Moderate deviations for the eigenvalue counting function of Wigner matrices
aa r X i v : . [ m a t h . P R ] J a n MODERATE DEVIATIONS FORTHE EIGENVALUE COUNTING FUNCTIONOF WIGNER MATRICES
Hanna D¨oring , Peter Eichelsbacher Abstract:
We establish a moderate deviation principle (MDP) for the numberof eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymp-totics of the variance of the eigenvalue counting function of GUE matrices dueto Gustavsson. The extension to certain families of Wigner matrices is basedon the Tao and Vu Four Moment Theorem and applies localization results byErd¨os, Yau and Yin. Moreover we investigate families of covariance matrices aswell.
AMS 2000 Subject Classification:
Primary 60B20; Secondary 60F10, 15A18
Key words:
Large deviations, moderate deviations, Wigner random matrices, covariancematrices, Gaussian ensembles, Four Moment Theorem Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Mathematik, NA 3/68, D-44780 Bochum, Germany, [email protected] Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Mathematik, NA 3/66, D-44780 Bochum, Germany, [email protected]
Both authors have been supported by Deutsche Forschungsgemeinschaft via SFB/TR 12. The first author wassupported by the international research training group 1339 of the DFG.
HANNA D ¨ORING AND PETER EICHELSBACHER
1. Introduction
Recently, in [4] the Central Limit Theorem (CLT) for the eigenvalue counting function ofWigner matrices, that is the number of eigenvalues falling in an interval, was established. This universality result relies on fine asymptotics of the variance of the eigenvalue counting function,on the Fourth Moment Theorem due to Tao and Vu as well as on recent localization resultsdue to Erd¨os, Yau and Yin. See also [3]. Our paper is concerned with the moderate deviationprinciple (MDP) of the eigenvalue counting function. We will start with the MDP for Wignermatrices where the entries are Gaussian (the so-called Gaussian unitary ensemble (GUE)), firstproven by the authors in [11]. Next we establish a MDP for individual eigenvalues in thebulk of the semicircle law (which is an MDP corresponding to the Gaussian behaviour provedin [14]). This MDP will be extended to certain families of Wigner matrices by means of theFour Moment Theorem (see [21], [20]). It seem to be the first application of the Four MomentTheorem to be able to obtain not only universality of convergence in distribution but also toobtain deviations results on a logarithmic scale, universally. Finally a strategy based on theMDP for individual eigenvalues in the bulk will be shown to imply the MDP for the eigenvaluecounting function, universally for certain Wigner matrices. In the meantime, we successfullyapply the Four Moment Theorem to obtain MDPs also at the edge of the spectrum as well asfor the determinant of certain Wigner matrices, see [9], [10].Consider two independent families of i.i.d. random variables ( Z i,j ) ≤ i 1) – strictly in the bulk –, and G denotes the distribution function of thesemicircle law) was established: E [ N I n ( W ′ n )] = n − k ( n ) + O (cid:0) log nn (cid:1) and V ( N I n ( W ′ n )) = (cid:0) π + o (1) (cid:1) log n. (1.1)The proof applied strong asymptotics for orthogonal polynomials with respect to exponentialweights, see [5]. In particular the CLT holds for N I ( W ′ n ) if I = [ y, ∞ ) with y ∈ ( − , N I ( W ′ n ) − n̺ sc ( I ) q π log n → N (0 , n → ∞ (called the CLT with numerics). These conclusions were extended to non-GaussianWigner matrices in [4].Certain deviations results and concentration properties for Wigner matrices were considered.Our aim is to establish certain moderate deviation principles. Recall that a sequence of laws( P n ) n ≥ on a Polish space Σ satisfies a large deviation principle (LDP) with good rate function I : Σ → R + and speed s n going to infinity with n if and only if the level sets { x : I ( x ) ≤ M } ,0 ≤ M < ∞ , of I are compact and for all closed sets F lim sup n →∞ s − n log P n ( F ) ≤ − inf x ∈ F I ( x )whereas for all open sets O lim inf n →∞ s − n log P n ( O ) ≥ − inf x ∈ O I ( x ) . We say that a sequence of random variables satisfies the LDP when the sequence of measuresinduced by these variables satisfies the LDP. Formally a moderate deviation principle is noth-ing else but the LDP. However, we speak about a moderate deviation principle (MDP) for a HANNA D ¨ORING AND PETER EICHELSBACHER sequence of random variables, whenever the scaling of the corresponding random variables isbetween that of an ordinary Law of Large Numbers (LLN) and that of a CLT.Large deviation results for the empirical measures of Wigner matrices are still only known forthe Gaussian ensembles since their proof is based on the explicit joint law of the eigenvalues,see [2] and [1]. A moderate deviation principle for the empirical measure of the GUE or GOE isalso known, see [6]. This moderate deviations result does not have yet a fully universal versionfor Wigner matrices. It has been generalised to Gaussian divisible matrices with a deterministicself-adjoint matrix added with converging empirical measure [6] and to Bernoulli matrices [8].Our first result is a MDP for the number of eigenvalues of a GUE matrix in an interval. It isa little modification of [11, Theorem 5.2]. In the following, for two sequences of real numbers( a n ) n and ( b n ) n we denote by a n ≪ b n the convergence lim n →∞ a n /b n = 0. Theorem 1.1. Let M ′ n be a GUE matrix and W ′ n := √ n M ′ n . Let I n be an interval in R . If V ( N I n ( W ′ n )) → ∞ for n → ∞ , then for any sequence ( a n ) n of real numbers such that ≪ a n ≪ p V ( N I n ( W ′ n )) (1.2) the sequence ( Z n ) n with Z n = N I n ( W ′ n ) − E [ N I n ( W ′ n )] a n p V ( N I n ( W ′ n )) satisfies a MDP with speed a n and rate function I ( x ) = x . Remark 1.2. Let I = [ y, ∞ ) with y ∈ ( − , Z n ) n with ˆ Z n = N I ( W ′ n ) − n̺ sc ( I ) a n q π log n satisfies the MDP with the same speed, the same rate function, and in the same regime (1.2)(called the MDP with numerics).In our paper we extend these conclusions to certain non-Gaussian Hermitian Wigner matrices. Tail-condition ( T ) : Say that M n satisfies the tail-condition ( T ) if the real part η and theimaginary part η of M n are independent and have a so-called stretched exponential decay:there are two constants C and C ′ such that P (cid:0) | η | ≥ t C (cid:1) ≤ e − t and P (cid:0) | η | ≥ t C (cid:1) ≤ e − t for all t ≥ C ′ .We say that two complex random variables η and η match to order k if E (cid:2) Re( η ) m Im( η ) l (cid:3) = E (cid:2) Re( η ) m Im( η ) l (cid:3) for all m, l ≥ m + l ≤ k . ODERATE DEVIATIONS FOR WIGNER MATRICES 5 The following theorem is the main result of our paper: Theorem 1.3. Let M n be a Hermitian Wigner matrix whose entries satisfy tail-condition ( T ) and match the corresponding entries of GUE up to order 4. Set W n := √ n M n . Then, for any y ∈ ( − , and I ( y ) = [ y, ∞ ) , with Y n := N I ( y ) ( W n ) , for any sequence ( a n ) n of real numberssuch that ≪ a n ≪ p V ( Y n ) (1.3) the sequence ( Z n ) n with Z n = Y n − E [ Y n ] a n p V ( Y n ) satisfies a MDP with speed a n and rate function I ( x ) = x . Moreover the sequence ˆ Z n = Y n − n̺ sc ( I ) a n q π log n satisfies the MDP with the same speed, the same rate function, and in the same regime (1.3) (called the MDP with numerics). Before we will prove the MDP for the GUE, we describe the organisation of the next sections.In a first step, we will apply Theorem 1.1 to obtain a MDP for eigenvalues in the bulk of thesemicircle law. Next we extend this result to certain families of Hermitian Wigner matricessatisfying tail-condition ( T ) by means of the Four Moment Theorem due to Tao and Vu. Thisis the content of Section 2. In Section 3, we show the MDP with numerics for the countingfunction of Wigner matrices. Moreover we apply recent results of Erd¨os, Yau and Yin [12] onthe localization of eigenvalues and of Dallaporta and Vu [4] in order to prove the MDP withoutnumerics. Section 4 is devoted to discuss the case of symmetric real Wigner matrices as wellas the symplectic Gaussian ensemble applying interlacing formulas due to Forrester and Rains,[13]. Finally, in Section 5 we present results for covariance matrices. We prove a universalMDP with numerics for the counting eigenvalue function of covariance matrices.In [11] we proved a MDP for certain determinantal point processes (DPP), including GUE.Theorem 1.1 follows immediately from an improvement of Theorem 5.2 in [11], which can beeasily observed applying the proof of [1, Theorem 4.2.25]. Let Λ be a locally compact Polishspace, equipped with a positive Radon measure µ on its Borel σ -algebra. Let M + (Λ) denotethe set of positive σ -finite Radon measures on Λ. A point process is a random, integer-valued χ ∈ M + (Λ), and it is simple if P ( ∃ x ∈ Λ : χ ( { x } ) > 1) = 0. Let χ be simple. A locallyintegrable function ̺ : Λ k → [0 , ∞ ) is called a joint intensity (correlation), if for any mutuallydisjoint family of subsets D , . . . , D k of Λ E (cid:0) k Y i =1 χ ( D i ) (cid:1) = Z Q ki =1 D i ̺ k ( x , . . . , x k ) dµ ( x ) · · · dµ ( x k ) , HANNA D ¨ORING AND PETER EICHELSBACHER where E denotes the expectation with respect to the law of the point configurations of χ . Asimple point process χ is said to be a determinantal point process with kernel K if its jointintensities ̺ k exist and are given by ̺ k ( x , . . . , x k ) = det (cid:0) K ( x i , x j ) (cid:1) i,j =1 ,...,k . (1.4)An integral operator K : L ( µ ) → L ( µ ) with kernel K given by K ( f )( x ) = Z K ( x, y ) f ( y ) dµ ( y ) , f ∈ L ( µ ) , is admissible with admissible kernel K if K is self-adjoint, nonnegative and locally trace-class(for details see [1, 4.2.12]). A standard result is, that an integral compact operator K withadmissible kernel K possesses the decomposition K f ( x ) = P nk =1 η k ϕ k ( x ) h ϕ k , f i L ( µ ) , where thefunctions ϕ k are orthonormal in L ( µ ), n is either finite or infinite, and η k > k , leadingto K ( x, y ) = n X k =1 η k ϕ k ( x ) ϕ ∗ k ( y ) , (1.5)an equality in L ( µ × µ ). Moreover, an admissible integral operator K with kernel K is called good with good kernel K if the η k in (1.5) satisfy η k ∈ (0 , K of a determinantalpoint process is (locally) admissible, then it must in fact be good, see [1, 4.2.21].The following example is the main motivation for discussing determinantal point processesin this paper. Let ( λ n , . . . , λ nn ) be the eigenvalues of the GUE (Gaussian unitary ensemble)of dimension n and denote by χ n the point process χ n ( D ) = P ni =1 { λ ni ∈ D } . Then χ n is adeterminantal point process with admissible, good kernel K ( n ) ( x, y ) = P n − k =0 Ψ k ( x )Ψ k ( y ), wherethe functions Ψ k are the oscillator wave-functions, that is Ψ k ( x ) := e − x / H k ( x ) √ √ πk ! , where H k ( x ) :=( − k e x / d k dx k e − x / is the k -th Hermite polynomial; see [1, Def. 3.2.1, Ex. 4.2.15].We will apply the following representation due to [15, Theorem 7]: Suppose χ is a deter-minantal process with good kernel K of the form (1.5), with P k η k < ∞ . Let ( I k ) nk =1 beindependent Bernoulli variables with P ( I k = 1) = η k . Set K I ( x, y ) = P nk =1 I k ϕ k ( x ) ϕ ∗ k ( y ) , andlet χ I denote the determinantal point process with random kernel K I . Then χ and χ I havethe same distribution, interpreted as stating that the mixture of determinental processes χ I has the same distribution as χ . In the following let K be a good kernel and for D ⊂ Λ wewrite K D ( x, y ) = 1 D ( x ) K ( x, y )1 D ( y ). Let D be such that K D is trace-class, with eigenvalues η k , k ≥ 1. Then χ ( D ) has the same distribution as P k ξ k where ξ k are independent Bernoullirandom variables with P ( ξ k = 1) = η k and P ( ξ k = 0) = 1 − η k . Theorem 1.4. Consider a sequence ( χ n ) n of determinantal point processes on Λ with goodkernels K n . Let D n be a sequence of measurable subsets of Λ such that ( K n ) D n is trace-class. ODERATE DEVIATIONS FOR WIGNER MATRICES 7 Assume that ( a n ) n is a sequence of real numbers such that ≪ a n ≪ (cid:0) n X k =1 η nk (1 − η nk ) (cid:1) / , where η nk are the eigenvalues of K n . Then ( Z n ) n with Z n := 1 a n χ n ( D n ) − E ( χ n ( D n )) p V ( χ n ( D n )) satisfies a moderate deviation principle with speed a n and rate function I ( x ) = x . Remark 1.5. In [16], functional moderate deviations for triangular arrays of certain indepen-dent, not identically distributed random variables are considered. Our result, Theorem 1.4,seem to follow from Proposition 1.9 in [16]. Anyhow we prefer to present a direct proof. Proof of Theorem 1.4. We adapt the proof of [1, Theorem 4.2.25]. We write K n for the kernel( K n ) D n and let S n := p V ( χ n ( D n )). χ n ( D n ) has the same distribution as the sum of inde-pendent Bernoulli variables ξ nk whose parameters η nk are the eigenvalues of K n . We obtain S n = P k η nk (1 − η nk ) and since K n is trace-class we can write, for any θ reallog E (cid:2) e θa n Z n (cid:3) = X k log E (cid:20) exp (cid:0) θa n ( ξ nk − η nk ) a n S n (cid:1)(cid:21) = − θa n P k η nk a n S n + X k log (cid:18) η nk (cid:0) e a n θ/ ( a n S n ) − (cid:1)(cid:19) . For any real θ and n large enough such that η nk ( e θa n /S n − ∈ [0 , 1] we apply Taylor for log(1+ x )and obtain 1 a n log E (cid:2) e θa n Z n (cid:3) = θ a n P k η nk (1 − η nk )2 a n S n + o (cid:18) a n P k η nk (1 − η nk ) a n S n (cid:19) . The last term is o (cid:0) a n S n (cid:1) . Applying the Theorem of G¨artner-Ellis, [7, Theorem 2.3.6], the resultfollows. (cid:3) Proof of Theorem 1.1. Now the first statement of Theorem 1.1 follows since V ( χ n ( D n )) → ∞ ,see [1, Cor. 4.2.27]. In particular for any I = [ y, ∞ ) with y ∈ ( − , Z n := N I ( W ′ n ) − E [ N I ( W ′ n )] a n p V ( N I ( W ′ n ))satisfies the MDP. The MDP with numerics (see Remark 1.2) follows, since the sequences ( ˜ Z n ) n and ( ˆ Z n ) n are exponentially equivalent in the sense of definition [7, Definition 4.2.10], and hencethe result follows from [7, Theorem 4.2.13]: Let Z ′′ n := N I ( W ′ n ) − n̺ sc ( I ) a n p V ( N I ( W ′ n )) . HANNA D ¨ORING AND PETER EICHELSBACHER Since | ˜ Z n − Z ′′ n | = (cid:12)(cid:12) E [ N I ( W ′ n )] − n̺ sc ( I ) a n √ V ( N I ( W ′ n )) (cid:12)(cid:12) → n → ∞ , ( ˜ Z n ) n and ( Z ′′ n ) n are exponentiallyequivalent. Moreover by Taylor we obtain | Z ′′ n − ˆ Z n | = o (1) a n N I ( W ′ n ) − n̺ sc ( I ) √ V ( N I ( W ′ n )) and the MDP for( Z ′′ n ) n implies lim sup n →∞ a n log P (cid:0) | Z ′′ n − ˆ Z n | > ε (cid:1) = −∞ for any ε > (cid:3) 2. Moderate deviations for eigenvalues in the bulk Under certain conditions on i it was proved in [14] that the i -th eigenvalue λ i of the GUE W ′ n satisfies a CLT. Consider t ( x ) ∈ [ − , 2] defined for x ∈ [0 , 1] by x = Z t ( x ) − d̺ sc ( t ) = 12 π Z t ( x ) − √ − x dx. Then for i = i ( n ) such that i/n → a ∈ (0 , 1) as n → ∞ (i.e. λ i is eigenvalue in the bulk), λ i ( W ′ n ) satisfies a CLT: r − t ( i/n ) λ i ( W ′ n ) − t ( i/n ) √ log nn → N (0 , 1) (2.1)for n → ∞ . Remark that t ( i/n ) is sometimes called the classical or expected location of the i -theigenvalue. The standard deviation is √ log nπ √ n̺ sc ( t ( i/n )) . Note that from the semicircular law,the factor n̺ sc ( t ( i/n )) is the mean eigenvalue spacing.The proof in [14] is achieved by the tight relation between eigenvalues and the countingfunction expressed by the elementary equivalence, for I ( y ) = [ y, ∞ ), y ∈ R , N I ( y ) ( W n ) ≤ n − i if and only if λ i ≤ y. (2.2)Hence the theorem due to Costin and Lebowitz as well as Soshnikov, see [18], can be applied.Moreover the proof in [14] relies on fine asymptotics for the Airy function and the Hermitepolynomials due to [5].Our first result in this Section is a corresponding MDP for λ i in the bulk: Theorem 2.1. Consider the GUE matrix W ′ n = √ n M ′ n . Consider i = i ( n ) such that i/n → a ∈ (0 , as n → ∞ . If λ i denotes the eigenvalue number i in the GUE matrix W ′ n it holds thatfor any sequence ( a n ) n of real numbers such that ≪ a n ≪ √ log n the sequence ( X ′ n ) n with X ′ n = r − t ( i/n ) λ i ( W ′ n ) − t ( i/n ) a n √ log nn satisfies a MDP with speed a n and rate function I ( x ) = x . Interesting enough, this result can be extended to large families of Hermitian Wigner matricessatisfying tail-condition ( T ) by means of the Four Moment Theorem of Tao and Vu: ODERATE DEVIATIONS FOR WIGNER MATRICES 9 Theorem 2.2. Consider a Hermitian Wigner matrix W n = √ n M n whose entries satisfy tail-condition ( T ) and match the corresponding entries of GUE up to order 4. Consider i = i ( n ) such that i/n → a ∈ (0 , as n → ∞ . If λ i denotes the eigenvalue number i of W n it holds thatfor any sequence ( a n ) n of real numbers such that ≪ a n ≪ √ log n , the sequence ( X n ) n with X n = r − t ( i/n ) λ i ( W n ) − t ( i/n ) a n √ log nn satisfies a MDP with speed a n and rate function I ( x ) = x . Remark 2.3. In [14] a CLT at the edge of the spectrum was considered also. The proof appliesthe result of Costin and Lebowitz as well as fine asymptotics presented in [5]. Consider i → ∞ such that i/n → n → ∞ and consider λ n − i , eigenvalue number n − i in the GUE orHermitian Wigner case. A CLT for the rescaled λ n − i is stated in [14, Theorem 1.2]. We wouldbe able to formulate and prove a MDP for eigenvalue λ n − i , but it is not the main focus of thispaper. We omit this. Proof of Theorem 2.1. The proof is oriented to the proof of [14, Theorem 1.1] and will apply theprecise asymptotic behaviour of the expectation and of the variance of the counting function N I ( W ′ n ), see (1.1), which is a reformulation of [14, Lemma 2.1-2.3]. Let P n denote the probabilityof the GUE determinantal point processes, and set I n := (cid:20) t ( i/n ) + ξ a n √ log nn √ p − t ( i/n ) , ∞ (cid:19) . Now we apply relation (2.2) and obtain P n (cid:18) λ i ( W ′ n ) − t ( i/n ) a n √ log nn √ √ − t ( i/n ) ≤ ξ (cid:19) = P n (cid:18) λ i ( W ′ n ) ≤ ξ a n √ log nn √ p − t ( i/n ) + t ( i/n ) (cid:19) = P n (cid:0) N I n ( W ′ n ) ≤ n − i (cid:1) = P n (cid:18) N I n ( W ′ n ) − E [ N I n ( W ′ n )] a n ( V ( N I n ( W ′ n ))) / ≤ n − i − E [ N I n ( W ′ n )] a n ( V ( N I n ( W ′ n ))) / (cid:19) . Since i/n → a ∈ (0 , t ( x ) we obtain t ( i/n ) ∈ ( − , a n ≪ √ log n we have ξ a n √ log nn √ √ − t ( i/n ) + t ( i/n ) ∈ ( − , 2) for n large. Therefore with (1.1)we have for n sufficiently large that E [ N I n ( W ′ n )] = n ̺ sc ( I n ) + O (cid:0) log nn (cid:1) and V ( N I n ( W ′ n )) = (cid:0) π + o (1) (cid:1) log n. With b ( n ) := a n √ log nn √ √ − t ( i/n ) and f n ( t ( i/n )) := t ( i/n ) + ξ b ( n ) we get from symmetry ̺ sc ( I n ) = Z ∞ f n ( t ( i/n )) ̺ sc ( x ) dx = 12 − Z f n ( t ( i/n ))0 ̺ sc ( x ) dx = 1 − Z f n ( t ( i/n )) − ̺ sc ( x ) dx. Now Z f n ( t ( i/n )) − ̺ sc ( x ) dx = Z t ( i/n ) − ̺ sc ( x ) dx + Z f n ( t ( i/n )) t ( i/n ) ̺ sc ( x ) dx = in + Z f n ( t ( i/n )) t ( i/n ) ̺ sc ( x ) dx and Z f n ( t ( i/n )) t ( i/n ) ̺ sc ( x ) dx = ξb ( n ) 12 π p − t ( i/n ) + O (cid:0) b ( n ) ) . Summarizing we obtain n ̺ sc ( I n ) = n − i − ξ a n p log n √ π + O (cid:0) a n log nn (cid:1) and therefore n − i − E [ N I n ( W ′ n )] a n ( V ( N I n ( W ′ n ))) / = ξ + ε ( n ) , where ε ( n ) → n → ∞ . By Theorem 1.1 we obtain for every ξ < n →∞ a n log P n (cid:18) λ i ( W ′ n ) − t ( i/n ) a n √ log nn √ √ − t ( i/n ) ≤ ξ (cid:19) = − ξ . (2.3)With P n (cid:18) λ i ( W ′ n ) − t ( i/n ) a n √ log nn √ √ − t ( i/n ) ≥ ξ (cid:19) = P n (cid:0) N I n ( W ′ n ) ≥ n − i + 1 (cid:1) the same calculations lead, for every ξ > 0, tolim n →∞ a n log P n (cid:18) λ i ( W ′ n ) − t ( i/n ) a n √ log nn √ √ − t ( i/n ) ≥ ξ (cid:19) = − ξ . (2.4)Hence the conclusion follows: see for example [7, Proof of Theorem 2.2.3]. To be more precisewe apply the preceding results (2.3) and (2.4) with Theorem 4.1.11 in [7], which allows us toderive a MDP from the limiting behaviour of probabilities for a basis of topology. For thelatter, we choose all open intervals ( a, b ), where at least one of the endpoints is finite and wherenone of the endpoints is zero. Denote the family of such intervals by U . From (2.3) and (2.4),it follows for each U = ( a, b ) ∈ U , L U := lim n →∞ a n log P (cid:0) X ′ n ∈ U (cid:1) = b / a < b < 00 : a < < ba / < a < b By [7, Theorem 4.1.11], ( X ′ n ) n satisfies a weak MDP with speed a n and rate function t sup U ∈U ; t ∈ U L U = t . With (2.4), it follows that ( X ′ n ) n is exponentially tight, hence by Lemma 1.2.18 in [7], ( X ′ n ) n satisfies the MDP with the same speed and the same good rate function. This completes theproof. (cid:3) ODERATE DEVIATIONS FOR WIGNER MATRICES 11 To extend the result of Theorem 2.1 to Hermitian Wigner matrices satisfying tail-condition( T ), we will apply the Four Moment Theorem (for the bulk of the spectra of Wigner matrices),see [21, Theorem 15]. Theorem 2.4 (Four Moment Theorem due to Tao and Vu) . There is a small positive constant c such that for every < ε < and k ≥ the following holds. Let M n and M ′ n be twoHermitian Wigner matrices satisfying tail-condition ( T ) . Assume furthermore that for any ≤ i < j ≤ n , Z ij and Z ′ ij match to order 4 and for any ≤ i ≤ n , Y i and Y ′ i match toorder 2. Set A n := √ nM n and A ′ n := √ nM ′ n , and let G : R k → R be a smooth functionobeying the derivative bounds |∇ j G ( x ) | ≤ n c for all ≤ j ≤ and x ∈ R k . Then for any εn ≤ i < i · · · < i k ≤ (1 − ε ) n , and for n sufficiently large depending on ε , k and the constants C, C ′ in tail-condition ( T ) , we have | E (cid:0) G ( λ i ( A n ) , . . . , λ i k ( A n )) (cid:1) − E (cid:0) G ( λ i ( A ′ n ) , . . . , λ i k ( A ′ n )) (cid:1) | ≤ n − c . (2.5)Applying this Theorem for the special case when M ′ n is GUE, one obtains [21, Corollary 18]: Corollary 2.5. Let M n be a Hermitian Wigner matrix whose atom distribution ξ satisfies E ξ = 0 and E ξ = and tail-condition ( T ) , and M ′ n be a random matrix sampled from GUE.Then with G, A n , A ′ n as in the previous theorem, and n sufficiently large, one has | E (cid:0) G ( λ i ( A n ) , . . . , λ i k ( A n )) (cid:1) − E (cid:0) G ( λ i ( A ′ n ) , . . . , λ i k ( A ′ n )) (cid:1) | ≤ n − c . (2.6)Now the universality of the MDP in Theorem 2.2 follows along the lines of the proof of [21,Corollary 21]. Proof of Theorem 2.2. Let M n be a Hermitian Wigner matrix whose entries satisfy tail-condition ( T ) and match the corresponding entries of GUE up to order 4. Let i, a and t ( · )be as in the statement of the Theorem, and let c be as in Corollary 2.5. Then [21, (18)] saysthat P n (cid:0) λ i ( A ′ n ) ∈ I − (cid:1) − n − c ≤ P n (cid:0) λ i ( A n ) ∈ I (cid:1) ≤ P n (cid:0) λ i ( A ′ n ) ∈ I + (cid:1) + n − c (2.7)for all intervals [ b, c ], and n sufficiently large depending on i and the constants C, C ′ of tail-condition ( T ). Here I + := [ b − n − c / , c + n − c / ] and I − := [ b + n − c / , c − n − c / ]. Wepresent the argument of proof of (2.7) just to make the presentation more self-contained. Onecan find a smooth bump function G : R → R + which is equal to one on the smaller interval I − and vanishes outside the larger interval I + . It follows that P n (cid:0) λ i ( A n ) ∈ I (cid:1) ≤ E G ( λ i ( A n )) and E G ( λ i ( A ′ n )) ≤ P n (cid:0) λ i ( A ′ n ) ∈ I (cid:1) . One can choose G to obey the condition |∇ j G ( x ) | ≤ n c for j = 0 , . . . , | E G ( λ i ( A n )) − E G ( λ i ( A ′ n )) | ≤ n − c . Therefore the second inequality in (2.7) follows from the triangle inequality. The first inequalityis proven similarly. Now for n sufficiently large we consider the interval I n := [ b n , c n ] with b n := b a n p log n √ p − t ( i/n ) + nt ( i/n ) and c n := c a n p log n √ p − t ( i/n ) + nt ( i/n )with b, c ∈ R , b ≤ c . Then for X n defined as in the statement of the Theorem we have P n (cid:0) X n ∈ [ b, c ] (cid:1) = P n (cid:18) λ i ( A n ) − nt ( i/n ) a n √ log n √ √ − t ( i/n ) ∈ [ b, c ] (cid:19) = P n (cid:0) λ i ( A n ) ∈ I n (cid:1) . With (2.7) and [7, Lemma 1.2.15] we obtainlim sup n →∞ a n log P n (cid:0) X n ∈ [ b, c ] (cid:1) ≤ max (cid:18) lim sup n →∞ a n log P n (cid:0) λ i ( A ′ n ) ∈ ( I n ) + (cid:1) ; lim sup n →∞ a n log n − c (cid:19) . For the first object we havelim sup n →∞ a n log P n (cid:0) λ i ( A ′ n ) ∈ ( I n ) + (cid:1) = lim sup n →∞ a n log P n (cid:18) λ i ( A ′ n ) − nt ( i/n ) a n √ log n √ √ − t ( i/n ) ∈ [ b − η ( n ) , c + η ( n )] (cid:19) with η ( n ) = n − c / (cid:0) a n √ log n √ √ − t ( i/n ) (cid:1) − → n → ∞ . Since c > n/a n → ∞ for n → ∞ by assumption, applying Theorem 2.1 we havelim sup n →∞ a n log P n (cid:0) X n ∈ [ b, c ] (cid:1) ≤ − inf x ∈ [ b,c ] x . Applying the first inequality in (2.7) in the same manner we also obtain the lower boundlim inf n →∞ a n log P n (cid:0) X n ∈ [ b, c ] (cid:1) ≥ − inf x ∈ [ b,c ] x . Finally the argument in the last part of the proof of Theorem 2.1 can be repeated to obtainthe MDP for ( X n ) n . (cid:3) 3. Universality of the Moderate deviations In this section we proof Theorem 1.3. As announced, we will show that the MDP behaviourof eigenvalues in the bulk of the GUE (Theorem 2.1) extended to Hermitian Wigner matrices(Theorem 2.2) leads to the MDP with numerics for the counting function of eigenvalues ofHermitian Wigner matrices. Proof of Theorem 1.3. For every ξ ∈ R we obtain that P n (cid:0) ˆ Z n ≤ ξ (cid:1) = P n (cid:0) N I ( y ) ( W n ) ≤ n − i n (cid:1) with i n := n̺ sc (( −∞ , y ]) − ξ a n q π log n . Hence using (2.2) it follows P n (cid:0) ˆ Z n ≤ ξ (cid:1) = P n (cid:0) λ i n ≤ y (cid:1) = P n (cid:18)r − t ( i n /n ) λ i n ( W n ) − t ( i n /n ) a n √ log nn ≤ ξ n (cid:19) ODERATE DEVIATIONS FOR WIGNER MATRICES 13 with ξ n := q − t ( i n /n ) y − t ( i n /n ) a n √ log nn . Now i n n = ̺ sc (( −∞ , y ]) − ξa n q π log nn → ̺ sc (( −∞ , y ]) ∈ (0 , n → ∞ . We will prove that ξ n = ξ + o (1). Applying Theorem 2.2, it follows thatlim n →∞ a n log P n (cid:0) ˆ Z n ≤ ξ (cid:1) = − ξ ξ < 0. With P n (cid:0) ˆ Z n ≥ ξ (cid:1) = P n (cid:0) N I ( y ) ( W n ) ≥ n − i n (cid:1) = P n (cid:0) λ i n +1 ≥ y (cid:1) the samecalculations will lead, for any ξ > 0, tolim n →∞ a n log P n (cid:0) ˆ Z n ≥ ξ (cid:1) = − ξ . The MDP for ( ˆ Z n ) n (the MDP with numerics) now follows along the lines of the proof ofTheorem 2.1 (topological argument).The MDP for ( Z n ) n follows by the arguments given in the proof of Theorem 1.1, using thedeep fact that the expectation E [ Y n ] and the variance V ( Y n ) of the eigenvalue counting functionhave identical behaviours to the ones for GUE matrices: E [ Y n ] = n̺ sc ( I ( y )) + o (1) and V ( Y n ) = (cid:0) π + o (1) (cid:1) log n. This result is established in [4, Theorem 2], applying strong localization of the eigenvalues ofWigner matrices, a recent result from [12].Finally we prove that lim n →∞ ξ n = ξ . We obtain t ( i n /n ) = t (cid:18) ̺ sc (( −∞ , y ]) − n ξ a n r π log n (cid:19) = t (cid:0) ̺ sc (( −∞ , y ]) (cid:1) − n ξ a n r π log n t ′ (cid:0) ̺ sc (( −∞ , y ]) (cid:1) + o (cid:0) a n √ log nn (cid:1) = y − ξ n a n p log n √ p − y + o (cid:0) a n √ log nn (cid:1) . Hence y − t ( i n /n ) a n √ lognn = ξ √ √ − y + o (1) and with lim n →∞ q − t ( i n /n ) = q − y it follows thatlim n →∞ ξ n = ξ . (cid:3) 4. Symmetric Wigner matrices and the GSE In this section, we indicate how the preceding results for Hermitian Wigner matrices canbe stated and proved for real Wigner symmetric matrices. Moreover we consider a Gaussiansymplectic ensemble (GSE). Real Wigner matrices are random symmetric matrices M n of size n such that, for i < j , ( M n ) ij are i.i.d. with mean zero and variance one, ( M n ) ii are i.i.d. with mean zero and variance 2. As already mentioned, the case where the entries are Gaussian isthe GOE. As in Section 1 and 2, the main issue is to establish our conclusions for the GOE.On the level of CLT, this was developed in [17] by means of the famous interlacing formulas due to Forrester and Rains, [13], that relates the eigenvalues of different matrix ensembles. Thefollowing relation holds between matrix ensembles:GUE n = even (cid:0) GOE n ∪ GOE n+1 (cid:1) . (4.1)The statement is: Take two independent (!) matrices from the GOE: one of size n × n andone of size ( n + 1) × ( n + 1). Superimpose the 2 n + 1 eigenvalues on the real line and then takethe n even ones. They have the same distribution as the eigenvalues of a n × n matrix fromthe GUE. If M R n denotes a GOE matrix and W R n := √ n M R n , first we will prove a MDP for Z R n := N I n ( W R n ) − E [ N I n ( W R n )] a n p V ( N I n ( W R n )) (4.2)for any 1 ≪ a n ≪ p V ( N I n ( W R n )), I n an interval in R , with speed a n and rate x / 2. Let withinthis section M C n denote a GUE matrix and W C n the corresponding normalized matrix. The niceconsequences of (4.1) were already suitably developed in [17]: applying Cauchy’s interlacingtheorem one can write N I n ( W C n ) = 12 (cid:2) N I n ( W R n ) + N ′ I n ( W R n ) + η ′ n ( I n ) (cid:3) , (4.3)where one obtains GOE ′ n in N ′ I n ( W R n ) from GOE n +1 by considering the principle sub-matrixof GOE n +1 and η ′ n ( I n ) takes values in {− , − , , , } . Note that N I n ( W R n ) and N ′ I n ( W R n )are independent because GOE n +1 and GOE n denote independent matrices from the GOE. Weobtain Z C n := N I n ( W C n ) − E [ N I n ( W C n )] a n p V ( N I n ( W C n )) = N I n ( W R n ) − E [ N I n ( W R n )] a n p V ( N I n ( W C n )) + N ′ I n ( W R n ) − E [ N ′ I n ( W R n )] a n p V ( N I n ( W C n ))+ η ′ n ( I n ) − E [ η ′ n ( I n )] a n p V ( N I n ( W C n )) =: X n + Y n + ε n (4.4)Now we can make use of the MDP for ( Z C n ) n , Theorem 1.1. Using the independence of X n and Y n in (4.4), as well as the fact that the third summand can be estimated by | ε n | ≤ a n √ V ( N In ( W C n )) , we obtain for every θ − | θ | a n p V ( N I n ( W C n )) + 2 1 a n log E e θa n X n ≤ a n log E e θa n Z n ≤ | θ | a n p V ( N I n ( W C n )) + 2 1 a n log E e θa n X n . Applying the Theorem of G¨artner-Ellis, [7, Theorem 2.3.6], the MDP for ( X n ) n with speed a n and rate x follows for all ( I n ) n with V ( N I n ( W C n )) → ∞ . Hence we have proved theMDP-version of [17, Lemma 2], that (cid:0) N In ( W R n ) − E [ N In ( W R n )] a n √ √ V ( N In ( W C n )) (cid:1) n satisfies an MDP with rate x / V ( N I n ( W C n )) → ∞ . The interlacing formula (4.3) leads to 2 V ( N I n ( W C n )) + O (1) = V ( N I n ( W R n ))if V ( N I n ( W C n )) → ∞ . Therefore ( Z R n ) n satisfies the MDP with speed a n and rate function x / ODERATE DEVIATIONS FOR WIGNER MATRICES 15 The proof of Theorem 2.1 can simply be adapted to the GOE. Since the Four Moment Theoremalso applies for real symmetric matrices, with an analog of [4, Lemma 5] in hand we obtain: Theorem 4.1. Consider a real symmetric Wigner matrix W n = √ n M n whose entries satisfytail-condition ( T ) and match the corresponding entries of GOE up to order 4. (1) Consider i = i ( n ) such that i/n → a ∈ (0 , as n → ∞ . Denote the eigenvalue number i of W n by λ i . Let ( a n ) n be a sequence of real numbers such that ≪ a n ≪ √ log n . Thenthe sequence ( X n ) n with X n := λ i − t ( i/n ) a n √ log nn √ − t ( i/n )2 universally satisfies a MDP with speed a n andrate function I ( x ) = x . (2) For any y ∈ ( − , and I ( y ) = [ y, ∞ ) , the rescaled eigenvaluecounting function N I ( y ) ( W n ) universally satisfies the MDP as in Theorem 1.3. Finally we consider the Gaussian Symplectic Ensemble (GSE). Here the following relationholds between matrix ensembles: GSE n = even (cid:0) GOE n +1 (cid:1) √ . The multiplication by √ denotesscaling the (2 n + 1) × (2 n + 1) GOE matrix by the factor √ . Let x < x < · · · < x n denotethe ordered eigenvalues of an n × n matrix from the GSE and let y < y < · · · < y n +1 denotethe ordered eigenvalues of an (2 n + 1) × (2 n + 1) matrix from the GOE. Then it follows that x i = y i / √ i -th eigenvalue of the GSE follows directlyfrom the GOE case. Theorem 4.2. Consider the GSE matrix W H n = √ n M H n . Consider i = i ( n ) such that i/n → a ∈ (0 , as n → ∞ . If λ i denotes the eigenvalue number i in the GSE matrix, it holds thatfor any sequence ( a n ) n of real numbers such that ≪ a n ≪ √ log n the sequence p − t ( i/n ) λ i ( W H n ) − t ( i/n ) a n √ log nn satisfies a MDP with speed a n and rate function I ( x ) = x . For any y ∈ ( − , and I ( y ) =[ y, ∞ ) , the rescaled eigenvalue counting function N I ( y ) ( W H n ) satisfies the MDP as in Theorem1.1. 5. Moderate deviations for covariance matrices In this section we briefly present the analogous results for covariance matrices. They relyon [19], where Gaussian fluctuations of individual eigenvalues in complex sample covariancematrices were considered, as well as on the Four Moment Theorem for random covariancematrices due to Tao and Vu ([22]). Let p = p ( n ) and n be integers such that p ≥ n andlim n →∞ pn = γ ∈ [1 , ∞ ). Let X be a random p × n matrix with complex entries X ij suchthat they are identically distributed and independent, have mean zero and variance 1. Assumemoreover that the following condition is fulfilled: Moment-condition ( M ): X satisfies moment-condition ( M ), if there exists C ≥ C > ij E [ | X ij | C ] ≤ C . Then W := W p,n := n X ∗ X is called covariance matrix . Hence it has at most p non zeroeigenvalues, which are real and nonnegative, denoted by 0 ≤ λ ( W ) ≤ · · · ≤ λ p ( W ). Weabbreviate λ i ( W ) as λ i . If the entries are Gaussian random variables, W p,n is called LaguerreUnitary Ensemble (LUE) or complex Gaussian Wishart ensemble. LUE matrices will be denotedby W ′ = W ′ p,n . In this case, the eigenvalues form a determinantal point process with admissible,good kernel given in terms of Laguerre polynomials; see for example [19]. With Theorem 1.4 weobtain a MDP for the counting function N I n ( W ′ p,n ) for intervals I n with V ( N I n ( W ′ p,n )) → ∞ for n → ∞ . The classical Marchenko-Pastur theorem states that as n → ∞ such that pn → γ ≥ n P ni =1 δ λ i → µ γ in distribution, where µ γ is the Marchenko-Pastur law withdensity dµ γ ( x ) = πx p ( x − α )( β − x )1 [ α,β ] ( x ) dx , where α = ( √ γ − and β = ( √ γ + 1) .Let α p,n := (cid:18)r pn − (cid:19) , β p,n := (cid:18)r pn + 1 (cid:19) and I n := [ t n , ∞ ) with t n ≤ β n,p − δ for some δ > 0. Then with [19, Lemma 3], the variance ofthe number of eigenvalues of W ′ p,n in I n satisfies V ( N I n ( W ′ p,n )) = 12 π log n (1 + o (1)) . (5.1)Moreover with [19, Lemma 1], the expected number of eigenvalues of W ′ p,n in I n = [ t n , ∞ ) with t n → t ∈ ( α, β ) satisfies E [ N I n ( W ′ p,n )] = n Z β p,n t n µ p,n ( x ) dx (1 + o (1)) , (5.2)where µ p,n ( x ) := πx p ( x − α p,n )( β p,n − x )1 [ α p,n ,β p,n ] ( x ). The proof of Theorem 1.4 leads to anMDP (with numerics) for (cid:18) N I n ( W ′ p,n ) − n R β p,n t n µ p,n ( x ) dxa n q π log n (cid:19) n for every I n = [ t n , ∞ ) with t n → t ∈ ( α, β ) and all 1 ≪ a n ≪ √ log n . Let G ( t ) := Z tα p,n µ p,n ( x ) dx for α p,n ≤ t ≤ β p,n . Arguing as in Section 2, the following MDP can be achieved: Theorem 5.1. Consider the LUE matrix W ′ p,n . Let t := t n,i = G − ( i/n ) with i = i ( n ) suchthat in → a ∈ (0 , as n → ∞ . Then as pn → γ ≥ the sequence ( X n ) n with X n := √ πµ p,n ( t )( λ i − t ) a n √ log nn (5.3) satisfies an MDP for any ≪ a n ≪ √ log n with rate x / . ODERATE DEVIATIONS FOR WIGNER MATRICES 17 Proof. Along the lines of the proof of Theorem 2.1, the main step is to calculate E (cid:2) N I n ( W ′ p,n ) (cid:3) for I n = [ t + ξb n , ∞ ) with b n = a n √ log nn π √ µ p,n ( t ) . Using the Taylor expansion for R · t + ξb n µ p,n ( x ) dx we obtain E (cid:2) N I n ( W ′ p,n ) (cid:3) = n Z β p,n t + ξb n µ p,n ( x ) dx (1 + o (1)) = n − i − ξa n p log n √ π + o (1) . The statement follows step by step along the proof of Theorem 2.1. (cid:3) As was done for Wigner matrices, one can extend the last Theorem to general covariancematrices W p,n whose entries satisfy the moment-condition ( M ) and match the correspondingentries of LUE up to order 4. Namely, Tao and Vu extended their Four Moment Theorem tothe case of covariance matrices in [22, Theorem 6]. We apply it in the same way as for Wignermatrices. Finally we end up with the following universality result: Theorem 5.2. Let W p,n be a covariance matrix whose entries satisfy moment-condition ( M ) and match the corresponding entries of LUE up to order 4. Then, for any I n = [ t n , ∞ ) with t n → t ∈ ( α, β ) and all ≪ a n ≪ √ log n the sequence ( ˆ Z n ) n with ˆ Z n = N I n ( W p,n ) − n R β p,n t n µ p,n ( x ) dxa n q π log n satisfies a MDP (with numerics) with speed a n and rate function I ( x ) = x . Remark 5.3. Since a version of the Erd¨os-Yau-Yin rigidity theorem for covariance matricesis not yet proved, the MDP “without numerics” for the eigenvalue counting function for non-Gaussian covariance matrices is not stated. Proof. For every ξ ∈ R we obtain with X n defined in (5.3) (where λ i = λ i ( W p,n )) that P n (cid:0) ˆ Z n ≤ ξ (cid:1) = P n (cid:0) X n ≤ ξ n (cid:1) with ξ n = √ πµ p,n ( G − ( i n /n )) t n − G − ( i n /n ) a n √ log nn and i n := n R t n α p,n µ p,n ( x ) dx − ξa n q π log n . Now i n /n → µ γ ( −∞ , t ]) ∈ (0 , t ∈ ( α, β ).Moreover Taylor expansion leads to G − ( i n /n ) = t n − ξ n a n p log n √ πµ p,n ( t n ) + o (1) . Hence √ πµ p,n ( t n ) t n − G − ( i n /n ) a n √ log nn = ξ + o (1)and we established the result. (cid:3) Real covariance matrices can be considered as well. The first step would be to establishour conclusions for the LOE, the Laguerre Orthogonal Ensemble. This can be done applying interlacing formulas , that relates the eigenvalues of LUE and LOE matrices. Forrester andRains proved in [13] the following relation: LUE p,n = even (cid:0) LOE p,n ∪ LOE p +1 ,n +1 (cid:1) . 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