Gaussian fluctuations of the determinant of Wigner Matrices

We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised by Tao and Vu.


Introduction
In this paper, we address the universality of the determinant of a class of random Hermitian matrices. Before discussing results specific to this symmetry assumption, we give a brief history of results in the non-Hermitian setting. In both settings, a priori bounds preceded estimates on moments of determinants, and the distribution of determinants for integrable models of random matrices. The universality of such determinants has been the source of recent active research.
1.1 Non-Hermitian matrices. Early papers on this topic treat non-Hermitian matrices with independent and identically distributed entries. More specifically, Szekeres and Turán first studied an extremal problem on the determinant of ±1 matrices [50]. In the 1950s, a series of papers [23,24,44,47,56]

calculated moments
The work of P.B. is partially supported by the NSF grant DMS#1513587 and a Poincaré Chair.
of the determinant of random matrices of fixed size (see [29]). In general, explicit formulae are unavailable for high order moments of the determinant except when the entries of the matrix have particular distribution (see, for example, [17] and the references therein). Estimates for the moments and the Chebyshev inequality give upper bounds on the magnitude of the determinant.
Along a different line of research, for an N × N non-Hermitian random matrix A N , Erdős asked whether det A N is non-zero with probability tending to one as N tends to infinity. In [34,35], Kolmós proved that for random matrices with Bernoulli entries, det A N = 0 with probability converging to 1 with N . In fact, this method works for more general models, and following [34], [11,33,51,52] give improved, exponentially small bounds on the probability that det A N = 0.
In [51], the authors made the first steps towards quantifying the typical size of |det A N |, proving that for Bernoulli random matrices, with probability tending to 1 as N tends to infinity, for any function ω(N ) tending to infinity with N . In particular, with overwhelming probability.
In [31], Goodman considered A N with independent standard real Gaussian entries. In this case, he was able to express |det A N | 2 as the product of independent chi-square variables. This enables one to identify the asymptotic distribution of log |det A N |. Indeed, one can prove that (see [48]). In the case of A N with independent complex Gaussian entries, a similar analysis yields log |det A N | − 1 2 log N ! + 1 4 log N 1 4 log N → N (0, 1).
Generalizing (1.2), Girko proved in [28] that the same result holds for A N with any independent entries having mean 0, variance 1, and fourth moment 3. Then in [42], the authors proved (1.2) holds under just an exponential decay hypothesis on the entries. Their method yields explicit rate of convergence and extends to handle the complex case. The convergence (1.2) was then extended under the sole bounded fourth moment assumption in [5].
The analysis of determinants of non-Hermitian random matrices relies crucially on the assumption that the rows of the random matrix are independent. The fact that this independence no longer holds for Hermitian random matrices forces one to look for new methods to prove similar results to those of the non-Hermitian case. Nevertheless, the history of this problem mirrors the history of the non-Hermitian case.

Hermitian matrices.
In the 1980s, Weiss posed the Hermitian analogs of [34,35] as an open problem. This problem was solved, many years later in [15], and then in [53] (Theorem 31) the authors proved the Hermitian analog of (1.1). This left open the question of describing the limiting distribution of the determinant.
In [16], Delannay and Le Caër used the explicit formula for the joint distribution of the eigenvalues to prove that for H N drawn from the GUE, log |det H N | − 1 2 log N ! + 1 4 log N Analogously, one has log |det H N | − 1 2 log N ! + 1 4 log N √ log N → N (0, 1) (1.4) when H N is drawn from the GOE. Proofs of these central limit theorems also appear in [7,13,18,54]. For related results concerning other models of random matrices, see [49] and the references therein.
While the authors of [54] give their own proof of (1.4) and (1.3), their main interest is to establish such a result in the more general setting of Wigner matrices. Indeed they show that if W N is a real Wigner matrix whose entries' first four moments match the first four moments of N (0, 1), then log |det W N | − 1 2 log N ! + 1 4 log N √ log N → N (0, 1).
They also prove the analogous result in the complex case. In this paper, we will relax this four moment matching assumption to a two moment matching assumption (see Theorem 1.2).
Finally, we mention that new interest in averages of determinants of random (Hermitian) matrices has emerged from the study of complexity of high-dimensional landscapes [4,27]. 2N (x ij + iy ij ) , 1 i < j N.
Here {x ii } 1 i N , {x ij } 1 i<j N , {y ij } 1 i<j N are independent identically distributed random variables satisfying E (x ij ) = 0, E x 2 ij = E y 2 ij = 1. (1.5) We assume further that the common distribution ν of {x ii } 1 i N , {x ij } 1 i<j N , {y ij } 1 i<j N , has subgaussian decay, i.e. there exists δ 0 > 0 such that In particular, this means that all the moments of the entries of the matrix are bounded. In the special case that ν = N (0, 1), W N is said to be drawn from the Gaussian Unitary Ensemble (GUE). We define a real Wigner matrix to have entries of the form As in the complex case, we assume the common distribution ν satisfies (1.6). In the special case ν = N (0, 1), W N is said to be drawn from the Gaussian Orthogonal Ensemble (GOE).
If W is a complex Wigner matrix satisfying (1.6), then Assumption (1.6) may probably be relaxed to a finite moment assumption, but we will not pursue this direction here. Similarly, it is likely that the matrix entries do not need to be identically distributed; only the first two moments need to match. However we consider the case of a unique ν in this paper.
Remark 1.3. Let H N be drawn from the GUE normalized so that in the limit as N → ∞, the distribution of its eigenvalues supported on [−1, 1], and let In [36], Krasovsky proved that for as N → ∞. Here C(·) is the Barnes function. Since the above estimate holds uniformly for ℜ (α k ) > − 1 2 , (1.9) shows that letting the vector D N (x 1 ) , . . . , D N (x m ) converges in distribution to a collection of m independent standard Gaussians. Our proof of Theorem 1.2 automatically extends this result to Hermitian Wigner matrices as defined above. If one were to prove an analogous convergence for the GOE, our proof of Theorem 1.2 would extend the result to real symmetric Wigner matrices as well.
Remark 1.4. We note that (1.9) was proved for fixed, distinct x k 's. If (1.9) holds for collapsing x k 's, this means that fluctuations of the log-characteristic polynomail of GUE become log-correlated for large dimension, as in the case of the Circular Unitary Ensemble [9]. More specifically, let D N (·) be as above, and let ∆ denote the distance between two points x, y in (−1, 1). For ∆ 1/N , we expect the covariance between D N (x) and D N (y) to behave like log(1/∆) log N , and for ∆ 1/N , we expect it to converge to 1.
As in Remark 1.3, our method automatically applies to mesoscopic scales (collapsing energy levels) for Wigner matrices, conditional on the knowledge of GOE and GUE cases. The exact statement is as follows, and we omit the proof, strictly similar to Theorem 1.2. We denote as N → ∞. Then converges in distribution to a Gaussian vector with covariance (min(1, c ij )) 1 i,j N (with diagonal 1 by convention), provided the same result holds for GOE.
The same result holds for Hermitian Wigner matrices, assuming it is true in the GUE case, up to a change in the normalization from 1 2 log N to √ log N in (1.10).
By Theorem 1.5, L N converges to a log-correlated field, provided this result holds for the Gaussian ensembles. It therefore suggests that the universal limiting behavior of extrema and convergence to Gaussian multiplicative chaos conjectured for unitary matrices in [25] extends to the general class of Wigner matrices. Towards these conjectures, [3,14,26,37,46] proved some asymptotics on the maximum of characteristic polynomials of circular unitary and invariant ensembles, and [6,43,57] established convergence to the Gaussian multiplicative chaos, for the same models. We refer to [2] for a survey on log-correlated fields and their connections with random matrices, branching processes, the Gaussian free field, and analytic number theory.
1.4 Statement of results: Fluctuations of Individual Eigenvalues. With minor modifications, the proof of Theorem 1.2 also extends the results of [32] and [45] which describe the fluctuations of individual eigenvalues in the GUE and GOE cases, respectively. Moreover, as proved in [45], the following theorem holds under the assumption of four moments of the matrix entries matching the Gaussian ones, by adapting the method from [53]. In Appendix B, we show that these individual fluctuations of the GOE (GUE) also hold for real (complex) Wigner matrices in the sense of Definition 1.1. In particular, the fluctuations of eigenvalues of Bernoulli matrices are Gaussian in the large dimension limit, which answers a question from [55].
To state the following theorem, we follow the notation of Gustavsson [32] and write k(N ) ∼ N θ to mean that k(N ) = h(N )N θ where h is a function such that for all ε > 0, for large enough N , (1.11) Theorem 1.6. Let W be a Wigner matrix satisfying (1.6) with eigenvalues λ 1 < λ 2 < · · · < λ N . Consider with β = 1 for real Wigner matrices, and β = 2 for complex Wigner matrices. Then as N → ∞, where Φ Λ is the cumulative distribution function for the m-dimensional normal distribution with covariance matrix Λ i,j = 1 − max {θ k : i k < j < m} if i < j, and Λ i,i = 1.
The above theorem has been known to follow from the homogenization developed in [8] (this technique gives simple expression for the relative individual positions for coupled eigenvalues from GOE and Wigner matrices) together with fluctuations of mesoscopic linear statistics, see [38] for a proof of eigenvalues fluctuations for Wigner and invariant ensembles. However, the technique from [8] is not enough for Theorem 1.2, as the determinant depends on the position of all eigenvalues.
1.5 Outline of the proof. In this section, we give the main steps of the proof of Theorem 1.2. Our outline discusses the real case, but the complex case follows the same scheme.
The main conceptual idea of the proof follows the three step strategy of [19,20]. With a priori localization of eigenvalues (step one, [12,22]), one can prove that the determinant has universal fluctuations after a adding a small Gaussian noise (this second step relies on a stochastic advection equation from [10]). The third step proves by a density argument that the Gaussian noise does not change the distribution of the determinant, thanks to a perturbative moment matching argument as in [21,53]. We include Figure 1 below to help summarize the argument.
First step: small regularization. In Section 2, with theorems 2.2 and 2.4, we reduce the proof of Theorem 1.2 to showing the convergence in probability where x 1 < x 2 < · · · < x N and y 1 < y 2 < · · · < y N are the (coupled) eigenvalues of a GOE matrix and W , respectively, and (1.14) For the reduction to (1.13), we use the fact that (1.7) holds for GOE matrices.
Second step: universality after coupling. Let M be a symmetric matrix which serves as the initial condition for the matrix Dyson's Brownian Motion (DBM) given by ii / √ 2 are independent standard Brownian motions. Note that M τ has the same distribution as e −τ /2 M + √ 1 − e −τ H where H is a GOE matrix independent of M . Therefore if M is a GOE matrix, then M τ is as well. Furthermore, the above matrix DBM induces a collection of independent standard Brownian motions (see [1]),B (k) t / √ 2, k = 1, . . . , N which we use to define the system of stochastic differential equations with initial condition given by the eigenvalues of M . It has been known since [41] that the system (1.16) has a unique strong solution. With this in mind, we follow [8], and introduce the following coupling scheme. First, run the matrix DBM takingW 0 , a Wigner matrix, as the initial condition. Using the induced Brownian motions, run the dynamics given by (1.16) using the eigenvalues y 1 < y 2 < · · · < y N ofW 0 as the initial condition. Call the solution to this system y(τ ). Using the very same (induced) Brownian motions, run the dynamics given by (1.16) again, this time using the eigenvalues of a GOE matrix, x(0), as the initial condition. Call the solution to this system x(τ ). Now fix ε > 0 and let are asymptotically equal in law. Here z τ is as in (3.4) with z = iη 0 . The significance of this is that since z τ ∼ iτ , we can bound (1.19) using Lemma A.8 and Theorem A.1.
Third step: moment matching. With this result in hand, we conclude the proof in Section 4. First, we choosẽ W 0 so thatW τ and H have entries whose first four moments are close, as in [21]. With this approximate moment matching, we use a perturbative argument, as in [54], to prove that if (1.7) holds forW τ , then it must hold for W . Finally, we use Theorem A.1, a straightforward consequence of the arguments in [40], and Lemma A.8, which identifies the size of the expectation of (1. 19), to prove that (1.19) converges to zero in probability. By the result of Section 3, this means that if holds for x(τ ), then it (1.7) must hold forW τ . Since x(τ ) is distributed as the eigenvalues of a GOE matrix, this concludes the proof.
Finally, throughout this paper, unless indicated otherwise, C (c) denotes a large (small) constant independent of all other parameters of the problem. It may vary from line to line.

Initial Regularization
Let µ 1 < µ 2 < · · · < µ N denote the eigenvalues of a GOE matrix, and let λ 1 < λ 2 < · · · < λ N denote the eigenvalues of a Wigner matrix satisfying (1.6). By (1.2), we know that and our goal is to prove that Re-writing this as we see that it is sufficient to prove that In this expression, the terms for which µ k and λ k are close to zero are difficult to control. However, we now prove that in order to prove Theorem 1.2, it is sufficient to show that with η 0 as in (1.14).
Proposition 2.1. Let x 1 < x 2 < · · · < x N , and y 1 < y 2 < · · · < y N denote the eigenvalues of two Wigner matrices, H 1 , H 2 , satisfying (1.6). Set and recall To prove Proposition 2.1, we will use Theorems 2.2 and 2.4 as input. In [12], Theorem 2.2 is stated for complex Wigner matrices, however, the argument there proves the same statement for real Wigner matrices.
Theorem 2.2 (Theorem 1 in [12]). Let W be a Wigner matrix and fixη > 0. For anyẼ > 0, there exist constants M 0 , N 0 , C, c, c 0 > 0 such that Remark 2.3. In [22], the authors proved that for some positive constant C, and N large enough, holds with high probability. Though this estimate is weaker than the estimate of Theorem 2.2, it holds for a more general model of Wigner matrix in which the entries of the matrix need not have identical variances.
On the other hand, we require the stronger the estimate in Theorem 2.2 in our proof of Proposition 2.1, and so we restrict ourselves to Wigner matrices as defined in Definition 1.1. The proof of Lemma A.8 also relies on Definition 1.1.
Theorem 2.4 (Theorem 2.2 in [8]). Let ρ (N ) denote the first correlation function for the eigenvalues of an N × N Wigner matrix. Then for any F : R → R continuous and compactly supported, and for any κ > 0, we have, Remark 2.5. In fact Theorem 2.2 in [8] makes a much stronger statement, namely it states the analogous convergence for all correlation functions in the case of generalized Wigner matrices.
For any c independent of N and β > c/N , let are the eigenvalues of any N × N Wigner matrix. Then for any δ > 0, we may choose N large enough so that where C is a constant independent of N and u. Proof Proof. Since we will prove P (|{x k ∈ I β }| 1) → 0. Let f I β denote a smoothed indicator function of I β and choose ε small. For N large enough, β < ε and so by Markov's inequality where the last bound holds by Theorem 2.4 for sufficiently large N .
Proof of Proposition 2.1. The idea is to chooseη so that we can use Theorem 2.2 to estimate and then take care of the remaining error using Corollaries 2.6 and 2.7. Let where c N = (log N ) 1 4 , and observe that In estimating the right hand side above, we will use the notation Taking N sufficiently large, by Theorem 2.2, we can write where in the last line, we have used q = 2 in Theorem 2.2, and C < ∞ is a constant. Therefore we have Next we estimate k (log |x k + iη| − log |x k |). The same arguments hold for k (log |y k + iη| − log |y k |), and so this will give us an estimate for E (|g (η)|). Taylor expansion yields Using integration by parts and Corollary 2.6, we have In the same way, we also have We now estimate |x k | η (log |x k + iη| − log |x k |). We consider two cases. First, let Integrating by parts, and noting that N 2 (A N ) = 0, we have Corollary 2.6 gives, for any δ > 0, by our choice of b N . We can of course make the same estimate for −η < x < −A N using the same calculation, so it remains to estimate |x k |<AN (log |x k + iη| − log |x k |). By Corollary 2.7, we have The estimates (2.3) and (2.4) along with Markov's inequality, and the estimate (2.5), conclude the proof.

Coupling of Determinants
In this section, we use the coupled Dyson Brownian Motion introduced in [8] to compare (1.19) and (1.18). DefineW τ by running the matrix Dyson Brownian Motion (1.15) with initial conditionW 0 whereW 0 is a Wigner matrix with eigenvalues y. Recall that this induces a collection of Brownian motionsB (k) t so that the system (1.16) with initial condition y has a (unique strong) solution y(·), and y(τ ) are the eigenvalues ofW τ . Using the same (induced) Brownian motions as we used to define y(τ ), define x(τ ) by running the dynamics (1.16) with initial condition given by the eigenvalues of a GOE matrix. Using the result of Section 2 as an input to Lemma 3.1, we now prove Proposition 3.2 which says that (1.19) and (1.18) are asymptotically equal in law.
To study the coupled dynamics x(t) and y(t), we follow [10,39]. For ν ∈ [0, 1], let where x is the spectrum of a GOE matrix, and y is the spectrum ofW 0 . With this initial condition, we denote the (unique strong) solution to (1.16) by λ (ν) (t). Note that Lemma 3.1 below, from [10], tells us that we may estimate f τ (z) by f 0 (z τ ), with z τ as in (3.4) and τ as in (1.17).
Lemma 3.1. Let κ > 0 and for any C > 0, define Let ε > 0. Then there exists C such that for any 0 < t < N −ε , any D > 0, and any z ∈ S C , we have In the above, z t is given by and that η 0 is large enough to make use of Lemma 3.1. Therefore, integrating both sides of (3.3), we have by Lemma 3.1 that with overwhelming probability, As a consequence, we have proved the following proposition.

Conclusion of the Proof
We will conclude the proof of Theorem 1.2 in two steps. The first step is to prove a Green's function comparison theorem, and the second is to establish Theorem 1.2 assuming Theorem A.1 and Lemma A.8. We prove both Theorem A.1 and Lemma A.8 in the Appendix.

Green's Function Comparison Theorem.
In this section, we first use Lemma 4.1 to choose aW 0 so that W τ given by (1.15) and initial conditionW 0 , matches W closely up to fourth moment. We will then prove Theorem 4.5, which by the result of Section 2, says that log | detW τ | and log |det W | have the same law as N → ∞.
Lemma 4.1 (Lemma 6.5 in [21]). Let m 3 and m 4 be two real numbers such that for some positive constant C 2 . Let ξ G be a Gaussian random variable with mean 0 and variance 1. Then for any sufficiently small γ > 0 (depending on C 2 ), there exists a real random variable ξ, with subexponential decay and independent of ξ G such that the first four moments of Remark 4.2. Let ξ be a random variable with E (ξ) = 0 and E ξ 2 = 1. Then, assuming they exist, m 3 = E ξ 3 and m 4 = E ξ 4 always satisfy the relation (4.1). To see this, write Now sinceW τ is defined by independent Ornstein-Uhlenbeck processes in each entry, it has the same distribution as where H is a GOE matrix independent ofW 0 . So choosing γ = 1 − e −τ , Lemma 4.1 says we can chooseW 0 so that the first three moments of the entries ofW τ match the first three moments of the entries of W , and the fourth moments of the entries of each differ by O(τ ). Our next goal is to prove Theorem 4.5 which says that withW τ constructed this way, if Theorem 1.2 holds forW τ , then it holds for W . We first introduce stochastic domination and state Theorem 4.4 which we will use in the proof.
: N ∈ N, u ∈ U N be two families of nonnegative random variables, where U N is a possibly N -dependent parameter set. We say that X is stochastically dominated by Y , uniformly in u, if for every ε > 0 and D > 0, we have for N (ε, D) sufficiently large. Stochastic domination is always uniform in all parameters, such as matrix indices and spectral parameters, that are not explicitly fixed. We shall use the notation Then we have uniformly for i, j = 1, . . . , N and z ∈ S. Note that |m sc (z)| 1, so we have Theorem 4.5. Let F : R → R be smooth with compact support, and let W and V be two Wigner matrices satisfying (1.6) such that for 1 i, j N , where τ is as in (1.17). Further, let where η 0 is as in (1.14).
Proof. The idea is to view W as obtained from V by N 2 operations in which we replace an entry of V by the corresponding entry of W, a method already used for fluctuations of determinants in [54]. We shall look at the effect of one such swapping operation, and see that its effect is negligible enough that by changing every entry of V to the corresponding entry of W , we may conclude the theorem.
Fix (i, j) and let E (ij) be the matrix whose elements are E (ij) kl = δ ik δ jl . Let W 1 and W 2 be two adjacent matrices in the swapping process described above. Since W 1 , W 2 differ in just the (i, j) and (j, i) coordinates, we may write where Q is a matrix with Q ij = Q ji = 0, and Importantly U,Ũ satisfy the same moment matching conditions we have imposed onW τ and W . Now by the fundamental theorem of calculus, we have for any symmetric matrix H, Furthermore, it is clear from its proof that Lemma A.8 holds with 1 in place of τ . Therefore to prove Theorem 4.5, it is sufficient to prove that The main idea now is to make expansions of s W1 and s W2 around s Q , and then to Taylor expand F . So let By the resolvent expansion we can write Tr (S(iη)) dη = This gives us an expansion of s W1 around s Q . Now Taylor expand F (R + ξ) as where 0 < ξ ′ < ξ, and introduce the notation A (m) in order to arrange terms according to powers of N . For example Making the same expansion for W 2 , we record our two expansions as with ξ i corresponding to W i . With this notation, we have Now only the first three moments of U,Ũ appear in the terms corresponding to m = 1, 2, 3, so by the moment matching assumption (4.2), all of these terms are all identically zero. Next, let us look at m = 4. Every term with first, second, and third moments of U andŨ is again zero, and what remains is is small. To see that this is in fact the case, we expand the traces, and apply Theorem 4.4 along with our fourth moment matching assumption (4.3). Specifically, Writing the corresponding Tr for W 2 and applying the moment matching assumption, we see that we can bound (4.7) by To bound the terms in the sum, we need to count the number of diagonal and off-diagonal terms in each product. To do this, let us say U pq ,Ũ pq and U qp ,Ũ qp are the only non-zero entries of U,Ũ . Then each of the sums over i 1 , . . . , i 8 are just sums over p, q, and when j / ∈ {p, q}, R ji1 and R i8j are certainly off-diagonal entries of R. This means we can apply Cauchy-Schwartz to write that for any γ > 0, Similarly, Since A (4) has a pre-factor of N −2 in (4.6), and the above holds for every choice of γ > 0, in our entire entry swapping scheme starting from V and ending with W , the corresponding error is o(1).
Lastly we comment on the error term A (5) . All terms in A (5) not involving Ω can be dealt with as above. The only term involving Ω is F ′ (R)Ω, and to deal with this, we can expand the expression for Ω as above. We do not have any moment matching condition for the fifth moments of U,Ũ , but (1.6) means that their fifth moments are bounded which is good enough because A (5) has a pre-factor of N −5/2 above.

Proof of Theorem 1.2.
In this section we will use Theorem A.1 to prove Proposition 4.8. Then, using Lemma A.8, we conclude the proof of Theorem 1.2.
We first observe that to prove (1.2), we can disregard the growth of log at infinity because the probability of finding an eigenvalue of a Wigner matrix outside of the interval [−3, 3], say, is overwhelmingly small. Indeed, introduce the notation φ where χ {|x|<100} is a smooth approximation of ½ {|x|<100} . Then by Theorem A.3, for any δ > 0, We now prove two lemmas that we will use as input to the proof of Proposition 4.8. The first lemma controls the Fourier transform of φ (100) N and the second gives us a formula which we will use to control its variance.
Lemma 4.6. Recall the notation τ = N −ε , let φ Proof. The bound is immediate for |ξ| < 1. For |ξ| > 1, we can integrate by parts since φ (100) N is smooth and compactly supported. The result is that for all ξ, where C is a constant independent of ξ.
Proof. Beginning on the left hand side, make the change of variables z = x − y, and let so that the left hand side can be re-written as R ||g z || 2 2 dz.
By Parseval's identity, Substituting this above, we have which concludes the proof.
For 0 < ξ < 100, the inequality | sin x| < x shows φ Using | sin y| < |y|, we see that the first term is O(1), and integrating by parts, we see that the second term is O(1) as well. This means holds with x(τ ) as defined in Section 3, and in Section 2, we proved that we can say that (1.7) holds forW τ if By Proposition 3.2, this is equivalent to proving that with z τ as in (3.4). In fact, by (3.5), we have that z τ = O (iτ ), and so it is sufficient to prove that To do this, first note that by Lemma A.8, we have and define Y analogously with y(0) in place of x(0). By Proposition 4.8, for any Lipschitz function F with Lipschitz constant ||F || Lip , we have Since this holds for any ε, we have proved (4.9), and so (1.7) holds forW τ . Finally, by Theorem 4.5, we conclude that (1.7) holds for W .

Appendix A: Central Limit Theorem for Regularized Determinants
In this appendix, we prove Theorem A.1 which is a slight improvement on Theorem 3.6 in [40].
where the random variables W Let φ be a test function that may depend on N , and whose Fourier transform for all 0 < ε < 1/4. Finally, let Here Remark A.2. [40] proves this theorem under the assumption that in place of (A.1). The essential consequence of our weaker assumption is that Theorem A.1 accepts φ which is on a mesoscopic scale. The fundamental input which allows us to make this improvement is Theorem A.3 which like Theorem 2.4 holds for generalized Wigner matrices. We replace (3.69) in [40] by Lemma A.6, and in doing so can omit the truncation argument that the authors of [40] use in their proof.
Our proof of Theorem A.1 closely follows the proof of Theorem 3.6 in [40], and in the following, where our proof is identical to [40], we will only cite the relevant sections of [40]. We begin by quoting the two following propositions which we will use.
Proposition A.4 (2.17, 2.72 in [40]). Let M be an N × N matrix, U (t) = e iMt , and let D jk denote d/dM jk . Then where we can re-write this as D jk e N (x) = 2iβ jk xe N (x) tU jk (t)φ(t)dt.
Proposition A.5 (Proposition 3.1 in [40]). Let ξ be a random variable such that E |ξ| p+2 < ∞ for a certain nonnegative integer p. Then for any function Φ : R → C of the class C p+1 with bounded derivatives, Φ (l) , l = 1, . . . , p + 1, we have and the cumulants κ j of ξ are defined by We now prove Theorem A.1.
Our goal is to show that Following [40], rewrite Z ′ N (x) as N (x)) . As in [40], we will derive a self-consistent equation for Y N (x, t), and this will imply (A.4). By Duhamel's formula, we have Applying Proposition A.5, we can write where D jk denotes d dM jk . Let The goal of the following computations is to analyze each T l to find a self-consistent equation for Y N (x, t).
Next, since κ 2 = var X (N ) jk = ω 2 (1 + δ jk ), we have Applying Proposition A.4, we have and substitute Then we have Recall Y N (x, t) = E (u • N (t)e N (x)). Therefore, by Cauchy-Schwartz, the fact that |e N (x)| = 1, and Lemma A.6, we have Y N (x, t) = O (N γ ) for any γ > 0. Applying Lemma A.6 again, and using the assumption (A.1) we see that r N (x, t) = O N −1/4 . We now analyze Here we have ignored the dependence κ 3 has on j, k because it is clear that this affects only N of the N 2 terms in the sum. Our analysis below is unaffected by this. Applying Proposition A.4, Therefore, in the expression for T 2 , we encounter the following two types of sums.
By Cauchy-Schwartz and the fact that we see that T 22 = O N −1/2 . Also, rewriting T 21 as Next we look at T 3 which is given by Since κ 4,jk = κ 4 − 9ω 2 δ jk , the modified variances of the diagonal entries of W only affect N terms in the sum above, and so it is clear we can replace κ 4,jk by κ 4 in what follows. Proposition A.4 allows us to expand D 3 jk Φ N , and applying the same argument as we used to bound |T 2 |, we see that any term of that has at least one off-diagonal term U jk is O N −1+10ε . The remaining terms arise from e • N (x)D 3 jk U jk (t) and 3D jk U jk (t)D 2 jk e • N (x). They are and respectively. Here we have omitted a factor of β 3 jk , but as before, it is clear that the effect of the N diagonal terms is negligible. Note that (A.7) does not involve φ, and so the analysis in [40] applies without modification here. The estimates following equation (3.110) in [40] show that (A.7) is O N − 1 4 . Now we consider (A.8). By Lemma A.7, we can replace U jj and U kk by v(t) = 1 2πω 2 e iλt (4ω 2 − λ 2 ) + dλ, and the computation from equations (3.121) to (3.125) in [40] gives Finally, we bound ε 3 . To do this, we need to bound D 4 jk Φ n , and in the expression for Y N (x, t), we have to sum ε 3 / √ N over j, k. On the other hand, recall that D jk is the derivative with respect to M jk , and so taking four such derivatives introduces a factor of N −2 . Therefore, for every term made up of products of D l jk e N (x) and D l jk U jk (t) for l = 1, 2, 3 we can use the analyses of T 1 , T 2 , T 3 to conclude that due to the extra factor of 1/ √ N we have in front of ε 3 , these terms are all negligible. To bound the terms corresponding to e • N (x)D 4 jk U jk , one can expand the fourth derivative to see that every term has a sum involving U jk . This means we can use the same argument as we used to bound |T 2 | to get a bound of O N −1/2 for the contribution of these terms. Lastly, we have to bound the terms corresponding to U jk D 4 jk e N (x). We can do this by expanding the expression for D 4 jk e N (x), applying the same argument as we used to bound |T 2 |, and assumption (A.1). In summary, we find Applying Lemma A.7, we can replacev N (t) by v(t) from above, and A N (t) by Proposition 2.1 in [40] gives the solution to this equation. The details of this calculation can be found in equations (2.82)-(2.87) and (3.128)-(3.129) of [40] and give This completes the proof of Theorem A.1.

By Theorem A.3, it follows that
with γ k as in (A.3). Therefore, again by Theorem A.3 (see (A.9)).
Lemma A.7. Let f : R → R have bounded first derivative. Then for any γ > 0, Proof. By Taylor's theorem, there exist ξ k , k = 1, . . . , N such that where γ k is defined as in (A.3). Now taking expectation we have Recall that by Theorem A.3, for any γ, D > 0, For the remaining term, by (A.3), we have which concludes the proof.
Below we will prove Lemma A.8 which we use both in the proof of Theorem 4.5, and to conclude the proof of Theorem 1.2.
Lemma A.8. Recall the notation τ = N −ε and let {x k } N k=1 , {y k } N k=1 denote the eigenvalues of two Wigner matrices, W 1 and W 2 . Then Proof. By the fundamental theorem of calculus, we can write with s W as in (1.21), and δ > 0. For the following argument, δ = 5 12 works. Writing the same expression for W 2 and taking the difference, we first note that by (A.9), we have that for any γ > 0, Therefore, we only need to bound Let z = E + iη be in S 1 100 (as defined in Theorem 4.4), and define f (z) = (N (s W1 (z) − s W2 (z))) .
We will first estimate |E (f (z))| for τ < η < 5, where we can use Theorem 4.4 to aid our analysis. We will then use complex analysis to estimate |E (f (z))| when 5 < η < N 5 12 .
Let τ < η < 5. Following the notation of [22], let W be a Wigner matrix and let We will use the notation W (i) to denote the (N − 1) × (N − 1) matrix obtained by removing the i th row and column from W , and w i to denote the i th column of W (i) . We will also denote the eigenvalues of W by where h i denotes the i th row of W , Note that by Theorem 4.4, so we can expand (A.13) around −z − m sc . Using (1.23), we find Summing over i and taking expectation, we get since the expectations of W ii and Z i are both zero. Because τ < η < 5, we may apply Theorem 4.4 to see that Now observe that by Theorem 4.4. Again applying Theorem 4.4, we have Putting together these estimates we have Next, we look at By Theorem 4.4 [v] = O ≺ 1 N η , so summing over i and integrating with respect to η, we get that In this expression, the terms for which k = l do not contribute to the sum by the independence of w i (l) and w i (k), and the independence of w i and G (i) . This means and therefore, we have (A.20) Expanding the first term on the left hand side above, we have The only terms which contribute to this sum are those for which at least two pairs of the indices amongst k, k ′ , l, l ′ coincide. Consider first the case k = l, k ′ = l ′ , k = k ′ . The contribution of these terms to the above sum is The first term on the right hand side here cancels the second term on the right hand side of (A.20). For the second term, by Theorem 4.4, we have Next consider the case where k = k ′ , l = l ′ , k = l. To estimate the contribution this makes, we consider separately the case when W has real entries, and the case when W has complex entries. In the first case, we can assume that the eigenvectors of W have real entries. Therefore, by the spectral decomposition, we have Using (A.17) and (A.22), this gives us If instead H has complex entries, this term is identically zero. Indeed the corresponding expression of interest becomes and because we have assumed that that for i = j, h ij is of the form x + iy where E(x) = E(y) = 0 and E x 2 = E y 2 , we have E (h ij ) 2 = 0. There remain two cases to consider. Suppose k ′ = l, l ′ = k, k = l. Then and we may estimate this as we did the first term on the right hand side of (A.16), taking the difference between the corresponding expressions for G 1 and G 2 .
Lastly, we consider the case k = k ′ = l = l ′ . By Definition 1.1 and Theorem 4.4, there exists a constant C such that Therefore by (1.22) we have in summary that Returning to (A.19), we have by Theorem 4.4 for any γ > 0. We also have that To bound the remaining term in (A.19), we first note that using the same argument as we did to prove (A.24), we have Applying Theorem 4.4, we therefore conclude that for any γ > 0. Putting together all of our estimates concerning (A.19), we have by Lemma that Returning to (A.18) we have for any γ > 0 that by Cauchy-Schwarz and Theorem 4.4. In total, we have And finally, we have using (A.14).
Let us summarize what we have achieved so far. For z = (E + iη) ∈ S 1 100 as defined in in Theorem 4.4, we have proved above that for any γ > 0, where C is as in (A.23). In particular, this means that To complete the proof of this lemma, we need to estimate 23), and defineq (z) = q 1 z .
By (A.29) and Morera's Theorem, this means and so with overwhelming probability, we can write where w = 1 z and C Γ is any curve which avoids −∞, − 1 3 ∪ 1 3 , ∞ . We are interested w close to the origin since this corresponds to z far away from the origin. By choosing C Γ to be image of under the transformation z → 1 z , we can estimate the right hand side of (A.30) using (A.28) and Theorem A.3. Indeed, w = 1/z can only be a bounded distance away from the origin since we are interested in z with |ℑ(z)| > 5. Therefore Furthermore, as ξ traverses C Γ , 1 ξ traverses C γ . So for any ξ such that ℑ 1 ξ > N −δ1 , we can estimateq(ξ) by (A.28). Here 0 < δ 1 < 1, and we shall specify it shortly. Now when z = 4 ± iN −δ1 , we have by Theorem A.3. Putting these observations together, we have that when |ℑ(z)| > 5, for any γ > 0. Therefore Since we may choose γ, δ, δ 1 , we may ensure that the right hand side of (A.31) is O(1). This completes the proof of Lemma A.8.

Appendix B: Fluctuations of Individual Eigenvalues
In this appendix, we prove Theorem 1.6. The main observation is that the determinant corresponds to linear statistics for the function ℜ log, while our individual eigenvalues fluctuations correspond to the central limit theorem for ℑ log. We build on this parallel below. The main step is Proposition B.1, which considers only the case m = 1, the proof for the multidimensional central limit theorem being strictly similar.
In analogy with (4.4), for any η 0, define with the convention that ℑ log (E + i∞) = π 2 . Then we can write and as η → 0 + , we have Proposition B.1. Let W be a real Wigner matrix satisfying (1.6). Then with ℑ log det(W − E) defined as If W is a complex Wigner matrix satisfying (1.6), then Proof. We discuss the real case, the complex case being identical. We use the notation with X k as in (1.12). Let The main observation is that This proves the claimed equivalence.
The proof of Proposition B.1 closely follows the proof of Theorem 1.2. In particular, the proof proceeeds by comparison with GOE and GUE. In the following, we first state what is known in the GOE and GUE cases.
Then we indicate the modifications to the proof of Theorem 1.2 required to establish Proposition B.1.
The GOE and GUE cases. Gustavsson first [32] established the followig central limit theorem in the GUE case, and O'Rourke [45] established the GOE case. Here the notation k(N ) ∼ N θ is as in (1.11).
By Lemma B.2, the real (complex) case in Proposition B.1 holds for the GOE (GUE) case. Therefore we can prove Proposition B.1 by comparison, presenting only what differs from the proof of Theorem 1.2. We only consider the real case, the proof in the complex case being similar. Each step below corresponds to a section in our proof of Theorem 1.2.
Then E |g (η 0 ) − g (η)| E  N ) 2 ), so that the contribution from this term is also o √ log N .
Step 2: Coupling of Determinants. With the notation of Section 3 we have, We can therefore use the proof of Proposition 3.2 to prove the following.
Step To analyze the second term, we can argue exactly as we did to prove (4.5). The proof of the following lemma, analogous to Lemma A.8, addresses the first term.
Finally, we consider Var We can verify both of these bounds by integrating by parts as in Lemma 4.6 and Proposition 4.8.