Quenched universality for deformed Wigner matrices

Following E. Wigner’s original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H+xA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H+xA$$\end{document} with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.

The second histogram shows the empirical normalised bulk eigenvalue gaps of a single complex Wigner matrix of size 5000×5000.Both distributions asymptotically approach the Gaudin-Mehta distribution p2 drawn as solid lines, see Section . .The basic guiding principle for establishing quenched universality of H x is to show that near a fixed energy E the eigenvalues of H x and H x ′ are essentially uncorrelated whenever x and x ′ are not too close.This provides the sufficient (asymptotic) independence along the sampling in the space of x.Following a similar idea in [ ] for a different setup, the independence of eigenvalues is proven by running the Dyson Brownian motion (DBM) for the matrix H.The corresponding stochastic differential equations for the eigenvalues of H x and H x ′ have almost independent stochastic differentials if the corresponding eigenvectors are asymptotically orthogonal.Therefore, independence of eigenvalues can be achieved by running the DBM already after a short time, provided we can understand eigenvector overlaps.The small Gaussian component added along the DBM flow can later be removed by fairly standard perturbation argument (Green function comparison theorem).
Thus the main task is to show that eigenvectors of H x become asymptotically orthogonal for different, sufficiently distant values of x.This orthogonality can be triggered by two quite different mechanisms that we now explain.
The first mechanism is present when A is not too close to a diagonal matrix, in other words if Å := A − A is nontrivial in the sense that Å2 ≥ c with some N -independent constant c > 0.Here A := 1 N Tr A denotes the normalized trace.In this case the entire eigenbasis of H x is rotated, i.e. it becomes essentially orthogonal to that of H x ′ whenever x and x ′ are not too close.As a consequence, the entire spectra of H x and H x ′ are essentially uncorrelated.To establish this effect of eigenbasis rotation, we use a multi-resolvent local law for the resolvents of H x and H x ′ ; this method currently requires |x − x ′ | ≥ N −ǫ for typical choices of x, x ′ .To ensure this, we assume that x = N −a χ where χ is an N -independent real random variable with some regularity and a ∈ [0, ǫ].
The second mechanism is the most transparent when A = I and x = N −a χ where χ is uniformly distributed on some small fixed interval; we call this mechanism the sampling in the spectrum.In this case the eigenbasis of H x actually does not depend on x.However, the eigenvectors corresponding to eigenvalues close to a fixed energy are algebraically orthogonal for sufficiently distant x, x ′ .We also prove that distant eigenvalue gaps of H, and hence the local spectral data of H x , H x ′ are essentially uncorrelated.
By the rigidity property of the eigenvalues, already a small change in x triggers this effect, so it works in the entire range of scales a ∈ [0, 1−ǫ].Moreover, the proof can easily be extended to more complicated random matrix ensembles well beyond the Wigner case.No multi-resolvent local law is needed in the proof.
Private communication via Stephen Shenker and Sourav Chatterjee in June .
A combination of these two mechanisms can be used in the situation when A = I, but A is still close to A times the identity in the sense that | A | ≥ C Å2 1/2 for some large C.This extension complements the main condition Å2 ≥ c needed in the first mechanism thus proving the result unconditionally for any A.

Notations and conventions.
We introduce some notations we use throughout the paper.For integers k ∈ N we use the notation [k] := {1, . . ., k}.For positive quantities f, g we write f g and f ∼ g if f ≤ Cg or cg ≤ f ≤ Cg, respectively, for some constants c, C > 0 which depend only on the model parameters appearing in our base Assumptions -.For any two positive real numbers ω * , ω * ∈ R+ by ω * ≪ ω * we denote that ω * ≤ cω * for some small constant 0 < c < 1/100.We denote vectors by bold-faced lower case Roman letters x, y ∈ C k , for some k ∈ N. Vector and matrix norms, x and A , indicate the usual Euclidean norm and the corresponding induced matrix norm.For vectors x, y ∈ C k we define x, y := i xiyi and for any N × N matrix A we use the notation A := N −1 Tr A to denote the normalized trace of A. We will use the concept of "with very high probability" meaning that for any fixed D > 0 the probability of an N -dependent event is bigger than 1 − N −D if N ≥ N0(D).Moreover, we use the convention that ξ > 0 denotes an arbitrary small constant which is independent of N .
Acknowledgement.The authors are indebted to Sourav Chatterjee for forwarding the very inspiring question that Stephen Shenker originally addressed to him which initiated the current paper.They are also grateful that the authors of [ ] kindly shared their preliminary numerical results in June .
Data availability.All data generated or analysed during this study are included in this manuscript. .
The Wigner-Dyson-Mehta conjecture for the bulk of Wigner matrices H asserts that for any i ∈ [ǫN, (1 − ǫ)N ] the distribution of the rescaled eigenvalue gap converges to a universal distribution with density p1 (for real symmetric Wigner matrices) or p2 (for complex Hermitian Wigner matrices) which can be computed explicitly from the integrable Gaussian GOE/GUE ensembles, see Section .later.This WDM conjecture was resolved in [ ] while similar results with a small averaging in the index i were proven earlier [ , ].
As a corollary to our main result Theorem .below on the monoparametric ensemble we prove a considerable strengthening of ( .), namely that with high probability the sampling of eigenvalues within a single fixed Wigner matrix generates WDM universality.
Corollary .(to Theorem . ).Let H be a Wigner matrix and I ⊂ (−2 + ǫ, 2 − ǫ) be an interval in the bulk of H of length |I| ≥ N −1+ξ for some ǫ, ξ > 0. Then there exist small κ, α > 0 and an event ΩI in the probability space PH of H with PH(Ω c I ) ≤ N −κ , such that for all H ∈ ΩI it holds that where the implicit constant in ( . ) and κ, α depend on ǫ, ξ.
Our main results are on the quenched (bulk) universality of monoparametric random matrices for general deterministic Hermitian matrices A of the same symmetry class as H, and independent scalar random variables x, just using the randomness of x for any fixed Wigner matrix H from a high probability set.For A = I the monoparametric universality of H x implies the spectral sampling universality as stated in Corollary ., see Section . .Our results extend beyond Wigner matrices, we also allow for arbitrary additive deformations (certain results even extend to Wigner matrices with correlated entries), and cover general sufficiently regularly distributed scalar random variables x.
Assumption (Deformed Wigner matrix).We consider deformed Wigner matrices of the form H = W + B, where W is a Wigner matrix as in ( .), and B = B * is an arbitrary deterministic matrix of bounded norm, i.e.B ≤ C0 for some N -independent constant C0.
Assumption .Assume that x = N −a χ with a ∈ [0, 1), where χ is an N -independent compactly supported real random variable such that for any small b1 > 0 there exists b2 > 0 such that for any interval To state the result, we now introduce the self-consistent density of states of and, in particular, the scDos of H by ρ := ρ 0 .It is well known that ρ x is a probability density which is compactly supported and real analytic inside its support [ , Proposition . ].For the special case E H = B = 0 the scDos of H is the standard Wigner semicircle law, i.e. ρ = ρsc.
We say that an energy E ∈ R lies in the bulk of the spectrum of H x if ρ x (E) ≥ c for some N -independent constant c > 0. For E in the bulk, the solution M x (z) can be continuously extended to the real line, M x (E) := lim η→0 + M x (E + iη), and M x (E + iη) for E in the bulk is uniformly bounded, cf.[ , Proposition . ]. Finally, we define the classical eigenvalue locations to be the quantiles of ρ x , i.e. we define γ x i by For clarity, in this section we only present single-gap versions of both mechanisms explained in the introduction that yield quenched universality.Subsequently we will present the multi-gap analogues in Section .
. .Monoparametric universality via eigenbasis rotation.The main universality result for the first mechanism (eigenbasis rotation) is the following quenched fixed-index universality result for the monoparametric ensemble.We denote the probability measure and expectation of x by Px, Ex in order to differentiate it from the probability measure PH of H.
Theorem .(Quenched universality for monoparametric ensemble).Let H be a deformed Wigner matrix satisfying Assumption , and let x = N −a χ be a scalar real random variable satisfying Assumption with a ∈ [0, a0], where a0 is a small universal constant .Fix any c0, c1 > 0 small constants and assume that Å2 ≥ c0, with Å : ) This restriction apparently excludes the case when A is complex Hermitian but H is real symmetric.With a slight modification of our proof (similar to the modification required in [ , Section ] compared to [ , Section ]), however, we can handle this case as well, but for brevity we refrain from presenting it.
Following the explicit constants along the proof, one may choose a0 = 1/100 To specify the c1-dependence, we often speak of c1-bulk index. 0.5 Here we show the histogram of a (rescaled) single eigenvalue gap λ N/2+1 − λ N/2 in the middle of the spectrum for N ∈ {2, 100, 1000} and for random matrices sampled from either the GUE or the monoparametric ensemble.For the GUE ensemble the histogram has been generated by sampling 2000 independent GUE matrices H.For the monoparametric ensemble only two GUE matrices H, A have been drawn at random, and the histogram has been generated by sampling 2000 standard Gaussian random variables x and considering the gaps of H x = H + xA.The solid black line represents the theoretical limit p2(s) which matches the empirical distribution very closely already for N = 2.In Appendix B we present numerical evidence for the speed of convergence, inspired by the observation on the slow convergence of the spectral form factor made in [ ].
Then there exist small α, κ > 0 and an event Ωi = Ωi,A with PH (Ω c i ) ≤ N −κ , so that for all H ∈ Ωi the statistics of the i-th rescaled gap of the eigenvalues for any smooth, compactly supported function f where the implicit constant in ( . ) depends on a0, c0, c1 and the diameter of supp f , and κ, α depend on a0.
Remark . .We mention a few simple observations about Theorem . .(i) By the regularity of f , ρ x and by rigidity of the bulk eigenvalues (see ( . ) later) we may replace the random scaling factor ρ x (λ x i ) with ρ x (γ x i ) at negligible error.(ii) For E H = 0 and a > 0 the condition ( . ) can simply be replaced by i ∈ [N ǫ ′ , N (1 − ǫ ′ )] for some ǫ ′ > 0 and the argument of f in ( . ) simplifies to f N ρsc(λ This mathematically rigorously answers to a question of Gharibyan, Pattison, Shenker and Wells [ ].While their original question referred to a standard Gaussian x, which is not compactly supported, a simple cut-off argument extends our proof to this case as well. The condition on Å2 is satisfied since A 2 = 1 + o(1) and A = o(1) with very high probability.Moreover, the scDos ρ x is very close to a rescaled semicircle law with radius 2 √ 1 + x 2 with very high probability in the joint probability space of H and A, hence the condition ( . ) holds for all i ∈ [N ǫ ′ , N (1 − ǫ ′ )] for some ǫ ′ > 0.
. .Monoparametric universality via spectral sampling.The main universality result for the second mechanism (spectral sampling) is the following quenched fixed-energy universality result for the monoparametric ensemble.We define i0 = i0(x, E) as the index such that γ x i 0 is the quantile of ρ x closest to E, i.e.
with ⌈•⌉ denoting rounding to the next largest integer.
For the special case A = I (formulated as Case ) in Theorem .below) we obtain quenched sampling universality for a much broader class of Hermitian random matrices H with slow correlation decay defined in [ ].In the second situation, Case ) in Theorem .below we consider deformed Wigner matrices H and general A with a condition complementary to the condition Å2 ≥ c0 of Theorem . .

Theorem . (Quenched monoparametric universality via spectral sampling mechanism).
There is a small universal constant a0 and for any small c1 > 0 there exists a c0 > 0 such that the following hold.Let x = N −a χ be a scalar random variable satisfying Assumption , and let H, A, a be such that either Case ) H is a correlated random matrix , A = I, and a ∈ [0, 1 − a1] for an arbitrary small a1, Case ) H is a deformed Wigner matrix (cf.Ass. ), c0| A | ≥ Å2 1/2 , | A | ≥ c0, and a ∈ [0, a0], and fix an energy E with ρ x (E) ≥ c1 > 0 for Px-almost all x.Then there exist small α, κ > 0 and an event ΩE = ΩE,A with PH (Ω c E ) ≤ N −κ such that for all H ∈ ΩE the matrix The exponents κ, α depend on a0, a1 while the implicit constant in ( . ) depends on a0, a1, c0, c1 and the diameter of supp f .
We remark that the condition c0| A | ≥ Å2 1/2 in Case ) is not really necessary for ( . ) to hold.Indeed, if Å2 ≥ c with any small positive constant, then we are back to the setup of Theorem .where the eigenbasis rotation mechanism is effective.One can easily see that the proof of ( . ) implies ( . ) in this case (see Remark .below).However, we kept the condition c0| A | ≥ Å2 1/2 with a sufficiently small c0 in the formulation of Theorem .since it is necessary for the spectral sampling mechanism to be effective which is the mechanism represented in Theorem . .Note that as long as Å = 0, i.e.Case ) is not applicable, the eigenbasis of H x changes with x and we have to rely on the multi-resolvent local law method.However, lacking an effective lower bound on Å2 , the effective asymptotic orthogonality still comes from the spectral sampling effect of A , the nontrivial tracial part of A. So along the proof of Case ), technically we follow the eigenbasis rotation mechanism, but morally the effect is similar to the spectral sampling mechanism as it still comes from a shift in the spectrum triggered by x A , the leading part of xA in H x = H + xA.Finally, a simple perturbation argument shows that x Å has no sizeable effect on the sampling, but its presence hinders the technically simpler orthogonality proof used in Case ).
. .Gaudin-Mehta distribution.For completeness we close this section by providing explicit formulas for the universal Gaudin-Mehta gap distributions p1, p2 which can either be defined as the Fredholm determinant of the sine kernel [ ] or via the solution to the Painlevé V differential equation [ ].Given the solution σ to the non-linear differential equation we have [ ] Remarkably, the Wigner surmise obtained by E. Wigner from explicitly computing the gap distribution for 2 × 2 matrices, is very close to the large N limit p2(s), more precisely T .
In Section .we prove Theorem .while in Section .we present the proof of Theorem .which structurally is analogous to the argument in Section . .For notational simplicity we introduce the discrete difference operator δ, i.e. for a tuple λ we set in order to express eigenvalue differences (gaps) more compactly.We also introduce the notation f gap for the expectation of test functions f with respect to the density p β from ( .), i.e.
. .Universality via eigenbasis rotation mechanism: Proof of Theorem . .To prove Theorem .we will show that the gaps λ i for sufficiently large |x1 − x2| are asymptotically independent in the sense of the following proposition whose proof will be presented in Section .In the following we will often denote the covariance of two random variables X, Y in the H-space by Proposition . .Under the conditions of Theorem .there exists a sufficiently small c * > 0 (depending on c0, c1) and for any small ζ1 there exists ζ2 > 0 such that the following holds.Pick real numbers x1, x2 with for any P1, P2 : R → R bounded smooth test functions, and where the implicit constant in O(•) may depend on c0, c1 at most polynomially.
Remark . .We stated the asymptotic independence of a single gap in Proposition .and only for two x1, x2 for notation simplicity.Exactly the same proof as in Section directly gives the result in ( . ) for test functions Pr : R p → R of several gaps, for some fixed p ∈ N. Additionally, by the same proof we can also conclude the asymptotic independence of several gaps for several x1, . . ., xq.For the same reason we also state Proposition .and Proposition .below only for two x1, x2 and test functions Pr : R → R.
Proof of Theorem . .We will first prove that without loss of generality we may assume that the linear size of the support of x is bounded by c * , where c * is from Proposition . .This initial simplification will then allow us to use perturbation in x when proving Proposition . .Suppose that Theorem . is already proved for random variables with such a small support with an error term N −α on sets of probability at least 1 − N −2κ and we are now given a random variable x with a larger support of size bounded by some constant C. Then we define the random variables where Ji's, for i = 1, 2, . . ., C/c * , are disjoint intervals of size c * such that supp(x) = i Ji.For any test function f we can then write , where we used Theorem .for the random variables xi in the last step and a union bound.
From now on we assume that the linear size of the support of x is bounded by c * .With ν(dx) denoting the measure of x we have for some sufficiently small ǫ2 so that we can apply Proposition .with ζ1 = ǫ2.In ( . ) we used that the regime |x1 − x2| ≤ N −ǫ 2 can be removed at the price of a negligible error by the regularity assumption on the distribution of x = N −a χ, with χ satisfying Assumption .For the cross-term in ( . ) we used that by gap universality for the deformed Wigner matrix H x with a fixed x (see e.g.[ , Corollary . ]) it follows that for some small fixed ζ3 > 0 depending only on the model parameters and on the constants a0, c1.
Applying Proposition .to the first term in ( . ) with Pr(t) := f (ρ xr (γ xr j )t) noting that ρ xr is uniformly bounded, so that for By using ( . ) and ( . ) in ( . ) it follows that From ( . ) and the Chebyshev inequality we obtain events Ω j,f on which ( . ) holds with probability PH (Ω c j,f ) ≤ N −κ for some suitably chosen κ, α > 0.
. .Universality via spectral sampling mechanism: Proof of Theorem . .The mechanism behind the proof of Theorem . is quite different compared to Theorem . .In particular, in order to prove Theorem .we will first show that under the assumptions of Theorem .the gaps δλ x i , δλ x j are asymptotically independent for any fixed x in the probability space of H as long as |i − j| is sufficiently large.This independence property for the A = I case has already been used as a heuristics without proof, e.g. in [ , ] (a related result for not too distant gaps for local log-gases can be deduced from the De Giorgi-Nash-Moser Hölder regularity estimate, see [ , [Section .]).More precisely, we have the following proposition: Proposition . .Under the conditions of Theorem .there exists a sufficiently small c * > 0 (depending on c0, c1) and for any sufficiently small ζ1 > 0 there exists ζ2 > 0 such that the following hold.Pick indices j1, j2 and real numbers x1, x2 such that the corresponding quantiles γ xr jr are in the c1-bulk of the spectrum of H xr , i.e. ρ xr (γ xr jr ) ≥ c1, for r = 1, 2. In the two different cases listed in Theorem .we additionally assume the following: Then in both cases it holds that for Pr : R → R bounded, smooth test functions.The implicit constant in O(•) may depend on c0, c1 at most polynomially.
Proof of Theorem . .We present the proof only for the more involved Case ), the Case ) is much easier and omitted.Similarly to ( . ) in the proof of Theorem .it is enough to consider the case when the linear size of the support of x is bounded by some c > 0 (determined later) and prove that for some small α, κ > 0.
Note that under the assumptions of Case ) in Theorem .for any x1, x2 in the support of the random variable ) with some θ, Θ (depending on c0, c1) as long as |x1 − x2| ≫ N −1 .The bound in ( . ) is a direct consequence of the following Lemma .(assuming that c ≤ c * ) whose proof is postponed to Appendix A.
where to go from the first to the second line we used that γ ) by the definition of i0(x, E) and that we are in the bulk.In the last inequality we used ( . ) and that its error terms are negligible by c0| A | ≥ Å2 1/2 and |x1 − x2| ≤ c assuming that c ≤ c0/10.Then by a similar chain of inequalities, and using ( . ) once more, we get the matching upper bound in ( .).
To prove ( .), we use the counterpart of ( . ) and that we can neglect the regime |x1 − x2| ≤ N −ǫ 2 for some sufficiently small ǫ2 > 0 so that we can apply Proposition .with ζ1 = ǫ2.We remove this regime to ensure that on its complement |i0(x1, E) − i0(x2, E)| is sufficiently large by ( .).More precisely, for any x1, x2 with ).Assuming c ≤ c * θ, we can apply Case ) of Proposition .by choosing j1 = i0(x1, E), j2 = i0(x2, E) and with exponent ζ2 to factorise the expectation in the equivalent of ( .).Using again the gap universality ( .), similarly to ( . ) we conclude ( . ) choosing α, κ > 0 appropriately.Remark . .The proof of ( . ) in the case Å2 ≥ c is analogous to the proof of Theorem .above.We note that Proposition .allows to also conclude the asymptotic independence of δλ i 0 (x 1 ,E) and δλ . .Proof of Corollary . .Picking E = 0 and the test function f (u) = 1(0 ≤ u ≤ y) in Case ) of Theorem ., and choosing the random variable x such that −x has density proportional to ρ|I , with ρ = ρsc, it follows that with very high probability in the space of H it holds that While f does not literally satisfy the regularity condition, one can easily extend the validity of ( . ) to interval characteristic functions f by a standard approximation argument.
where in the first and third step we used rigidity (see e.g.[ , Lemma . , Theorem . ] or [ , Section ]), i.e. that for any small ξ > 0 we have ) for all γi in the bulk, with very high probability.
In this section we first present the proof of Proposition . in details and later in Section .we explain the very minor changes that are required to prove Proposition . . . .Proof of Proposition . .By standard Green function comparison (GFT) argument (see e.g.[ , Section ]) it is enough to prove Proposition .only for matrices with a small Gaussian component.More precisely, consider the DBM flow with Bt being a real symmetric or complex Hermitian standard Brownian motion (see e.g.[ , Section . ] for the precise definition) independent of the initial condition H0 = H # , where H # is a deformed Wigner matrix specified later.Throughout this section we fix T > 0 and analyse the DBM for times 0 ≤ t ≤ T .We denote the ordered collection of eigenvalues of Ht + xA by λ . The main result of this section is the asymptotic independence of We note that in this entire section we do not use the randomness of x, in the statement of Propositions .and .x1, x2 are fixed parameters.Hence all probabilistic statements, such as covariances etc., are understood in the probability space of the random matrices and the driving Brownian motions in ( .).
Proposition . .Let H # be a deformed Wigner matrix satisfying Assumption , let Ht be the solution of ( .), and let A be a deterministic matrix such that Å2 ≥ c0 and A 1. Then there exists a small c * > 0 (depending on c0, c1) and for any small ζ1, ω1 > 0 there exists some ζ2 > 0 such that the following hold.) with t1 = N −1+ω 1 for any P, Q : R → R bounded smooth test functions.
Using Proposition .as an input we readily conclude Proposition . .
Proof of Proposition . .Let H be the deformed Wigner matrix from Proposition ., and consider the Ornstein-Uhlenbeck flow with Bt being a real symmetric or complex Hermitian standard Brownian motion independent of H0.
Let H # t 1 , with t1 from Proposition ., be such that with U a GOE/GUE matrix independent of H # t 1 and c = c(t1) = 1 + O(t1) is an appropriate constant very close to one.Then by ( . ) it follows that with Hct 1 being the solution of ( . ) with initial condition H0 = H # t 1 .Then, by a standard GFT argument [ , Section ], we have that ( . ) Finally, by ( . ) together with ( . ) and Proposition .applied to H # := H # t 1 we conclude the proof of Proposition . .
. .Proof of Proposition . .This proof is an adaptation of the proof of [ , Proposition .] (which itself is based upon [ ]) with two minor differences.First, the DBM in this paper is for eigenvalues (see ( . ) below) while in [ , Eq. ( .)] it was for singular values.Second, in [ , Section ] it was sufficient to consider singular values close to zero hence the base points j1 and j2 were fixed to be 0; here they are arbitrary.Both changes are simple to incorporate, so we present only the backbone of the proof that shows the differences, skipping certain steps that remain unaffected.The flow ( . ) induces the following flow on the eigenvalues of Ht + x l A: with r ∈ [2] and β = 1, 2 in the real and complex case, respectively.Here (omitting the time dependence) we used the notation , for fixed r, consists of i.i.d standard real Brownian motions.However, the families b x 1 , b x 2 are not independent for different r's, in fact their joint distribution is not necessarily Gaussian.The quadratic covariation of these two processes is given by We remark that in ( . ) we used a different notation for the quadratic covariation compared to [ , Section . .].
. . .Definition of the comparison processes for λ xr .To make the notation cleaner we only consider the real case (β = 1).
To prove the asymptotic independence of the processes λ x 1 , λ x 2 , realized on the probability space Ω b , we will compare them with two completely independent processes µ (r) (t) = {µ (r) i being the eigenvalues of two independent GOE matrices H (r) , and β (r) = {β (r) i (t)} N i=1 being independent vectors of standard i.i.d.Brownian motions.
We now define two intermediate processes λ (r) (t), µ (r) (t) so that for t ≫ N −1 the particles λ (r) i (t) will be close to λ xr i (t) and µ xr i (t), respectively, for indices i close to jr, with very high probability (see Lemmas .-. below).Additionally, the processes λ (r) (t), µ (r) (t), which will be realized on two different probability spaces, will have the same joint distribution: ( . ) Fix any small ωA > 0 (later ωA will be chosen smaller than ωE from ( .)) and define the process λ (r) (t) to be the unique strong solution of with initial data λ (r) (0) being the eigenvalues of independent GOE matrices, which are also independent of H # in ( .).
Here the Brownian motions for indices close to jr are exactly the ones in ( .).For indices away from jr we define the driving Brownian motions to be an independent family ) of standard real i.i.d.Brownian motions which are also independent of b in .The Brownian motions b out are defined on the same probability space of b in , which we will still denote by Ω b , with a slight abuse of notation.
For any i, j ∈ [4N ω A + 2] we use the notation with r, m ∈ [2] and i, j ∈ [2N Similarly, on the probability space Ω β we define the comparison process µ (r) (t) to be the solution of with initial data µ (r) (0) being the eigenvalues of independent GOE matrices defined on the probability space Ω β , which are also independent of H (r) .We now construct the driving Brownian motions in ( . ) so that ( . ) is satisfied.
For indices away from jr the standard real Brownian motions are i.i.d. and they are independent of β (1) , β (2) in ( .).For indices |i − jr| ≤ N ω A the collections will be constructed from the independent families as follows.
Since the original process λ xr (t) and the comparison processes µ (r) (t) are realized on two different probability spaces, we construct a matrix valued process C # (t) and a vector-valued Brownian motion β in on the probability space Ω β such that (C # (t), β in (t)) have the same joint distribution as (C(t), θ(t)) with C, θ from ( .).This β in is the driving Brownian motion of the µ (r) (t) process in ( .).Define the process on the probability space Ω β .By construction we see that the processes b in and ζ in have the same distribution, and that the two collections b out and ζ out are independent of b in , β in and among each other.Hence we conclude that Finally, by the definitions in ( .), ( . ) and by ( .), we conclude that the processes λ (r) (t), µ (r) (t) have the same joint distribution (see ( .)), since their initial conditions and their driving processes ( . ) agree in distribution. . . .Proof of the asymptotic independence of the eigenvalues.In this section we use that the processes λ xr (t), λ (r) (t) and µ (r) (t), µ (r) (t) are close pathwise at time t1 = N −1+ω 1 as stated below in Lemma .and Lemma ., respectively, to conclude the proof of Proposition . .The proof of these lemmas is completely analogous to the proof in [ , Lemmas .
-. ], [ , Eq. ( .), Theorem .], hence we will only explain the very minor differences required in this paper.First, we compare the processes λ xr (t), λ (r) (t), in particular this lemma shows that for i far away from j1, j2 the Brownian motions b x 1 i , b x 2 i can be replaced by the independent Brownian motions from b out at a negligible error.Lemma . .Let λ xr (t), λ (r) (t), with r ∈ [2], be the processes defined in ( . ) and ( .), respectively.For any small ω1 > 0 there exists ω > 0, with ω ≪ ω1, such that it holds for any jr in the c1-bulk, with very high probability on the probability space Ω b , where t1 := N −1+ω 1 .Here by γj r we denoted the jr-quantile of the semicircular law.
Second, we compare the processes µ (r) (t), µ (r) (t), i.e. we control the error made by replacing the weakly correlated Brownian motions ζ in by the independent Brownian motions β in .
The key ingredient for the proof of Lemma . is the following fundamental bound on the eigenvector overlaps in ( . ) proven in Section , which ensures that the correlation Θ xr ,xm ij in ( . ) is small.
Using Lemmas .-. as an input we conclude Proposition . .
Before concluding this section with the proof of Lemmas .-. , in Proposition .below we state the main technical result used in their proofs.The proofs of these lemmas rely on extending the homogenisation analysis of [ , Theorem . ] to two DBM processes with weakly coupled driving Brownian motions.We used a very similar idea in [ , Section .] for DBM processes for singular values.We now first present the general version of this idea before applying it to prove Lemmas .-. .In Proposition .below we compare the evolution of two DBMs whose driving Brownian motions are nearly the same for indices close to a fixed index i0 and are independent for indices away from i0.Proposition . is the counterpart of [ , Proposition . ], where a similar analysis is performed for DBMs describing the evolution of particles satisfying slightly different DBMs.
Define the processes si(t), ri(t) to be the solution of . ) and ) with initial conditions si(0) = si being the eigenvalues of a deformed Wigner matrix H satisfying Assumption , and ri(0) = ri being the eigenvalues of a GOE matrix.Here we used the same notations of [ , Eqs. ( .)-( .)] to make the comparison with [ ] easier.For simplicity in ( .)-( .) we consider the DBMs only in the real case (the complex case is completely analogous).
Remark . .In [ , Eqs. ( .)] we assumed that the initial condition si(0) = si were general points satisfying [ , Definition . ], and not necessary the singular values of a matrix.Here we choose si(0) = si to be the eigenvalues of a deformed Wigner matrix to make the presentation shorter and simpler, however Proposition .clearly holds also for collections of particles satisfying similar assumptions to [ , Definition . ].
We now formulate the assumptions on the driving Brownian motions in ( .)-( .).Set an N -dependent parameter K = KN := N ω K , for some small fixed ωK > 0.

Assumption . Suppose that the families {b
) and ( . ) are realised on a common probability space.Let denote their quadratic covariation (in [ , Eqs. ( . )] we used a different notation to denote the covariation).Fix an index i0 in the bulk of H, and let the processes satisfy the following assumptions: with very high probability for any fixed t ≥ 0.
Let ρ denote the self-consistent density of H, and recall that ρsc denotes the semicircular density.By ρt, ρsc,t we denote the evolution of ρ and ρsc, respectively, along the semicircular flow (see e.g.[ , Eq. ( .)]) and let γi(t), γi(t) denote the quantiles of ρt and ρsc,t.

Proposition . . Let the processes s
be the solutions of ( . ) and ( .), and assume that the driving Brownian motions in ( .)-( .) satisfy Assumption .Let i0 be the index fixed in Assumption .Then for any small ω1, ω ℓ > 0 such that ω1 ≪ ω ℓ ≪ ωK ≪ ωQ there exist ω, ω > 0 with ω ≪ ω ≪ ω1, and such that it holds for any |i| ≤ N ω , with very high probability, where t1 := n −1+ω 1 and pt(x, y) is the fundamental solution (heat kernel) of the parabolic equation ) Proof.The proof of ( . ) is nearly identical to that of [ , Theorem . ] up to a straightforward modification owing to the fact that the driving Brownian motions in ( .)-( .) are not exactly identical but they are very strongly correlated, see ( .).A similar modification to handle this strong correlation was explained in details in a closely related context in [ , Proof of Proposition . in Section .], with the difference that in [ ] singular values were considered instead of eigenvalues hence the corresponding DBMs are slightly different.Furthermore, Proposition . is stated in a simpler form than [ , Proposition .] since the initial conditions are already eigenvalues and not arbitrary points hence they automatically satisfy certain regularity assumptions.The precise changes due to this simplification are described in the technical Remark .below.
Remark . .There a few differences in the setup of Proposition . and [ , Proposition . ].These are caused by the fact that we now consider si(0) = si to be the eigenvalues of a deformed Wigner matrix H, instead of a collection of particles satisfying [ , Definition . ].In particular, ν in [ , Definition .] can be chosen equal to zero, then, since the eigenvalues of H are regular ([ , Eq. ( .)]) on an order one scale, we can choose g = N −1+ξ , for an arbitrary small ξ > 0, and G = 1 in [ , Definition . ]. Additionally, t f = N −1+ω f is replaced by t1 = N −1+ω 1 , and ρ fc,t f in is replaced by ρ.Finally, we remark that in [ , Proposition . ] for ω f we required that ωK ≪ ω f ≪ ωQ, instead in Proposition .we required that ω1 ≪ ωK ≪ ωQ.This discrepancy is caused by the fact that in the proof of [ , Proposition . ] we first needed to run the DBM for si(t) for an initial time t0 = N −1+ω 0 to regularise the particles si(0) = si, with ωK ≪ ω0 ≪ ωQ, then run both DBMs for an additional time N −1+ω 1 , with ω1 ≪ ωK ≪ ω0 ≪ ωQ (see below [ , Eq. ( .)]).Finally, in [ , Proposition . ] we have t f := t0 + t1 ∼ t0, hence the reader can think ω f = ω0.In the current case we do not need to run ( . ) for an initial time t0 since si(0) = si are already regular being the eigenvalues of a deformed Wigner matrix.
Analogously, we observe that the processes µ (r) (t), µ (r) (t) satisfy the assumptions of Proposition .with i0 = jr, i = 0, ρ = ρsc, ωK = ωA, ωQ = ωE due to ( . ) (in particular ( . ) is needed to check Assumption -(c)), and thus we obtain From now on we focus only on the precesses λ xr (t1), λ (r) (t1) and so on the proof of Lemma . .The proof to conclude Lemma . is completely analogous and so omitted.Combining ( . ) with another application of Proposition ., this time for i0 = jr and i = 1, we readily conclude that with very high probability, where we used rigidity a similar rigidity bound for λ i .Additionally, to go to the last line of ( . ) we used the following properties of the heat kernel pt 1 (x, y): ( . ) The bound in the second line of ( . ) follows by [ , Eq. ( . )].The second relation of ( . ) follows by [ , Eqs. ( . ), ( . )].The bound in ( . ) concludes the proof of Lemma . . . .Proof of Proposition . .We now turn to the proof of Proposition . .We first present Case ) which is structurally very similar to the proof of Proposition . .Afterwards we turn to Case ) which is easier but additionally requires to modify the flow ( . ) to account for the correlations among entries of H.
. . .Case ).Proceeding as in ( .)-( .) and using the notations and assumptions from Case ) of Proposition ., it is enough to prove with t1 = N −1+ω 1 , for some small ω1 > 0.Here λ(t) are the eigenvalues of Ht, which is the solution of ( . ) with initial condition H0 = H # t 1 , where H # t 1 is from in ( .).The proof of ( . ) follows by a DBM analysis very similar to the one in Section . .More precisely, all the processes λ xr (t), λ (r) (t), µ (r) (t), and µ (r) (t) are defined exactly in the same way; the only difference is that Proposition .has to be replaced by the following bound on the eigenvector overlap (its proof will be given at the end of Section ).
Proposition . .We are in the setup of Case ) of Proposition . .For any small c1 > 0 there exists a c0 > 0 and a c * depending on c0, c1 such that the following hold for any ζ1 > 0 sufficiently small.Assume For notational convenience we introduce the commonly used notion of stochastic domination.For some family of non-negative random variables X = X(N ) ≥ 0 and a deterministic control parameter ψ > 0 we write X ≺ ψ if for each ǫ > 0, D > 0 there exists some constant C such that , for r = 1, 2, be the orthonormal eigenbasis of the matrices H + xrA and fix indices i1, i2 in the bulk i.e. with ℑM xr (γ xr ir ) 1. Then it holds that whenever N −1/6 δ 1.
Proof of Proposition . .This proof is an adaptation of a similar argument from [ , Theorem . ], so here we only give a short explanation.From ( . ) obtain where W G1G2 := W G1G2 + G1 G1G2 + G1G2 G2.Thus we have Star in bracket M ( * ) indicates that the statement holds for both M and its adjoint M * Recall that it was proven in [ , Proposition .] that if | G1G2A | ≺ A θ for some constant θ ≤ η −1 uniformly in A, then also again uniformly in A. Strictly speaking [ , Proposition . ] was stated in the context of Hermitized i.i.d.random matrices.However, a simpler version of the same proof clearly applies to deformed Wigner matrices.The main simplification compared to [ ] is that due to the constant variance profile of Wigner matrices summations as the one in [ , Eq. ( .a)] can be directly performed, without introducing the block matrices E1, E2.The remainder of the proof apart from the simplified resummation step verbatim applies to the present case.Using ( . ) in ( . ) and θ ≤ η −1 , η 1 it follows that and therefore ( . ) We now iterate ( . ) using that N δ12η ≥ N ξ starting from θ0 = 1/η (which follows trivially from Cauchy-Schwarz).
In doing so we obtain a decreasing sequence of θ's and after finally many steps conclude that where θ * is the unique positive solution to the equation Asymptotically we have ( . ) and using ( . ) once more with θ * concludes the proof.
. .Proof of Propositions .and . .Both proofs rely on Proposition .and proving that the lower bound on the stability factor given in Lemma .with Er = γ xr ir , r = 1, 2, is bounded from below by N −ǫ with some small ǫ.This will be done separately for the two propositions.
For Proposition .we use that |E1 − E2| |x1 − x2| ≤ c * with a small c * and that Å2 1, hence For Proposition .we have ( . ) In estimating the first term we used that γ x 1 i 1 , γ x 1 i 2 are in the bulk, while we used ( . ) for the second term.Notice that Choosing c0 sufficiently small, depending on c1, and recalling that |i1 − i2| ≥ N 1−ζ 1 , we can achieve that . ).This shows the required lower bound for the leading (first) term in ( .).The second term is non-negative.
The first error term is negligible, |x1 − x2| 3 ≤ N −3ζ 1 .For the second error term we have This proves that in the setup of Proposition .as well. .

M
The following results are the multi-gap versions of Theorems .and . .The gaps will be tested by functions of k variables, so we define the set of k-times differentiable and compactly supported test functions F with some large constants L, B > 0. In the following we will often use the notation i := (i1, . . ., i k ) for a k-tuple of integer indices i1, . . ., i k .The gap distribution for H x will be compared with that of the Gaussian Wigner matrices, we therefore let {µi} i∈[N] denote the eigenvalues of a GOE/GUE matrix corresponding to the symmetry class of H.
Theorem .(Quenched universality via spectral sampling mechanism).Under the conditions of Theorem .for any c1bulk-energy E we have the following multi-gap version of Wigner-Dyson universality.There exists ǫ = ǫ(a0, c0, c1) > 0 and an event ΩE,A with P(Ω c E,A ) ≤ N −ǫ such that for all H ∈ ΩE,A the matrix where K := N ζ , and some ζ = ζ(a0, c0, c1) > 0, c = c(k) > 0. The constant C in ( . ) may depend on k, L, B, a0, c0, c1 and all constants in Assumptions and at most polynomially, but it is independent of N .
First, to handle the supremum over the uncountable family F k,L,B of test functions F we reduce the problem to a finite set of test functions so that the union bound can be taken.Notice that for sufficiently smooth test functions F , which are compactly supported on some box [0, L] k of size L, we can expand F in partial Fourier series as (see e.g.[ , Remark ] and [ , Eq. ( )]) with integer n1, . . ., n k , where ϕ : R → R is a smooth cut-off function such that it is equal to one on [0, L] and it is equal to zero on [−L/2, 3L/2] c .Here ζ * > 0 is a small fixed constant that will be chosen later.Introduce the notation fn(x) := e inx/L ϕ(x), n ∈ Z.
Proceeding exactly as in the proof of Theorem . in Section ., and using the fact that ( . ) holds for test functions P1, P2 of k variables (see Remark .), we conclude that for any fixed i1, . . ., i k and n1, . . ., n k there exists a probability event Ω i 0 ,i,n , with P(Ω c i 0 ,i,n ) ≤ N −κ , on which fn j N ρ x (γ x i 0 )δλ x i 0 +i j − Eµ Proof of Theorem . .Given ( .), the proof of Theorem ., following Section .instead of Section .and using that Proposition .holds for P1, P2 of k variables (see Remark .), is completely analogous and so omitted.Then differentiating the MDE in x and E we find that Hence, by we conclude where in the last equality we wrote A = A + Å.This concludes the proof of ( . ) in case when the adjoint is present.The estimate of |1 − M1M2 | is much easier, it follows directly from (A. ).
Proof of Lemma . .To make the presentation clearer we just consider the case x1 = x and x2 = 0, the general case is analogous and so omitted.For any fixed real parameters x, y consider the MDE . ) Note that for y = 0 (A. ) is the MDE for H and for y = x (A. ) is the one for H x = H + xA.We denote the unique solution of (A. ) by M x,y = M x,y (z), the associated scDos by ρ x,y and the corresponding quantiles by γ x,y i .We will use that where we used Schwarz inequality and the bounds in (A. ).The important fact about the second bound in (A. ) is that it is integrable in E since it has a |E − E0| −1/2 singularity near an edge point E0 and a |E − E0| −2/3 singularity near a cusp point E0.Here we also used that |x| = |x1 − x2| ≤ c * is sufficiently small so that γ s,y i is in the bulk not only for s = y = x, but for all s, y ∈ [0, x].From (A. ) and (A. ) we readily conclude ( .).

B. N
Here we present numerical evidence quantifying the speed of convergence of the single gap distribution to its theoretical limit for the monoparametric ensemble, cf. Figure .This numerics was inspired by the observation made in [ ] on the slow convergence of the spectral form factor.
We thank Stephen Shenker for communicating preliminary numerical results supporting this observation in June .

F.
λ k (H))[λ k+1 (H) − λ k (H)]The two types of universality: The first histogram shows the normalised gaps of the two middle eigenvalues in the spectrum of 5000 complex Wigner matrices of size 100 × 100.
Gaussian case was introduced by H. Gharibyan, C. Pattison, S. Shenker and K. Wells who coined it as the monoparametric ensemble [ ].
bigger than the Wigner matrix W defined in Assumption .This notational inconsistency occurs only in this description of the correlated ensemble where we follow the convention of [ ].We assume that E H ≤ C and that W satisfies Assumptions (B)-(E) of [ ].We recall that Assumption (B) requires that all moments of the matrix elements of W are finite, i.e.E |W ab | q ≤ Cq with some constant Cq for any q integer, uniformly in the indices a, b ∈ [N ] 2 , while Assumption (E) requires that the covariance operator satisfies the so called flatness condition c R ≤ S[R] ≤ C R for any positive semi-definite matrix R, where c, C are some fixed positive constants.Finally, Assumptions (C), (D) or their simplified version (CD) impose decay conditions on the cumulants of different entries of W , we refer the reader to [ , Eqs. ( .a)-( .b)] for the precise condition.The self-consistent density of states ρ is defined analogously to ( .), where M solves the MDE ( . ) with M replaced by S[M].
using the upper bound on the density ρ x 2 in the first term and ( . ) in the second term.In the last step we used |x1 − x2| |i1 − i2|/N from the conditions of Proposition . .This shows that the error term |E1 − E2| 3 (|i1 − i2|/N ) 3 is negligible compared with the main term A. of order at least (|i1 − i2|/N ) 2 since we also assumed |i1 − i2|/N ≤ c * which is small.
term we use that that ∂xγ x,s i = A , giving the leading term x A in Lemma . .To estimate ∂yγ s,y i , we differentiate the defining equation of the quantiles γ ℑM s,y (E) dE = 0 for any s, y ∈ [0, x].Then, using that in the bulk | ℑM s,y (γ s,y i ) | ≥ c, we conclude|∂yγ sM s,y (E)) 2 • M s,y (E) ÅM s,y (E) dE Å2 1/2 , (A. )

F.
The figure shows the Kolmogorov-Smirnov distance D(F, F ′ ) := sup s |F (s)−F ′ (s)| of the empirical cumulative distribution function (CDF) of the (rescaled) eigenvalue gap λ N/2+1 − λ N/2 to the CDF F2 corresponding to p2 for various values of N for both GUE and the monoparametric ensemble.The empirical CDF for the GUE has been generated by sampling 100 GUE matrices H.For the monoparametric ensemble H x = H + xA typical GUE random matrices H, A have been fixed and 100 Gaussian random variables x have been sampled.The error bars represent the standard deviation of the obtained Kolmogorov-Smirnov distance for independent repetitions.In accordance with Figure we find that the gap distribution for GUE matches its theoretical limit very well for any value of N , while for the monoparametric ensemble the KS-distance seems to decay only slowly with N .

Empirically we find that the convergence towards the universal gap statistics in ( . ) is much slower for the monopara
ω A + 1].The covariance matrix C(t) of the increments of b in , consisting of four blocks of size 2N ω A + 1, is given by Ht + xrA has multiple eigenvectors, however, without loss of generality, we can assume that almost surely Ht + x l A does not have multiple eigenvectors for any r ∈ [2] for almost all t ∈ [0, T ] by [ , Proposition .] together with Fubini's theorem.By Doob's martingale representation theorem [ , Theorem .] there exists a real standard Brownian motion θ