scaling bound (6.5) for κ. In order to estimate (6.21) we recall [22,Lemma 5.8].
..j k we denote generic products of resolvents G j1 , . . . , G j k , or their adjoints/transpositions (in that order, each G ji appears exactly once) with bounded deterministic matrices in between, e.g.
(ii) For j 1 , . . . , j k and any 1 ≤ s < t ≤ k we have the averaged bound Since only η 1 , η i play a role within the proof of (6.19), we drop the indices from η 1i * and use the notation η * = η 1i * . For the first term in (6.22) we use (6.23b) to obtain Similarly for the second term we use (6.14) and again (6.23b) to bound it by since η * ≥ 1/n. Finally, for the last term of (6.22) we estimate the presence of an additional factor of G 1 or G i is compensated by the fact that the same type of G appears two additional times as the first or last factor in some product. For the k = 2 case we observe that by parity at least one factor will be off-diagonal in the sense that it has two distinct summation indices from {a, b, c} giving rise to an additional factor of (nη * ) −1/2 by summing up one of the indices with the Ward identity. For example, for the term with Thus, in general we obtain a bound of By combining (6.24)-(6.26) we obtain a bound of (nη 1 η i η * ) −2 on the additional term (6.21).
The remaining terms can be estimated as in [22, Eq. (6.32)] and we conclude the proof of (6.20) and thereby Proposition 3.3.
Proof of Lemma 6.1. The claim (6.3) is equivalent to the claim (6.27) where τ, τ * are the eigenvalues of the matrix from which (6.29) follows by choosing M to be the largest eigenvalue of (A + A * )/2.
where in the last step we used (5.21).
We now assume that for some 0 ≤ 1 we have i.e. that all inequalities in (6.30a)-(6.30i) are in fact equalities up to an 2 error. The assertion (6.27) is then equivalent to the proof of which we present now. The fact that (6.30h)-(6.30i) is 2 -saturated implies the saturation i.e. the bound on the last two terms in (6.32).

Asymptotic independence of resolvents: proof of Proposition 3.4
For any fixed z ∈ C let H z be defined in (3.1). Recall that we denote the eigenvalues of H z by {λ z ±i } i∈[n] , with λ z −i = −λ z i , and by {w z ±i } i∈[n] their corresponding orthonormal eigenvectors. As a consequence of the symmetry of the spectrum of H z with respect to zero, its eigenvectors are of the form w z ±i = (u z i , ±v z i ), for any i ∈ [n]. The eigenvectors of H z are not well defined if H z has multiple eigenvalues. This minor inconvenience can be easily solved by a tiny Gaussian regularization (see (7.17) and Remark 7.5 later). In the remainder of this section we always assume that all the indices are not zero, e.g we use the notation   Section 7 is divided as follows: in Section 7.1 we state the main technical results needed to prove Proposition 7.2 and conclude its proof. In Section 7.2 we prove Theorem 2.8, which will follow by the results stated in Section 7.1. In Section 7.3 we estimate the overlaps of eigenvectors, corresponding to small indices, of H z l , H zm for l = m; this is the main input to prove the asymptotic independence in Proposition 7.2. In Section 7.4 and Section 7.6 we prove several technical results stated in Section 7.1. In Section 7.5 we present Proposition 7.14 which is a modification of the path-wise coupling of DBMs close to zero from [22,Proposition 7.14] to the case when the driving martingales in the DBM have a small correlation. This is needed to deal with the (small) correlation of λ z l , the eigenvalues of H z l , for different l's.

Overview of the proof of Proposition 7.2
The main result of this section is the proof Proposition 7.2, which is essentially about the asymptotic independence of the eigenvalues λ z l i , λ zm j , for l = m and small indices i and j. We do not prove this feature directly, instead we will compare λ z l i , λ zm j with similar eigenvalues µ (see (7.14) later) differs from the standard DBM in two related aspects: (i) the driving martingales are weakly correlated, (ii) the interaction term has a coefficient slightly deviating from one. Note that the stochastic driving terms b i in (7.14) are martingales but not Brownian motions (see Appendix B for more details). Both effects come from the small but non-trivial overlap of the eigenvectors w z l i with w z l j . They also influence the well-posedness of the DBM, so an extra care is necessary. We therefore define two comparison processes. First we regularise the DBM by (i) setting the coefficient of the interaction equal to one, (ii) slightly reducing the diffusion term, and (iii) cutting off the possible large values of the correlation. The resulting process, denoted byλ(t) (see (7.22) later), will be called the regularised DBM. Second, we artificially remove the correlation in the driving martingales for large indices. This partially correlated DBM, defined in (7.27) below, will be denoted by λ(t). We will show that in both steps the error is much smaller than the relevant scale 1/n. After these preparations, we can directly compare the partially correlated DBM λ(t) with its Ginibre counterpart µ(t) (see (7.29) later) since their distribution is the same. Finally, we remove the partial correlation in the process µ(t) by comparing it with a purely independent Ginibre DBM µ(t), defined in (7.24) below. Now we define these processes precisely. From now on we assume that p = 2 in Proposition 7.2 to make our presentation clearer. The case p ≥ 3 is completely analogous.

Consider the Ornstein-Uhlenbeck (OU) flow
for a time t f := n ω f n , for some explicit constant c > 0 very close to 1, and U is a real Ginibre matrix independent of q X t f . Using a simple Green's function comparison argument (GFT), by [22,Lemma 7.5], we conclude the following lemma.
Lemma 7.3. The eigenvalues of H z l and the eigenvalues of H z l t f , with t f = n −1+ω f obtained from replacing X by X t f , are close in the sense that for any sufficiently small ω f , δ 0 , δ 1 > 0 it holds where η l ∈ [n −1−δ0 , n −1+δ1 ].
Next, we consider the matrix flow EJP 26 (2021), paper 24. and denote by H z t the Hermitisation of X t − z. Here B t is a real standard matrix valued Brownian motion independent of X 0 and B t . Note that by construction X ct f is such that and the resolvent by G z t (w) := (H z t − w) −1 for w ∈ H. For any w = (u, v), with u, v ∈ C n define the projections P 1 , P 2 : C 2n → C n by P 1 w = u, P 2 w = v, (7.9) and, for any z, z ∈ C, define the eigenvector overlaps by , |i|, |j| ≤ n.
for any |i|, |j| ≤ n. The coefficients Θ z,z ij (t) are small with high probability due to the following lemma whose proof is postponed to Section 7.3.

Lemma 7.4 (Eigenvectors overlaps are small). For any sufficiently small constants
with very high probability for any fixed T ≥ 0.
Most of the DBM analysis is performed for a fixed z ∈ {z 1 , z 2 }, with z 1 , z 2 as in Proposition 7.2, for this purpose we introduce the notation Λ z ij (t) := Θ z,z ij (t), (7.13) for any |i|, |j| ≤ n. In particular, note Θ z,z ij = Θ z,z ij and so that by (7.11) it follows that Λ z ij (t) = Λ z ji (t). By the derivation of the DBM in Appendix B, using the fact that w z = w z , for z = z l with l ∈ [2], it follows that (7.7) induces the flow on the eigenvalues {λ z i (t)} |i|≤n of H z t . Here {λ z i } |i|≤n are the eigenvalues of the initial matrix H z . The martingales {b z i } i∈[n] , with b z i (0) = 0, and Λ z ij (t), the overlap of eigenvectors in (7.13), (7.10), are defined on a probability space Ω b equipped with the filtration (F b,t ) 0≤t≤T := σ(X 0 , (B s ) 0≤s≤t ) 0≤t≤T , (7.15) where B s is defined in (7.7). The martingale differentials in (7.14) are such that  and db z −i = − db z i for i ∈ [n]. Here we used the notation Ω b for the probability space to emphasize that is the space where the martingales b z are defined, since in Section 7.1.2 we will introduce another probability space which we will denote by Ω β .
In the remainder of this section we will apply Lemma 7.4 for z = z 1 , z = z 2 and z = z 1 , z = z 2 and z = z l , z = z l , for l ∈ [2], with z 1 , z 2 fixed as in Proposition 7.2. We recall that throughout this section we assumed that p = 2 in Proposition 7. Using this result, we may, without loss of generality, assume that the eigenvalues of H z l t are almost surely distinct for any fixed time t ≥ 0. Indeed, if this were not the case then we replace H z l 0 by (t)} |i|,|j|≤n are well-defined for almost all t ≥ 0; we set them equal to zero whenever they are not well defined.
Remark 7.5. The perturbation of X in (7.17) is exponentially small, hence does not change anything in the proof of the local laws in Theorem 3.1 and Theorem 3.5 or in the Green's function comparison (GFT) argument in Lemma 7.3, since these proofs deal with scales much bigger than e −n . This implies that any local law or GFT result which holds for H z l t then holds true for H z l t,reg as well. Hence, in the remainder of this section we assume that [19, Lemma 6.2] holds true for H z l t (the unperturbed matrix). The process (7.14) is well-defined in the sense of Proposition 7.6, whose proof is postponed to Section 7.6. Proposition 7.6 (The DBM in (7.14) is well-posed). Fix z ∈ {z 1 , z 2 }, and let H z t be defined by the flow (7.7). Then the eigenvalues λ(t) of H z t are the unique strong solution to (7.14) on [0, T ], for any T > 0, such that λ(t) is adapted to the filtration (F b,t ) 0≤t≤T , λ(t) is γ-Hölder continuous for any γ ∈ (0, 1/2), and P λ −n (t) < · · · < λ −1 (t) < 0 < λ 1 (t) < · · · < λ n (t), for almost all t ∈ [0, T ] = 1.
In order to prove that the term Λ z ij in (7.14) is irrelevant, we will couple the driving martingales in (7.14) with the ones of a DBM that does not have the additional term Λ z ij (see (7.22) below). For this purpose we have to consider the correlation of {b z1 i } |i|≤n , {b z2 i } |i|≤n for two different z 1 , z 2 ∈ C as in Proposition 7.2. In the following we will focus only on the driving martingales with positive indices, since the ones with negative indices are defined by symmetry. The martingales b z l = {b z l i } i∈[n] , with l = 1, 2, are defined on a common probability space equipped with the filtration (F b,t ) 0≤t≤T from (7.15).
We consider b z1 , b z2 jointly as a 2n-dimensional martingale (b z1 , b z2 ).  Fluctuation around the circular law Note that C(t) is a positive semi-definite matrix. In particular, taking also negative indices into account, for a fixed z ∈ {z 1 , z 2 }, the family of martingales b z = {b z i } |i|≤n is such that dt, |i|, |j| ≤ n. ij are typically small for any z, z ∈ C such that |z|, |z | ≤ 1 − n −ω h and |z − z | ≥ n −ω d . We now define their cut-off versions (see (7.21)

below).
We only consider positive indices, since negative indices are defined by symmetry. Throughout this section we use the convention that regularised objects will be denoted by circles. Let z l , with l ∈ [2] be fixed throughout Section 7 as in Proposition 7.2. Define the 2n × 2n matrixC(t) bẙ for any l, m ∈ [2], recalling that Λ z l ij = Θ z l ,z l ij . Note that by Lemma 7.4 it follows that C(t) = C(t) on a set of very high probability, andC(t) = 1 2 I, with I the 2n × 2n identity matrix, on the complement of this set, for any t ∈ [0, T ]. In particular,C(t) is positive semi-definite for any t ∈ [0, T ], since C(t), defined as a covariance in (7.18), is positive semi-definite. The purpose of the cut-off in (7.20) it is to ensure the well-posedness of the process (7.22) below.
We compare the processes λ z l (t) in (7.14) with the regularised processesλ z l (t) defined, for z = z l , by , |i| ≤ n, (7.22) with ω r > 0 such that ω f ω r ω E . We organise the martingales b z1 , b z2 with positive indices into a single 2n-dimensional vector b = (b z1 , b z2 ) with a correlation structure given by (7.18). Then by Doob's martingale representation theorem [41,Theorem 18.12] there exists a standard Brownian motion w = (w (1) , w (2) ) ∈ R 2n realized on an extension to the GOE case, i.e. β = 1). The well-posedness of the process (7.22) is proven in Appendix A. The main result of this section is the following proposition, whose proof is deferred to Section 7.4.
Proposition 7.7 (The regularised processλ is close to λ). For any sufficiently small with very high probability, where t f = n −1+ω f and c > 0 is defined in (7.5).

Definition of the partially correlated processes λ, µ
The construction of the partially correlated processes forλ z l (t) is exactly the same as in the complex case [22,Section 7.2]; we present it here as well for completeness. We want to compare the correlated processesλ z l (t), with l = 1, 2, defined on a probability space Ω b equipped with a filtration F b,t with carefully constructed independent processes µ (l) (t), l = 1, 2 on a different probability space Ω β equipped with a filtration F β,t , which is defined in (7.25) below. We choose µ (l) (t) to be a complex Ginibre DBM, i.e. it is given as the solution of dµ (l) The comparison ofλ z l (t) and µ (l) (t) is done via two intermediate partially correlated processes λ (l) (t), µ (l) (t) so that for a time t ≥ 0 large enough λ (l) i (t) for small indices i will be close toλ z l i (t) and µ (l) i (t), respectively, with very high probability. Additionally, the processes λ (l) , µ (l) will be constructed such that they have the same joint distribution: (7.26) for any T > 0.
Fix ω A > 0 such that ω h ω A ω f , and for l ∈ [2] define the process λ (l) (t) to be the solution of d λ n ω A < |i| ≤ n, (7.27) with initial data λ (l) (0) being the singular values, taken with positive and negative sign, of independent complex Ginibre matrices Y (l) independent of λ z l (0). Here db z l i is the martingale differential from (7.22) which is used for small indices in (7.27). For large indices we define the driving martingales to be an independent collection {{ b } are defined on a common probability space that we continue to denote by Ω b with the common filtration F b,t , given by The well-posedness of (7.27), and of (7.29) below, readily follows by exactly the same arguments as in Appendix A.
Notice thatλ(t) and λ(t) differ in two aspects: the driving martingales with large indices for λ(t) are set to be independent, and the initial conditions are different. Lemma 7.8 below states that these differences are negligible for our purposes (i.e. after time ct 1 the two processes at small indices are closer than the rigidity scale 1/n). Its proof is postponed to Section 7.5.1. Let ρ sc (E) = 1 2π √ 4 − E 2 denote the semicircle density.
Similarly to Lemma 7.8 we also have that µ(t) and µ(t) are close thanks to the carefully designed relation between their driving Brownian motions. The proof of this lemma is postponed to Section 7.5.1.

Proof of Proposition 7.2
In this section we conclude the proof of Proposition 7.2 using the comparison processes defined in Section 7.1.1 and Section 7.1.2. We recall that p = 2 for simplicity. More precisely, we use that the processes λ z l (t),λ z l (t) andλ z l (t), λ (l) (t) and µ (l) (t), µ (l) (t) are close path-wise at time t 1 , as stated in Proposition 7.7, Lemma 7.8, and Lemma 7.9, respectively, choosing ω, ω as the minimum of the ones in the statements of this three results. In particular, by these results and Lemma 7.3 we readily conclude the following lemma, whose proof is postponed to the end of this section. Lemma 7.10. Let λ z l be the eigenvalues of H z l , and let µ (l) (t) be the solution of (7.24). Let ω, ω, ω h > 0 given as above, and define ν z l := ρ sc (0)/ρ z l (0), then for any small ω f > 0 such that ω h ω f there exists δ 0 , δ 1 such that ω h δ m ω, for m = 0, 1, and that (7.32) where t f = n −1+ω f , η l ∈ [n −1−δ0 , n −1+δ1 ], and the error term is given by n ω . (7.33) We remark that Ψ in (7.33) denotes a different error term compared with the error terms in (3.9) and (6.10).
By the definition of the processes µ (l) (t) in (7.24) it follows that µ (l) (t), µ (m) (t) are independent for l = m and so that Then, similarly to Lemma 7.10, we conclude that the correlation between these processes for different l's plays no role. Hence, already the usual coupling (see e.g. [15,18,45]) between the processes λ z l (t), µ (l) (t) defined in (7.14) and (7.24), respectively, would be sufficient to prove (7.35). On the other hand, the comparison processesλ z l (t) are anyway needed in order to remove the coefficients Λ ij (which are small with very high probability) from the interaction term in (7.14).
We conclude this section with the proof of Lemma 7.10.
Proof of Lemma 7.10. In the following, to simplify notations, we assume that the scaling factors ν z l are equal to one. First of all, we notice that the summation over the indices n ω < |i| ≤ n in (7.6) can be removed, using the eigenvalue rigidity (7.2) similarly to [22, Eq. (7.6)-(7.7)], at a price of an additional error term n 2(δ1+δ0)− ω : The error term is negligible by choosing δ 0 , δ 1 to be such that ω h δ m ω, for m = 0, 1. Then, from the GFT Lemma 7.3, and (7.8), using (7.36) again, this time for λ z l i l (ct f ), we have that We remark that the rigidity for λ z l i l (ct f ) is obtained by Theorem 3.1 exactly as in (7.2).
Next, by the same computations as in [22,Lemma 7.8] by writing the difference of l.h.s. and r.h.s. of (7.38) as a telescopic sum and then using the very high probability bound from Proposition 7.7 we get i l (ct f )) 2 + η 2 l + O(Ψ). (7.38) Similarly to (7.38), by Lemma 7.8 it also follows that (7.39) By (7.26) it readily follows that Moreover, by (7.31), similarly to (7.38), we conclude Combining (7.37)-(7.41), we conclude the proof of (7.32).
Finally, we conclude Section 7.1 by listing the scales needed in the entire Section 7 and explain the dependences among them. EJP 26 (2021), paper 24.

Relations among the scales in the proof of Proposition 7.2
Throughout Section 7 various scales are characterized by exponents of n, denoted by ω's, that we will also refer to scales for simplicity.
All the scales in the proof of Proposition 7.2 depend on the exponents ω d , ω h , ω f 1. We recall that ω d , ω h are the exponents such that Lemma 7.4 on eigenvector overlaps holds under the assumption |z l − z m |, |z l − z m |, |z l − z l | ≥ n −ω d , and |z l | ≤ 1 − n −ω h . The exponent ω f determines the time t f = n −1+ω f to run the DBM so that it reaches its local equilibrium and thus to prove the asymptotic independence of λ z l i (ct f ) and λ zm j (ct f ), with c > 0 defined in (7.5), for small indices i, j and l = m.
The most important scales in the proof of Proposition 7.2 are ω, ω, δ 0 , δ 1 , ω E . The scale ω E is determined in Lemma 7.4 and it controls the correlations among the driving martingales originating from the eigenvector overlaps in (7.11)-(7.13). The scale ω gives the n −1−ω precision of the coupling between various processes while ω determines the range of indices |i| ≤ n ω for which this coupling is effective. These scales are chosen much bigger than ω h and they are determined in Proposition 7.7, Lemma 7.8 and Lemma 7.9, that describe these couplings. Each of these results gives an upper bound on the scales ω, ω, at the end we will choose the smallest of them. Finally, δ 0 , δ 1 describe the scale of the range of the η's in Proposition 7.2. These two scales are determined in Lemma 7.10, given ω, ω from the previous step. Putting all these steps together, we constructed ω, ω, δ 0 , δ 1 claimed in Proposition 7.2 and hence also in Proposition 3.4.
These scales are related as for m = 0, 1. Along the proof of Proposition 7.2 four auxiliary scales, ω L , ω A , ω r , ω c , are also introduced. The scale ω L describes the range of interaction in the short range approximation processes x z l (t, α) (see (7.60) later), while ω A is the scale for which we can (partially) couple the driving martingales of the regularized processesλ z l (t) with the driving Brownian motions of Ginibre processes µ (l) (t). The scale ω c is a cut-off in the energy estimate in Lemma 7.13, see (7.68). Finally, ω r reduces the variance of the driving martingales by a factor (1 + n −ωr ) −1 to ensure the well-posedness of the processesλ z l (t), λ (l) (t), µ (l) , x z l (t, α) defined in (7.22), (7.27), (7.29), and (7.48), respectively. These scales are inserted in the chain (7.42) as follows Note that there are no relations required among ω A and ω, ω, δ m . In the following we present only the proof of Theorem 2.10, since the proof of Theorem 2.8 proceeds exactly in the same way. Universality of the joint distribution of the singular values of X − z 1 and X − z 2 follows by universality for the joint distribution of the eigenvalues of H z1 and H z2 , which is defined in (1.2), since the eigenvalues of H z l are exactly the singular values of X − z l taken with positive and negative sign. From now on we only consider the eigenvalues of H z l , with z l ∈ C such that | z l | ∼ 1, |z 1 − z 2 |, |z 1 − z 2 | ∼ 1, and |z l | ≤ 1 − for some small fixed > 0.

Universality and independence
For l ∈ [2], denote by {λ z l i } |i|≤n the eigenvalues of H z l and by {λ z l i (t)} |i|≤n their evolution under the DBM flow (7.14). Define {µ (l) i (t)} |i|≤n , for l ∈ [2], to be the solution of (7.24) with initial data {µ (l) i } |i|≤n , which are the eigenvalues of independent complex EJP 26 (2021), paper 24.

Proof of Proposition 7.7
Throughout this section we use the notation z = z l , with l ∈ [2], with z 1 , z 2 fixed as in Proposition Proposition 7.7.
Remark 7.11. In the remainder of this section we assume that |z| ≤ 1 − , with some positive > 0 instead of n −ω h , in order to make our presentation clearer. One may follow the -dependence throughout the proofs and find that all the estimates deteriorate with some fixed −1 power, say −100 . Thus, when |z| ≤ 1 − n −ω h is assumed, we get an additional factor n 100ω h but this does not play any role since ω h is the smallest exponent (e.g. see Proposition 7.7) in the analysis of the processes (7.14), (7.22).
The proof of Proposition 7.7 consists of several parts that we first sketch. The process λ z (t) differs from λ z (t) in three aspects: (i) the coefficients Λ z ij (t) in the SDE (7.14) for λ z (t) are removed; (ii) large values of the correlation of the driving martingales is cut off, and (iii) the martingale term is slightly reduced by a factor (1 + n ωr ) −1/2 . We deal with these differences in two steps. The substantial step is the first one, from Section 7.4.1 to Section 7.4.4, where we handle (i) by interpolation, using short range approximation and energy method. This is followed by a more technical second step in Section 7.4.5, where we handle (ii) and (iii) using a stopping time controlled by a well chosen Lyapunov function to show that the correlation typically remains below the cut-off level.
A similar analysis has been done in [17,Section 4] (which has been used in the singular value setup in [19,Eq. (3.13)]) but our more complicated setting requires major modifications. In particular, (7.14) has to be compared to [17, Eq. (4.1)] with dM i = 0, Z i = 0, and identifying Λ z ij with γ ij , using the notations therein. One major difference is that we now have a much weaker estimate |Λ z ij | ≤ n −ω E than the bound |γ ij | ≤ n −1+a , for some small fixed a > 0, used in [17]. We therefore need to introduce an additional cut-off function χ in the energy estimate in Section 7.4.4.

Interpolation process
In order to compare the processes λ z andλ z from (7.14) and (7.22) we start with defining an interpolation process, for any α ∈ [0, 1], as dt, x z i (0, α) = λ z i (0), |i| ≤ n. (7.48) We recall that ω f ω r ω E . We use the notation x z i (t, α) instead of z i (t, α) as in [17,Eq. (4.12)] to stress the dependence of x z i (t, α) on z ∈ C. The well-posedness of the process (7.48) is proven in Appendix A for any fixed α ∈ [0, 1]. In particular, the particles keep their order x z i (t, α) < x z i+1 (t, α). Additionally, by Lemma A.2 it follows that the differentiation with respect to α of the process x z (t, α) is well-defined.
Note that the process x z (t, α) does not fully interpolate betweenλ z (t) and λ z (t); it handles only the removal of theΛ ij term. Indeed, it holds x z (t, 0) =λ z (t) for any t ∈ [0, T ], but x z (t, 1) is not equal to λ z (t). Thus we will proceed in two steps as already explained: Step 1 The process x z (t, α) does not change much in α ∈ [0, 1] for particles close to zero (by Lemma 7.13 below), i.e. x z i (t, 1) − x z i (t, 0) is much smaller than the rigidity scale 1/n for small indices; Step 2 The process x z (t, 1) is very close to λ z (t) for all indices (see Lemma 7.14 below).
We start with the analysis of the interpolation process x z (t, α), then in Section 7.4.5 we state and prove Lemma 7.14.

Local law for the interpolation process
In order to analyse the interpolation process x z (t, α), we first need to establish a local law for the Stieltjes transform of the empirical particle density. This will be used for a rigidity estimate to identify the location of x i (t, α) with a precision n −1+ , for some small > 0, that is above the final target precision but it is needed as an a priori bound. Note that, unlike for λ z (t), for x z (t, α) there is no obvious matrix ensemble behind this process, so local law and rigidity have to be proven directly from its defining equation (7.48).
Define the Stieltjes transform of the empirical particle density by m n (w, t, α) = m z n (w, t, α) := 1 2n (7.49) and denote the Stieltjes transform of ρ z , the self-consistent density of states (scDOS) of H z , by m z (w). Moreover, we denote the Stieltjes transform of ρ z t , the free convolution of ρ z with the semicircular flow up to time t, by m z t (w). Using the definition of the quantiles γ z i (t) in (7.45), by Theorem 3.1 we have that with very high probability for any ξ > 0, uniformly in |z| ≤ 1 − , for some small fixed c 1 , c 2 , γ > 0. We recall that C ≤ −100 . The rigidity bound in the second line of (7.50) follows by a standard application of Helffer-Sjöstrand formula.
In Lemma 7.12 we prove that (7.50) holds true uniformly in 0 ≤ t ≤ t f . For its proof, similarly to [17,Section 4.5], we follow the analysis of [40, Section 3.2] using (7.50) as an input.
Proof. Differentiating (7.49), by (7.48) and Itô's formula, we get dt. (7.52) Note that by (7.20)-(7.21) it follows that The equation (7.52) is the analogue of [40,Eq. (3.20)] with some differences. First, the last two terms are new and need to be estimated, although the penultimate term in (7.52) already appeared in [17,Eq. (4.62)] replacingΛ ij by γ ij , using the notation therein. Second, the martingales in the second term in the r.h.s. of (7.52) are correlated. Hence, in order to apply the results in [40,Section 3.2] we prove that these additional terms are bounded as in [17,Eq. (4.64)]. Note that in [17,Eq. (4.64)] the corresponding term to the penultimate term in the r.h.s. of (7.52) is estimated using that γ ij ≤ n −1+a , for some small a > 0. In our case, however, the bound on |Λ| is much weaker and a crude estimate by absolute value is not affordable. We will use (7.53) and then the explicit form of Λ ij in (7.10)- (7.13), that enables us to perform the two summations and write this term as the trace of the product of two operators (see (7.57) later).
Since |Λ ii | ≤ n −ω E by its definition below (7.21), the last term in (7.52) is easily bounded by Next, we proceed with the estimate of the penultimate term in (7.52). Define the operators (7.55) where {w i (t)} |i|≤n are the orthonormal eigenvectors in the definition of Λ ij (t) in (7.10), and for any fixed w ∈ H the functions f, g : R → C are defined as g(x) := 1 x − w . (7.56) Then, using the definitions (7.55)-(7.56) and (7.53), we bound the last term in the first line of (7.52) as with very high probability uniformly in 0 ≤ t ≤ t f . Note that in the first equality of (7.57) we used thatΛ ij (t) = Λ ij (t) for any 0 ≤ t ≤ t f with very high probability by (7.53).
Finally, in order to conclude the proof, we estimate the martingale term in (7.52). For this purpose, using that E[db i db j | F b,t ] = (δ i,j − δ i,−j +Λ ij )/2 dt and proceeding similarly EJP 26 (2021), paper 24. (7.58) where the operator T is defined in (7.55), and in the penultimate inequality we used that Λ ij (t) = Λ ij (t) for any 0 ≤ t ≤ t f with very high probability. Combining (7.54), (7.57), and (7.58) we immediately conclude the proof of the first bound in (7.51) using the arguments of [40,Section 3.2]. The rigidity bound in the second line of (7.51) follows by a standard application of Helffer-Sjöstrand (see also below (7.50)).

Short range approximation
Since the main contribution to the dynamics of x z i (t, α) comes from the nearby particles, in this section we introduce a short range approximation process x z (t, α), which will very well approximate the original process x z (t, α) (see (7.63) below). The actual interpolation analysis comparing α = 0 and α = 1 will then be performed on the short range process x z (t, α) in Section 7.4.4.
Fix ω L > 0 so that ω f ω L ω E , and define the index set A := {(i, j) | |i − j| ≤ n ω L } ∪ {(i, j) | |i|, |j| > 5c 2 n}, (7.59) with c 2 > 0 defined in (7.51). We remark that in [17, Eq. (4.69)] the notation ω l is used instead of ω L ; we decided to change this notation in order to not create confusion with ω l defined in [22,Eq. (7.67)]. Then we define the short range approximation x z (t, α) of the process x z (t, α) by dt, The well-posedness of the process (7.60) follows by nearly identical computations as in the proof of Proposition A.1.
In order to check that the short range approximation x z (t, α) is close to the process x z (t, α), defined in (7.48), we start with a trivial bound on |x z i (t, α) − x z i (t, 0)| (see (7.61) below) to estimate the difference of particles far away from zero in (7.62), for which we do not have the rigidity bound in (7.51). Notice that by differentiating (7.48)   similarly to [17,Lemma 4.3]. By the rigidity estimate (7.51), the weak global estimate (7.61) to estimate the contribution of the far away particles for which we do not know rigidity, and the bound |Λ ij | ≤ n −ω E from (7.21) it follows that 1 2n Note that (7.63) implies that the second estimate in (7.51) holds with x z i replaced by x z i . In order to conclude the proof of Proposition 7.7 in the next section we differentiate in the process x z in α and study the deterministic (discrete) PDE we obtain from (7.60) after the α-derivation. Note that the α-derivative of x z is well defined by Lemma A.2.

Energy estimate
, for any |i| ≤ n. In the remainder of this section we may omit the z-dependence since the analysis is performed for a fixed z ∈ C such that |z| ≤ 1 − , for some small fixed > 0. By (7.60) it follows that v is the solution of the |i| ≤ n,  (7.65) and Before proceeding with the optimal estimate of the ∞ -norm of v in (7.67), we give the following crude bound that will be needed as an a priori estimate for the more precise result later. The bound (7.66) immediately follows by exactly the same computations as in [17,Lemma 4.7] using that |Λ ij | ≤ n −ω E .
The main technical result to prove Step 1 towards Proposition 7.7 is the following lemma. In particular, after integration in α, Lemma 7.13 proves that the processes x z (t, 1) and x z (t, 0) are closer than the rigidity scale 1/n. This lemma is based upon the finite speed of propagation mechanism for the dynamics (7.64) [31,Lemma 9.6]. Our proof follows [13, Lemma 6.2] that introduced a carefully chosen special cut-off function.
Proof. In order to bound |v i (t)| for small indices we will bound vχ ∞ for an appropriate cut-off vector χ supported at a few coordinates around zero. More precisely, we will use an energy estimate to control vχ 2 and then we use the trivial bound vχ ∞ ≤ vχ 2 .
This bound would be too crude without the cut-off.
We start estimating the terms in the second line of the r.h.s. of (7.73). The most critical term is the first one because of the ( x i − x j ) −2 singularity of B ij . We write this term as Then, using (7.70), v ∞ 1 by (7.66), |Λ ij | ≤ n −ω E by (7.21), the rigidity (7.72), and that ω L ω c , we bound the first sum by with very high probability. In the last inequality we trivially inserted ϕ to reproduce χ, using that ϕ((2c 2 ) −1 | x i |) = ϕ((2c 2 ) −1 | x j |) = 1 with very high probability uniformly in 0 ≤ t ≤ t f if |i|, |j| ≤ c 2 n by the rigidity estimate in (7.72).
Next, we consider the second term in the second line of the r.h.s. of (7.73). Using (7.70), and that |Λ ij | ≤ n −ω E , proceeding similarly to (7.75)-(7.76), we bound this term as 1 n Fluctuation around the circular laẘ Θ z1,z2 ij (t) in (7.18), (7.10),(7.13) and (7.20)-(7.21), respectively. In the following we may omit the z-dependence. Introduce the stopping times (7.82) for some large R > 0, and τ := τ 1 ∧ τ 2 ∧ t f . (7.83) Note that |λ i (t)| ≤ R with very high probability, since λ(t) are the eigenvalues of H z t , whose norm is typically bounded. Furthermore, by (7.61) and the fact that the process x(t, 0) stays bounded by [40,Section 3] it follows that |x i (t, α)| ≤ R for any t ∈ [0, t f ] and α ∈ [0, 1]. We remark that the analysis in [40, Section 3] is done for a process of the form (7.48), with α = 0, when it has i.i.d. driving Brownian motions, but the same results apply for our case as well since the correlation in (7.20) does not play any role (see (7.58)). This, together with Lemma 7.13 applied for z = z 1 , z = z 2 and z = z 1 , z = z 2 and z = z l , z = z l , implies that τ = t f with very high probability. In particular,Θ ij (t) = Θ ij (t) for any t ≤ τ , hence C(t) =C(t) (7.84) for any t ≤ τ .
In the remainder of the proof, omitting the time-and z-dependence, we use the notation x = x z (t, 1), λ = λ(t). Define then, as a consequence of (7.84), subtracting (7.14) and (7.48), it follows that for any 0 ≤ t ≤ τ , where > 0, (7.86) since |Λ ij (t)| = |Λ ij (t)| ≤ n −ω E , and hence, the last term in the r.h.s. of (7.89) is always non positive. This implies that for any ξ > 0. Then, since we conclude the upper bound in (7.80). Then noticing that u −i = −u i for i ∈ [n], we conclude the lower bound as well.

Path-wise coupling close to zero: proof of Lemmata 7.8-7.9
This section is the main technical result used in the proof of Lemmata 7.8-7.9. In Proposition 7.17 we will show that the points with small indices in the two processes become very close to each other on a certain time scale t f = n −1+ω f , for any small ω f > 0.
The main result of this section (Proposition 7.17) is stated for general deterministic initial data s(0) satisfying a certain regularity condition (see Definition 7.16 later) even if for its applications in the proof of Proposition 7.2 we only consider initial data which are eigenvalues of i.i.d. random matrices. The initial data r(0), without loss of generality, are assumed to be the singular values of a Ginibre matrix (see also below (7.91) for a more detailed explanation). For notational convenience we formulate the result for two general processes s and r and later we specialize them to our application.
Fix a small constant 0 < ω r 1, and define the processes s i (t), r i (t) to be the solution and dt, 1 ≤ |i| ≤ n, For convenience we also assume that {r ±i } n i=1 are the singular values of X, with X a Ginibre matrix. This is not a restriction; indeed, once a process with general initial data s is shown to be close to the reference process with Ginibre initial data, then processes with any two initial data will be close. On the correlation structure between the two families of i.i.d. Brownian motions and the initial data {s ±i } i∈[n] we make the following assumptions. Assumption 7.15. Fix ω K , ω Q > 0 such that ω K ω r ω Q 1, with ω r defined in (7.90)-(7.91), and define the n-dependent parameter K = K n = n ω K . Suppose that the families {b s ±i } n i=1 , {b r ±i } n i=1 in (7.90)-(7.91) are realised on a common probability space with a common filtration F t . Let denote the covariance of the increments conditioned on F t . The processes satisfy the following assumptions: for any i, j ∈ [n], q 1 , q 2 ∈ {s, r}. The quantities in (7.93) for negative i, j-indices are defined by symmetry.
are very strongly dependent in the sense that for any |i|, |j| ≤ K it holds |L ij (t)| ≤ n −ω Q (7.94) with very high probability for any fixed t ≥ 0. . Fix a very small ν > 0, and choose g, G such that n −1+ν ≤ g ≤ n −2ν , G ≤ n −ν .
A set of 2n-points s = {s i } |i|≤n on R is called (g, G)-regular if there exist constants c ν , C ν > 0 such that for any |E| ≤ G, η ∈ [g, 10], and if there is a constant C s large enough such that s ∞ ≤ n Cs . Moreover, c ν , C ν ∼ 1 if η ∈ [g, n −2ν ] and c ν ≥ n −100ν , C ν ≤ n 100ν if η ∈ (n −2ν , 10]. Proposition 7.17 (Path-wise coupling close to zero). Let the processes s(t) = {s ±i (t)} i∈[n] , r(t) = {r ±i (t)} i∈[n] be the solutions of (7.90) and (7.91), respectively, and assume that the driving martingales in (7.90)-(7.91) satisfy Assumption 7.15 for some ω K , ω Q > 0. Additionally, assume that s(0) is (g, G)-regular in the sense of Definition 7.16 and that r(0) are the singular values of a Ginibre matrix. Then for any small ω f , ν > 0 such that ν ω K ω f ω Q and that gn ν ≤ t f ≤ n −ν G 2 , there exist constants ω, ω > 0 such that ν ω ω ω f , and Proof. The proof of Proposition 7.17 is nearly identical to the proof of [22, Proposition 7.14], which itself follows the proof of fixed energy universality in [15,45], adapted to the block structure (3.1) in [18] (see also [14] for a different technique to prove universality, adapted to the block structure in [62]). We will not repeat the whole proof, just explain the modification. The only difference of Proposition 7.17 compared to [22,Proposition 7.14] is that here we allow the driving martingales in (7.90)-(7.91) to have a (small) correlation (compare Assumption 7.15 with a non zero Ξ q1,q2 ij to [22,Assumption 7.11]). The additional pre-factor (1 + n −ωr ) −1/2 does not play any role.
The correlation of the driving martingales in (7.90)-(7.91) causes a difference in the estimate of [22,Eq. (7.83)]. In particular, the bound on n ω A < |i| ≤ n, (7.98) for ω A = ω K (with ω K defined in Assumption 7.15), and for any D > 0 with very high probability, we bound the quadratic variation of (7.97) by Here we estimated the regime when |i| or |j| are larger than n ω A differently compared to [ Then, by (a)-(b) of Assumption 7.15, for |i|, |j| ≤ n ω A we have E[dC i (t, α) dC j (t, α) | F t ] = δ ij + α 2 Ξ s,s ij (t) + (1 − α) 2 Ξ r,r ij (t) 2n(1 + n −ωr ) dt + α(1 − α) 2n(1 + n −ωr ) E db s i db r j + db r i db s j F t , (7.100) and that E db s i db r j F t = E (db s i − db r i ) db r j F t + (δ ij + Ξ r,r ij (t)) dt (|L ii (t)| 1/2 + |Ξ r,r ij (t)| + δ ij ) dt, (7.101) where in the last step we used Kunita-Watanabe inequality for the quadratic variation (db s i − db r i ) db r j .

Proof of Lemma 7.8 and Lemma 7.9
The fact that the processesλ(t), λ(t) and µ(t), µ(t) satisfy the hypotheses of Proposition 7.17 for the choices ν = ω h , ω K = ω A , ω Q = ω E , and Ξ q1,q2 ij = Θ z1,z2 ij follows by Lemma 7.4 applied for z = z 1 , z = z 2 and z = z 1 , z = z 2 and z = z l , z = z l , and exactly the same computations as in [22,Section 7.5]. We remark that the processes µ (l) (t) do not have the additional coefficient (1 + n −ωr ) in the driving Brownian motions, but this does not play any role in the application of Proposition 7.17 since it causes an error term n −1−ωr that is much smaller then the bound n −1−ω in (7.31). Then, by Proposition 7.17, the results in Lemma 7.8 and Lemma 7.9 immediately follow.

Proof of Proposition 7.6
First of all we notice that λ(t) is γ-Hölder continuous for any γ ∈ (0, 1/2) by Weyl's inequality. Then the proof of Proposition 7.6 consists of two main steps, (i) proving that the eigenvalues λ(t) are a strong solution of (7.14) as long as there are no collisions, and (ii) proving that there are no collisions for almost all t ∈ [0, T ]. The proof that the eigenvalues λ(t) are a solution of (7.14) is deferred to Appendix B. The fact that there are no collisions for almost all t ∈ [0, T ] is ensured by [19, Lemma 6.2] following nearly the same computations as in [17,Theorem 5.2] (see also [19,Theorem 6.3] for its adaptation to the 2 × 2 block structure). The only difference in our case compared to the proof of [17,Theorem 5.2] is that the martingales dM i (t) (cf. [17, Eq. (5.4)]) are defined as dM i (t) := db z i (t) √ n , |i| ≤ n, (7.104) with {b z i } i∈[n] having non trivial covariance (7.16). This fact does not play any role in that proof, since the only information about dM = {dM i } |i|≤n used in [17,Theorem 5.2] is that it has bounded quadratic variation and that M (t) is γ-Hölder continuous for any γ ∈ (0, 1/2), which is clearly the case for dM defined in (7.104).

A The interpolation process is well defined
We recall that the eigenvectors of H z are of the form w z ±i = (u z i , ±v z i ) for any i ∈ [n], as a consequence of the symmetry of the spectrum of H z with respect to zero. Consider the matrix flow dX t = dB t √ n , X 0 = X, Hermitisation of X t − z, and {w z i (t)} |i|≤n its eigenvectors. We recall that the eigenvectors {w z i (t)} |i|≤n are almost surely well defined, since H z t does not have multiple eigenvalues almost surely by (7.17). We set the eigenvectors equal to zero where they are not EJP 26 (2021), paper 24. well defined. Recall the definitions of the coefficients Λ z ij (t),Λ z ij (t) from (7.10), (7.13) and (7.21), respectively. Set ∆ n := (x i ) |i|≤n ∈ R 2n 0 < x 1 < · · · < x n , x −i = −x i , ∀i ∈ [n] , and let C(R + , ∆ n ) be the space of continuous functions f : R + → ∆ n . Let ω E > 0 be the exponent in (7.21), and let ω r > 0 be such that ω r ω E . In this appendix we prove that for any α ∈ [0, 1] the system of SDEs dx z i (t, α) = db z i (t) n(1 + n −ωr ) + 1 2n dt, x z i (0, α) = x i (0), |i| ≤ n, dt, |i|, |j| ≤ n. (A. 3) The main result of this section is Proposition A.1 below. Its proof follows closely [17,Proposition 5.4], which is inspired by the proof of [5,Lemma 4.3.3]. We nevertheless present the proof of Proposition A.1 for completeness, explaining the differences compared with [17,Proposition 5.4] as a consequence of the correlation in (A.3).
We will mostly omit the z-dependence since the analysis of (A.2) is done for any fixed z ∈ C; in particular, we will use the notationΛ ij =Λ z ij . By (7.10), (7.13) and (7.21) it follows thatΛ ij (t) =Λ ji (t), and that |Λ ij (t)| ≤ n −ω E , for any t ≥ 0.
In the following we will often omit the time dependence. Note that the term in (A.10) In the remainder of the proof we show that the termΛ ii −Λ ij is negligible using the fact that |Λ ij | ≤ n −ω E , and so that this term can be absorbed in the negative term coming from the first sum in the r.h.s. of (A.10) for l = j. We now prove that A(x(t, α)) ≤ 0 if a > 0 is sufficiently large. Firstly, we write A(x(t, α)) as A(x(t, α)) = −2 l =i,j =i j =l (A.11) EJP 26 (2021), paper 24.
Then, using that the first sum in (A.11) is non-positive for (i − l)(i − j) > 0, and that c 1 ≤ θ ij ≤ c 2 , with c 1 = n −ωr , we bound A(x(t, α)) as follows A(x(t, α)) ≤ −2(1 + c 2 ) (A. 12) In (A.12) we used that since θ ij ≥ c 1 = n −ωr and |Λ ij | ≤ n −ω E , where ω r ω E . This shows that the correlations of the martingales {b i } |i|≤n is negligible. Note that the r.h.s. of (A.12) has exactly the same form as [17,Eq.  which is negative for a sufficiently large.
Additionally, by a similar argument as in [17,Proposition 5.5], we conclude the following lemma.

B Derivation of the DBM for singular values in the real case
Let X be an n × n real random matrix, and define Y z := X − z. Consider the matrix flow (A.1) defined on a probability space Ω equipped with a filtration (F t ) 0≤t≤T , and denote by H z t the Hermitisation of X t − z. We now derive (7.14), under the assumption that the eigenvalues are all distinct. This derivation is easily made complete by the argument in the proof Proposition 7.6 in Section 7.6.