Extreme gaps between eigenvalues of Wigner matrices

This paper proves universality of the distribution of the smallest and largest gaps between bulk eigenvalues of generalized Wigner matrices, from the symmetric and Hermitian classes. The assumptions on the distribution of the matrix elements are a subexponential decay and some smoothness of the density. The proof relies on the Erd{\H o}s-Schlein-Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle.

1.1 Extreme statistics in random matrix theory. The study of extreme spacings in random spectra was initially limited to integrable models. Vinson [40] showed that the smallest gap between eigenvalues of the N × N Circular Unitary Ensemble, multiplied by N 4/3 , has limiting distribution function e −x 3 , as the size N increases. In his thesis, similar results for the smallest gap between eigenvalues of a generalization of the Gaussian Unitary Ensemble were obtained. With a different method Soshnikov [38] computed the distribution of the smallest gap for general translation invariant determinantal point processes in large boxes: properly rescaled the smallest gap converges, with the same limiting distribution function e −x 3 . Vinson also gave heuristics suggesting that the largest gap between eigenvalues in the bulk should be of order √ log N /N , with Poissonian fluctuations around this limit, a problem popularized by Diaconis [13]. Ben Arous and the author addressed this problem in [2] concerning the first order asymptotics for the maximum gap, and described the limiting process of small gaps, for CUE and GUE. These results were extended by Figalli and Guionnet to some invariant multimatrix Hermitian ensembles [24]. The convergence in distribution of the largest gap was recently solved by Feng and Wei, also for CUE and GUE [23]. Finally, Feng and Wei also investigated the smallest gaps beyond the determinantal case, characterizing their asymptotics for the circular β ensembles [21]. For the Gaussian orthogonal ensemble, they proved that the smallest gap rescaled by N 3/2 converges with limiting distribution function e −x 2 [22].
The intuition for all results above are (i) the Poissonian ansatz, namely the eigenvalues gaps are asymptotically independent, (ii) weak convergence of the spacings holds with good convergence rate, so that the finite N gap density asymptotics at 0 + and ∞ are close to the limiting Gaudin density asymptotics.
The above limit theorems and heuristic picture holds beyond invariant ensembles. Indeed, the gap universality for Wigner matrices by Erdős and Yau [17] extends to submicroscopic scales. We informally state it as follows (see Theorem 1.2 for details, in particular the smoothness assumption).
Theorem. Let λ 1 < · · · < λ N be the eigenvalues of a symmetric Wigner matrix with entries satisfying some weak smoothness assumption. Then there exists c > 0 such that The same result holds for the Hermitian class, with rescaling N 4/3 and limiting distribution function e −x 3 . Our work also applies to universality of the largest gaps (see Theorem 1.4), under similar assumptions. Does the above theorem require our slight smoothness hypothesis on the matrix entries? For the largest gaps, which are essentially on the microscopic scale 1/N , this assumption is unnecessary as shown by Landon, Lopatto and Marcinek in the simultaneous work [30] . The scale of the smallest gaps is much harder to access: the current best lower bound on separation of eigenvalues for Wigner matrices with atomic distribution is N −2+o (1) , by Nguyen, Tao and Vu [34] (see also [33] for the case of sparse matrices).
Motivations for the extreme eigenvalues gaps statistics include relaxation time for diagonalization algorithms [2,12], conjectures in analytic number theory (e.g. the extreme gaps between zeros of the Riemann zeta function [2,10]), conjectures in algorithmic number theory (the Poisson ansatz for large gaps suggests the complexity of an algorithm to detect square free numbers [4]), and quantum chaos in the complementary Poissonian regime [3].
Another motivation for extreme value statistics in random matrix theory emerged after the work of Fyodorov, Hiary and Keating [25]: the maximum of the characteristic polynomial of random matrices predicts the scale and fluctuations of the maximum of the Riemann zeta function on typical intervals of the critical line. Recent progress about their conjecture verified the size of the maximal maximum of the characteristic polynomial, for integrable random matrices [1,11,29,36]. We expect that the observable (1.10) will also help understanding universality for such extreme statistics. Indeed it was an important tool in the recent proof of fluctuations of determinants of Wigner matrices [7].
1.2 Main results. We will use the notation In this work, we consider the following class of random matrices. Definition 1.1. A generalized Wigner matrix H = H(N ) is a Hermitian or symmetric N × N matrix whose upper-triangular elements H ij = H ji , i j, are independent random variables with mean zero and variances σ 2 ij = E(|H ij | 2 ) that satisfy the following two conditions: (i) Normalization: for any j ∈ 1, N , (ii) Non-degeneracy: σ 2 ij ∼ N −1 for all i, j ∈ 1, N . In the Hermitian case, we assume Var Re(H ij ) ∼ Var Im(H ij ) and independence of Re(H ij ), Im(H ij ) 1 .
We also suppose for convenience (this could be replaced by a large finite moment assumption) that the matrix elements √ N H ij satisfies the following tail estimate: there exists constants c > 0, such that for any i, j, N and x > 0 we have In some of the following results, we additionally assume non-atomicity for the matrix entries. A sequence (H) N of random matrices is said to be smooth on scale η if √ N H ij has density e −V , where V = V N,i,j satisfies the following condition uniformly in N, i, j. For any k 0 there exists C > 0 such that (1.2) Finally, we define the process of small gaps and their position as where β = 1 for the generalized Wigner symmetric ensemble and β = 2 for Hermitian one. The following theorem generalizes (and relies on comparison with) the GUE and GOE cases [2,23]. Theorem 1.2 (Small gaps process). Let (H N ) be generalized Wigner matrices satisfying (1.1). Let κ > 0.
(i) Symmetric class. Assume (H N ) is smooth on scale η = N −1/4+ε for some fixed ε > 0, in the sense of (1.2). The point process χ (N ) converges as N → ∞ to a Poisson point process χ with intensity given, for any measurable sets A ⊂ R + and I ⊂ (−2 + κ, 2 − κ), by (ii) Hermitian class. Assume (H N ) is smooth on scale N −1/3+ε for some fixed ε > 0 The point process χ (N ) converges to a Poisson point process χ with intensity N −1 be the reordered gaps between eigenvalues, i.e. {t We abbreviate t k the k-th smallest gap and t N −k the k-th largest. Corollary 1.3 (Smallest gaps). Assume (H N ) N is as in Theorem 1.2, κ > 0 and consider a non-empty interval I ⊂ (−2 + κ, 2 − κ).
There are at least two ways to understand the above scaling of the smallest spacings ( = N −3/2 for β = 1, = N −4/3 for β = 2). First, in the Gaussian integrable case, the eigenvalues interaction i<j |λ i −λ j | β suggests P(N (λ i+1 − λ i ) < x) ∼ x β+1 uniformly in small x and i, so that decorrelation of spacings would give N (N ) β+1 ∼ 1. Second, the resolvent method gives Wegner estimates for Wigner matrices with smooth entries [16]. For example, [6,Corollary B.2] shows P(N (λ i+1 − λ i ) < x) CN ε x 2 . A union bound on these level repulsion estimates provides a lower estimate on the smallest gaps, which matches our order.
For the largest gaps, Gumbel fluctuations are expected, with heuristics also relying on decoupling, and the asymptotics e −cx 2 for the upper tail distribution of N (λ i+1 − λ i ). However, for the integrable Gaussian ensembles these facts have been established only for β = 2, thanks to the determinantal structure. We therefore only state the following theorem for the Hermitian class. It proceeds by comparison with the GUE case from [22]. May the analogue for GOE be known, the universality would follow.
As in [22], we denote S(I) = inf I √ 4 − x 2 and rescale the kth largest gaps as For any interval J, we have Moreover, the rate of convergence is given bounded by d TV (τ * k (H), τ * k (GUE)) N c /(N η 2 ) for any c > 0.
As explained in Subsection 1.3, the previous theorems rely on a short and quantitative proof of relaxation of Dyson Brownian motion. Therefore this gives new results also for typical fluctuations, illustrated below with Gaussian fluctuations of eigenvalues. Theorem 1.5 (Eigenvalues fluctuations close to the edge). Let (H N ) be generalized Wigner matrices satisfying (1.1) and γ i be defined through (2.5). Consider where c = (3/2) 1/3 πβ 1/2 , with β = 1 for the symmetric class, 2 for the Hermitian one. Fix δ ∈ (0, 1). Then for any sequence i = i N → ∞, with i N δ , we have X i → N (0, 1) in distribution.
Let m 1 and k 1 < · · · < k m satisfy Then (X k1 , . . . , X km ) converges to a Gaussian vector with covariance matrix These anomalous small Gaussian fluctuations were first shown in [27] for GUE and [35] for GOE. Our proof proceeds by comparison with these results. Fluctuations of eigenvalues around their typical locations are known in the bulk of the spectrum for Wigner matrices [7,31]. Theorem 1.5 extends to any δ ∈ (0, 1) a previous result from [5] which was limited to δ < 1/4, and therefore completes the proof of eigenvalues fluctuations anywhere in the spectrum 2 .
More generally, the proof sketch below explains edge statistics for general obervables of and eigenvalues with indices in 1, N 1−ε , i.e. almost up to the bulk. As another example, for any fixed ε > 0 and diverging converges to the Gaudin distribution, a result proved in [5] for i < N 1/4 .

Sketch of the proof.
In this paper the small and large constants c, C do not depend on N but may vary from line to line. We denote κ(z) = inf(|z − 2|, |z + 2|) and a subpolynomial error parameter, for some fixed large enough C 0 > 0. Finally, we restrict the following outline and the full proof to the symmetric class, the Hermitian one requiring only changes in notations.
As already mentioned, our work and proceeds by interpolation with the integrable models, following the general method from [15]. This dynamic approach requires (i) a priori bounds on the eigenvalues locations, (ii) local relaxation for the eigenvalues dynamics after a short time, (iii) a density argument based on the matrix structure, to show that eigenvalues statistics have not changed after short-time dynamics.
In this work, the necessary estimate for (i) is the rigidity from [20]. Concerning the density argument (iii), for Theorem 1.5 we follow the Lindeberg exchange method [39] for Green's functions [19]. For theorems 1.2 and 1.4, (iii) is obtained through the inverse heat flow from [15] (this is where smoothness is required).
Our contribution is about (ii), for which we give a short and quantitative proof. The Dyson Brownian motion dynamics are defined as follows. Let B be a N × N matrix such that B ij (i < j) and B ii / √ 2 are independent standard Brownian motions, and B ij = B ji . Consider the matrix Ornstein-Uhlenbeck process If λ 1 (0) < · · · < λ N (0), the eigenvalues λ(t) of H t are given by the strong solution of the system of stochastic differential equations (the β k 's are some Brownian motions distributed as the B kk 's) The coupling method introduced in [6] proceeds as follows. Consider y(t) the solution of the same SDE (1.4) with another initial condition y(0) = {y 1 (0) < · · · < y N (0)}, the spectrum of a GOE matrix. Then the differences δ k (t) = e t/2 (λ k (t) − y k (t)) satisfy the long-range parabolic differential equation .
Smoothing of this equation for bulk indices means that for t Such estimates were proved in [17,32], with a weak error term (N −1−ε with some non-explicit ε > 0). In this work, we obtain the essentially optimal (up to subpolynomial orders) estimate This gives the relaxation step (ii) for the smallest gaps. The proof for the large gaps proceeds identically and only requires t 1/N . More importantly than the error estimate (1.5), its short proof reduces Hölder regularity to a simple maximum principle, and it also applies to edge universality. In details, for any ν ∈ [0, 1], let be interpolating between the Wigner and GOE initial conditions, as in [32]. Define From now we set ν ∈ (0, 1) and omit it from the notations. Let (1.10) The above function is the main idea in this work. A key observation is that the quadratic singularities from the denominator in (1.9) disappear when combined with the Dyson Brownian motion evolution itself, so that the time evolution of f has no shocks. This is reminiscent of a similar argument in [8, Lemma 6.2], for a different observable. More precisely, f follows dynamics close to the advection equation and suggest the approximation This estimate holds with a small error term (see Proposition 3.3) because there are no eigenvalues shocks in the equation guiding f , contrary to (1.8). The approximation (1.13) has two applications.
First application: relaxation at the edge. (1.14) Edge universality follows from the shape of the characteristics (1.12), which take points around the edge further away from the bulk. More precisely, we choose z = z 0 = E + iη with E ∈ [−2, 0] and η > 0. By a straightforward calculation based on the rigidity estimates from [20] In particular, integrating the above equation in 0 ν 1, we obtain so local edge relaxation is proved for any t > ϕ C N −1/3 , with an optimal error term. Such quantitative bounds can be similarly extended to any Second application: relaxation in the bulk. For relaxation in the bulk, we directly work with f instead of f . Fix some time scales t 1 < t 0 , a length scale t 1 < r < t 0 and bulk index k 0 . We are interested in evaluating u i (t) for some t 0 t t 0 + t 1 and |i − k 0 | N r. Assume that for any such t the maximum value of u t occurs at some index k = k(t) with |k − k 0 | N r (this is generally wrong but the conclusion will remain thanks to a finite speed of propagation estimate from [8]). We follow the maximum principle as in the analysis of the the eigenvector moment flow from [8]: for any η > 0 to be chosen, denoting z = x k (t) + iη, we have In the bulk of the spectrum, (1.13) holds with the good error term ϕ C /(N η) (see Proposition 3.3), so that the previous equation behaves similarly to If r is small enough, Im f 0 (z t ) ≈ Im f 0 ((γ k0 + iη) t0 ) (remember z a priori depends on k). Denote m = Im f 0 ((γ k0 + iη) t0 )/(N Im m sc (γ k0 + i0 + )). The above equation therefore behaves as so that for any η much smaller than t 1 we obtain max |i−k0| N r (u i (t 1 ) − m) = O( ϕ C N 2 η ). The same estimate holds for the minimum, so that max |i−k0| N r |u i (t 1 ) − m| = O( ϕ C N 2 t1 ), and in particular The estimate (1.5) follows by integration over ν ∈ (0, 1).
Acknowledgement. The idea developed in this paper benefited from discussions with students of graduate classes at the Courant Institute in 2015, 2018, the Saint Flour summer school in 2016 and the IHES summer school in 2017. The author also thanks the organizers and participants of the workshop [41] where the questions of universality of extreme gaps, and rate of convergence in universality, were raised. This work is partially supported by the Poincaré chair and the NSF grant DMS-1812114.
where our branch choice will always be Im √ z 2 − 4 > 0 for Im(z) > 0, above and in (1.12). More generally than (1.7), consider x(t) the strong solution of where the B k 's are standard Brownian motions and x(0) is still given by (1.6), and β = 1 (resp. β = 2) corresponds to the spectral dynamics with equilibrium measure GOE (resp. GUE). We still define u (ν) Then the function (1.10) satisfies the following dynamics. Lemma 2.1. For any Im z = 0, we have Proof. It is a simple application of Itô's formula. We omit the time index. First, Applying again the Itô formula d( . Combining with (II), we obtain All singularities have now disappeared and we have proved
To estimate f t or f t (see (1.14)), we first need some estimates on the characteristics (z t ) t 0 from (1.12), and the initial values f 0 , f 0 . For this, we define the curve Lemma 2.2. For 0 < t < 1 and z = z 0 satisfying η = Im z > 0, we have

In particular, for any
Proof. Without loss of generality assume that Re z > 0. Let w = z − 2. We have (z 2 − 4) 1/2 ∼ 2w 1/2 so that On S , we always have b(z) ∼ κ(z) and a(z) ∼ η so the last estimate follows immediately.
We now define the typical eigenvalues location and the set of good trajectories such that rigidity holds: wherek = min(k, N + 1 − k). The following important a priori estimates were proved in [20], for fixed t and ν = 0 or 1. The extension in these parameter is straightforward, by time discretization in t and ν first, then by Weyl's inequality to bound increments in small time intervals, and the fact that |u (ν) k (t)| < u (ν) (0) ∞ to bound increments in some small ν-intervals. Lemma 2.3. There exists a fixed C 0 > 0 (remember ϕ = ϕ(C 0 )) large enough such that the following holds. For any D > 0, there exists N 0 (D) such that for any N > N 0 we have Moreover, we have the following estimates on the initial condition f 0 , f 0 . Lemma 2.4. For any λ ∈ A and z = E + iη ∈ R, we have Im f 0 (z) Cϕ 1/2 if η > max(E − 2, −E − 2), and Im f 0 (z) Cϕ 1/2 η κ(z) otherwise. The same upper bound (naturally) holds for f 0 . Proof. The rigidity estimate on A easily implies that and the estimates follow. Note that we used z ∈ R to justify approximation of eigenvalues by typical location: in R the imaginary part of z is always greater than the eigenvalues fluctuation scale.
Finally, the following is an elementary calculation. We write z t = r(z, t), for r given by the right hand side of (1.12).

Relaxation at the edge.
For the following important estimate towards edge universality, remember the notation (1.14).
Proof. For any 1 , m N 10 , we define t = N −ε−10 and z (m We also defie the stopping times (with respect to F t = σ(B k (s), 0 s t, 1 k N )) We will prove that for any D > 0 there exists N 0 (ε, D) such that for any N N 0 (ε, D), we have We first explain why the above equation implies the expected result by a grid argument in t and z.
On the one hand, we have the sets inclusion Indeed, for any given z and t, chose t , z (m) such that t t < t +1 and |z − z m | < 5N −10 . Then | f t (z) − f t (z (m) )| < N −2 , say, as follows directly from the definition of f t and the estimate |v k (s)| < ϕN −2/3 (obtained by maximum principle). Moreover, under the event ∩ k A ,m,k , we have | f t (z (m) )− f t (z (m) )| < N −2 as follows easily from (2.2).
On the other hand, for some fixed universal c > 0 and arbitrary small a > 0, for any martingale M we have (see e.g. [37, Appendix B.6]) .
Together with the deterministic estimate We now prove (2.6). We abbreviate t = t , z = z (m) for some 1 , m N 10 . Let g s (z) = f s (z t−s ). From lemmas 2.2 and 2.4, so that we only need to bound the increment of g. Using lemmas 2.1 and 2.5, Itô's formula gives where we used Lemma 2.2, κ(E) = κ(z) = b(z) on S , and the fact that, for s < t ∧ τ , we have We also have Finally, we want to bound sup 0 u t |M u | where For some fixed universal c > 0 and arbitrary a > 0, we have P sup with overwhelming probability.
Let k j = jϕ 2 and I j = k j , k j+1 ∩ 1, N , 0 j N/ϕ 2 . Then (2.9) For each 0 j N/ϕ 2 , pick a m = m j such that |z (m) − γ kj | < N −9 . First, as v k (s) 0 for any k and s, for s t ∧ τ we have k∈Ij v k (s) η m Im f s (z (m) ). To estimate Im f s (z (m) ), introduce such that t s < t +1 . On the event ∩ k A ,m,k , we have | f s∧τ (z (m) ) − f t ∧τ (z (m) )| < N −2 as seen easily from (2.2). We therefore proved and in particular the same estimate holds for sup k∈Ij v k (s). Lemma A.1 allows us to bound sup k∈Ij where for the last inequality, we evaluate this deterministic integral in Lemma A.2 We now state quantitative relaxation of the dynamics at the edge. Remember that λ and y satisfy the same equation (1.7), with respective initial conditions a generalized Wigner and GOE spectrum.
Theorem 2.7. Let ε, ε > 0 be fixed small constants. For any D > 0 there exists N 0 and such that for any Proof. Assume first that k ∈ ϕ 5 , N 1−ε . Then define Note that κ(γ k ) 1/2 ∼ (k/N ) 1/3 . Therefore, by Lemma 2.3 and Proposition 2.6, In particular, for any k ∈ ϕ 5 , N 1−ε and t ∈ [0, N − ε ], we have Note that all our estimates have been uniform in ν ∈ (0, 1), so that the above equation holds for any N greater than some N 0 independent of ν.
The above equation easily implies that for any p 1, E(|v Markov's inequality concludes the proof, when k ∈ ϕ 5 , N 1−ε For k ∈ 1, ϕ 5 we repeat the same reasoning with z = z 0 = γ k0 + i yields to the same estimate up to the deteriorated ϕ 10 exponent, say.
2.3 Proof of Theorem 1.5. For a test function F , we rely on [27,35] so that we only need to prove for any diverging k ∈ 1, N 1−ε . From Theorem 2.7, for t > (k/N ) 1/3 ϕ 11 , we have so that (2.10) holds for any Gaussian divisible ensemble of type H t = e −t/2 H 0 + (1 − e −t ) 1/2 U , where H 0 is any initial generalized Wigner matrix and U is an independent standard GOE matrix. We now construct a generalized Wigner matrix H 0 such that the first three moments of H t match exactly those of the target matrix H and the differences between the fourth moments of the two ensembles are less than N −c for some c positive. This existence of such a initial random variable is give for example by [18,Lemma 3.4]. By the following Theorem 2.8, we have E Ht F (X k ) = E H F (X k ) + o(1). The previous two equations conclude our proof of (2.10), and therefore Theorem 1.5 (the proof in the multidimensional case is analogue).
The following theorem is a slight extension of the Green's function comparison theorem from [19], (see for example [8, theorem 5.2] for an analogue statement for eigenvectors). Compared to [19], we include the following minor modifications: (1) We state it for energies in the entire spectrum. (2) We allow the test function to be N -dependent.
Theorem 2.8 can be proved exactly as in [19], so we don't repeat repeat it. Note that at the edge, the 4 moment matching can be replaced by 2 moments. For our applications, this improvement is not necessary.
for all 1 i j N and 1 k 3. Assume also that there is an a > 0 such that Then there is ε > 0 depending on a such that for any integer k, any choice of indices 1 j 1 , . . . , j k N and smooth bounded Θ : R k → R, 3 Relaxation from a maximum principle 3.1 Result. The main result of this section is the following. Again, remember that λ and y satisfy the same equation (1.7), with respective initial conditions a generalized Wigner and GOE spectrum. We denotē Theorem 3.1. Let α, δ > 0 be fixed, arbitrarily small, and N −1+δ < t < N −δ . For any fixed (small) ε > 0, (large) D > 0, there exists N 0 such that for any N N 0 and any k ∈ αN, (1 − α)N we have Corollary 3.2. Let α, δ > 0 be fixed, arbitrarily small, and N −1+δ < t < N −δ . Then for any fixed (small) ε > 0, (large) D > 0, for large enough N for any k ∈ αN, (1 − α)N we have Proof. Note that From Theorem 3.1, the above two terms do not exceed N ε /(N 2 t) with probability 1 − N −D . For the third term, we haveδ (3.1) As u k (0) = y k (0) − λ k (0), using Lemma 2.3 we obtain that the main contribution from (3.1) is of order with overwhelming probability, where we used that γ k is int he bulk. This concludes the proof.
3.2 Approximation along characteristics. Proposition 2.6 gave some useful a priori bounds on f t (z), especially useful for universality at the edge of the spectrum. The following estimate, in the bulk, goes further by justifying (1.13) in the bulk of the spectrum.
Proposition 3.3. Consider the dynamics (2.2) for some fixed β > 0. Let ε, κ > 0 be fixed (small) constants. Then for any D > 0 there exists N 0 (ε, D) such that for any N N 0 (ε, D) we have Proof. We strictly follow the proof of Proposition 2.6. Actually, the only differences are (i) the observable, now f instead of f (but the equations are the same), (ii) simplifications, as subtle edge estimates are not required anymore for our bulk estimates. Details are left to the reader.
3.3 Localized maximum principle. The maximum principle was used in random matrix theory for the relaxation of eigenvector statistics along the Dyson Brownian motion dynamics in [8,9]. We follow the method these works, with the following analogue of [9, Propostion 3.7], which we will use iteratively. We first need to introduce a few notations analogous to [9]. Let k 0 be a fixed index in the bulk, ψ = N ω be an error parameter where w > 0 is small and fixed. We also definē Proof. The proof proceeds as [9, Propostion 3.7], with only substantial difference the a priori estimate in the approximation by short range dynamics: we have the following analogue (3.6) of [9, Lemma 3.5]. Let c jk = 1/(N (x j − x k ) 2 ) and write B = S + L , Denote by U S (s, t) the semigroup associated with S from time s to time t, i.e. ∂ t U S (s, t) = S (t)U S (s, t) for any s t, and U S (s, s) = Id. The notation U B (s, t) is analogous. Then, for large enough N , the following approximation holds with overwhelming probability: for any t/2 < u < v < t and |k − k Compared to [9,Lemma 3.5], note that the above estimate is simpler because it consists in only one particle, but it contains the extra term d N t + |u−v| N t due to the error between the local meansū k (s) andū k0 (v). The proof of (3.6) is the same as [9, Lemma 3.5], with main tool Proposition 3.3.
With the a priori estimate (3.6), the rest of the proof is identical to that of [9, Propostion 3.7].
3.4 Proof of Theorem 3.1.The quantitative relaxation of Dyson's Brownian motion in the bulk will follow from the proposition below, itself obtained from iterations of the maximum principle from the previous subsection.
Proposition 3.5. Let α, δ > 0 be fixed, arbitrarily small, and N −1+δ < t < N −δ . For any fixed (small) ε > 0, (large) D > 0, there exists N 0 such that for any N N 0 and any k ∈ αN, (1 − α)N we have P |u Proof. Our time horizon parameter t is fixed, and we consider t = t/ψ 10 , d = t/ψ. Remember that the definition (3.3) depends on a fixed bulk index k 0 . From Proposition 3.4, for any m 1 we have Iterations of this equation give, at m 0 = min{m : t/2 m < ψ 100 /N }, This concludes the proof, as the center k 0 is arbitrary.
From Proposition 3.5, the proof of Theorem 3.1 is straightforward: we just notice that evaluate the 2p-th moment of δ k (t) −δ k (t) and use Markov's inequality as in the proof of Theorem 2.7.
4 Extreme gaps 4.1 Reverse heat flow. We first state a quantitative analogue of [15,Proposition 4.1]. Its proof is essentially the same as in [15]. In the following dγ denotes the standard Gaussian measure which is reversible for the Ornstein-Uhlenbeck dynamics with generator A = 1 2 ∂ xx − x 2 ∂ x . Lemma 4.1. Let 0 < 2a < b < 1. Assume e −V is a centered probability density, with V smooth on scale η = N −a in the sense of (1.2) and [−x,x] c e −V θ −1 e −x θ for some θ > 0. Denote u = de −V /dγ. Let t = N −b . Then for any D > 0 there exists C > 0 and a probability density g t w.r.t. γ such that (i) |e tA g t − u|dγ CN −D (ii) g t dγ is centered, has same variance as udγ, and satisfies [−x,x] c g t dγ θ −1 e −x θ for some θ > 0.
Still using (1.2), we easily have t k |A k u|dγ C k t k η −2k . We assume H is smooth on scale η. From Corollary 3.2, there exists a generalized Wigner matrix H such that if H t denotes under the Dyson Brownian Motion dynamics with initial condition H, the total variation distance between H t and H is of order N −D for any D, provided t N −ε η 2 . In particular, the total variation distance between their spectra is also at most N −D .
On the other hand, for such t, from Corollary 3.2 the gaps between eigenvalues of H can all be coupled with some GUE gaps up to an error at most N ε /(N 2 t).
The total variation between the bulk spectra of H and GUE is therefore at most N ε /(N 2 t). The choice t = N −ε η 2 concludes the proof.