Local semicircle law, Wegner estimate and level repulsion for Wigner random matrices László Erdős University of Munich Oberwolfach, 2008 Dec Joint work with H.T. Yau (Harvard), B. Schlein (Cambrigde) Goal: Spectral statistics of large Wigner matrices, i.e. random hermitian matrices with i.i.d. entries
WIGNER ENSEMBLE H = (h jk ) j,k N is a hermitian N N matrix, N. h jk = N (x jk + iy jk ), (j < k), h jj = 2 N x jj where x jk, y jk (j < k) and x jj are independent with distributions x jk, y jk dν := e g(x) dx, Normalization: E x jk = 0, E x 2 jk = 2. Example: g(x) = x 2 is Gaussian Unitary Ensemble (GUE) Scaling ensures that the evalues µ µ 2... µ N remain finite E N α= µ 2 α = ETr H2 = N j,k= and actually Spec(H) = [ 2, 2] + o(). h jk 2 = N 2
ρ (x) = 4 x 2 2π 2 2 Eigenvalues: µ µ 2...... µ N Typical eigenvalue spacing is µ i+ µ i N. 3
MAIN QUESTIONS ) Density of states (DOS): one-point correlation function. lim lim {µ η 0 N Nη # j [E η2, E + η2 } ] = sc (E) sc (E) := 4 E 2 ( E 2) Wigner semicircle law 2π 2) Eigenvalue spacing in the bulk: n-point correlation function ( K (n) E+ µ N sc (E),..., E+ µ ) n det [ S(µ i µ j ) ] as N S(µ) = sin(πµ) πµ N sc (E) Wigner-Dyson sine-kernel In particular, this implies level repulsion (in the bulk) P (N [ ] ) µ j+ µ j [x, x + dx] x 2 dx, x 3) (De)localization properties of eigenvectors. 4
UNIVERSALITY CONJECTURES ) Local eigenvalue statistics depends only on the symmetry class of the random matrix. Hermitian matrices follow the Wigner- Dyson sine kernel, similar universal laws are known for symmetric and symplectic matrices. 2) Level repulsion occurs together with delocalization. In fact, level statistics can be used as a signature for the delocalized regime (used in numerics). Motivation in background: The level repulsion statistics for random Schrödinger operators (in a big finite domain) distinguishes between Anderson localization and delocalization. Other motivations: nuclear physics, quantum chaos, Riemann ζ-function. 5
KNOWN RESULTS FOR UNITARY ENSEMBLES Unitary ensemble: hermitian matrices with density P(H) e Tr V (H) where V (x) x 2 i) Invariant under H UHU for any unitary U ii) V (x) = x 2 [GUE] is the only ensemble with iid entries Joint density function of the eigenvalues is explicitly known p(µ,..., µ n ) = const. (µ i µ j ) 2 e j V (µ j) i<j Local eigenvalue statistics follow Wigner-Dyson sine kernel [Dyson] [Pastur-Shcherbina] [Deift] Eigenvectors are delocalized: v j 4 4 = O(N ) (if v j 2 = ) [unitary symmetry] (Similar results hold for symmetric and symplectic ensembles) 6
KNOWN RESULTS FOR WIGNER MATRICES Semicircle law on the macroscopic scale [Wigner, 955] Universality at the edge [Soshnikov, 999] Tracy-Widom distribution for the largest eigenvalue lim N P( N 2/3 (µ N 2) t ) = F 2 (t) and determinental process with Airy-kernel for the k largest eigenvalues. Wigner-Dyson sine kernel is proven in the bulk if the entries have Gaussian component [Johansson, 200] OPEN: Wigner-Dyson universality in the bulk 7
OUR RESULTS FOR WIGNER MATRICES Wigner semicircle law holds on the smallest possible O(N ) scale. Eigenfunctions are fully delocalized Tail of the ev gap distribution is at least subexponential Averaged density of state is bounded on any scale (Wegner estimate) Level repulsion for Wigner matrices is at least as strong as for GUE. (Indicates sine-kernel) 8
WIGNER SEMICIRCLE LAW FOR THE DENSITY OF STATES N(I) := #{µ n I} no. of evalues in I R. Then for any δ > 0 P N lim lim η 0 N(E η 2, E + η 2 ) Nη sc (E) δ = 0 I.e. the empirical counting measure, N(E) = N converges weakly to sc (E)dE in probability: f(e) N(E)dE f(e) sc (E)dE j δ(µ j E), for any fixed f. This is LLN for O(N) ev s. What if the scale of the testfunction η depends on N, η = N α? LLN should still hold as long as Nη. Note that Wigner-Dyson law is an information on the finest scale η = O(N ). 9
Theorem : Let the single entry distribution dν be rotationally symmetric (or real and imaginary parts i.i.d) with a Gaussian tail, E ν exp δ X 2 <. Fix κ, ε > 0 and E [ 2+κ,2 κ]. Then there is a K = K(κ, ε), s.t. P N[E η 2, E + η 2 ] Nη sc (E) ε for all N, η such that Nη K(κ, ε), N N(ε). e cε Nη = semicircle law holds for energy windows /N. Smallest possible scale (improves our earlier work) LDE rate is not optimal (fluctuation should be (Nη) ) Earlier work [Khorunzhy] [Bai-Miao-Tsay] [Guionnet-Zeitouni] work on scales at least N /2 η. 0
UPPER BOUND FOR THE DENSITY OF STATES Lemma: Let I = η /N, then for all suff. large K, we have P{N(I) KN I } e ckn I Proof: Let I = [E η 2, E + η 2 ], z = E + iη and G z = (H z). N(I) Nη 2 N ImTr(H E iη) = 2 N N k= Im G z (k, k) To estimate G z (,), decompose ( ) h a H =, h C, a C a N, B C (N ) (N ) B Let λ α,u α be the ev s of B, thus G z (,) = h z a (B z) a = h z N α= a u α 2 λ α z
Introducing we have G z (,) ξ α := N a u α 2, with E ξ α = Im[ h z N α ξ ] α λ α z (Note we used only the imaginary part!) + N α η ξ α (λ α E) 2 +η 2 Nη α : λ α I ξ α Need a lower bound on α A ξ α with A := {α : λ α I}. The ξ α s are not independent, but almost. FACT : For the decomposition H = of H and λ α of B are interlaced: ( h a a B ), the eigenvalues µ α µ λ µ 2 λ 2... = N(I) A N(I) + 2
LEMMA [ Small deviation for α A ξ α ] P A α A ξ α δ e c A moreover, if δ N ν, A N β, ν + 8β <, then P A α A ξ α δ (Cδ) A Thus α A ξ α c A cn(i) with high probability, therefore N(I) Nη 2 N k G z (k, k) CNη N(I) = N(I) CNη The actual proof uses a high moment Chebyshev to avoid a log N factor for the bad exceptional sets for each k. 3
Proof of a weaker small deviation bound P A α A ξ α δ e c A [In our earlier proofs we used the bound e c A that requires more assumption on the single site distribution] Theorem [Hanson-Wright] (Thanks to M. Ledoux) Let b j a sequence of complex i.i.d. such that either both real and imaginary parts are i.i.d. or the distribution is rotationally symmetric. Assume E exp(δ 0 b 2 ) <. Let d jk C, and define X = N j,k= then there is c = c(δ 0 ) > 0 such that where D 2 := jk d jk 2. d jk [ bj b k E b j b k ] P( X δ) 4exp( cmin{δ/d, δ 2 /D 2 }) 4
X = N j,k= d jk [ bj b k E b j b k ], D 2 = d jk 2 jk P( X δ) 4exp( cmin{δ/d, δ 2 /D 2 }) Let (b, b 2,... b N ) i.i.d with distribution dν, then α A ξ α = α A u α b 2 = jk [ ū α (j)u α (k) ] } α {{ } =:d jk and since E ξ α =, we have α ξ α = X + A. Thus P ( α ξ α δ A ) P ( X A 2 since, by orthonormality of u α we have ) b j b k Ce c A /D = Ce c A D 2 = jk d jk 2 = N α,β A j,k= ū α (j)u α (k)ū β (j)u β (k) = A 5
PROOF OF THE LOCAL SEMICIRCLE LAW m(z) = N Tr (x)dx H z = x z Stieltjes transform Here = N is the empirical counting measure. Note that η(e) = Im m(e + iη) is the measure smoothed on scale η. The Stieltjes tr. of the semicircle law satisfies m sc (z) + m sc (z) + z = 0 m sc(z) = z 2 z 2 4 This fixed point equation is stable away from the spectral edge. We show that m(z) approximately satisfies this equation, thus it is close to m sc (z). 6
Let m (k) (z) be the Stieltjes tr. for the minor B (k) : m(z) = N Tr H z, m(k) (z) = N Tr B (k) z Then from the expansion obtain with m(z) = N X k = a (k) N k= m = N G z (k, k) = N N k= B (k) z a(k) E k N k= h kk z a (k) h kk z ( N ) m (k) X k [ a (k) B (k) z a(k) } {{ } =( N )m(k) ] = N B (k) z a(k) N α= ξ (k) α λ (k) α z (recall ξ (k) α = N a (k) u (k) α 2, E k ξ (k) α = ) 7
m = N N k= h kk z ( N ) m (k) X k (i) P{h kk ε} e δ 0ε 2 N (ii) By interlacing property m ( N ) m (k) = o() (iii) P{ X k ε} e cε Nη [Proof: Hanson-Wright + upper bound on the density] Then, away from an event of tiny prob, we have m = N N k= m + z + δ k where the random variables δ k satisfy δ k ε. From stability of the equation m sc = m sc +z, we get m m sc Cε. 8
EXTENDED STATES: EIGENVECTOR DELOCALIZATION No concept of absolutely continuous spectrum. v C N, v 2 = is extended if v p N p 2, p 2. E.g. For GUE, all eigenvectors have v 4 N /4 (symmetry) Question: in general for Wigner? [T. Spencer] Theorem 2: Fix κ > 0 and I [ 2 + κ,2 κ] with I = /N, then for p < P v, v 2 =, Hv = µv, µ I, v p MN p 2 The same result holds for M (log N) 2 with p =. e c M 9
Proof. Decompose as before H = ( h a a B Let Hv = µv and v = (v,w), w C N. Then hv +a w = µv, av +Bw = µw = w = (µ B) av From the normalization, = w 2 + v 2, we have v 2 = + N α ξ α (µ λ α ) 2 where recall λ α,u α are the ev s of B and let N ),, (ξ α := N a u α 2 ) (q/n) 2 α A ξ α A = { α : λ α µ = q q (log N) 4 N Concentration ineq. and lower bound on the local DOS imply α A with very high probability, thus } ξ α c A cq v 2 q N = v N /2 modulo logs 20
TAIL OF THE GAP DISTRIBUTION Theorem 3. Let E [ 2 + κ,2 κ] and µ α be the largest eigenvalue below E. Then P (µ α+ E KN ), α N Ce c K Remark: Expected decay is Gaussian. Proof: Too big gap would mean that the local density of states in the middle of the gap is small, contradicting the lower bound on the density of states (local semicircle law) 2
WEGNER ESTIMATE Theorem 4. Assume the single site distribution dν is smooth ˆν(t) ( + Ct 2 ) 5, t Let E 2 κ, I ε = [E 2N ε, E + 2N ε ], then P(N Iε ) Cε unif. in ε, E, N = averaged density of states E N(E) is an ac measure with uniformly bounded density sup E 2 κ sup E N(E) C N N 0 Remarks: i) On scales η /N, the empirical density wildly fluctuates, but its expectation is bounded. No sticking to E. ii) Unlike for the local semicircle law, we need smoothness. 22
Proof of Wegner: From the basic inequality with η = ε/n by Chebyshev N Iε ε N N j= P(N Iε ) Cε 2 E Im Im h jj z N h z N α ξ(j) α λ (j) α z α ξ() λ () α z 2 Cε 2 E λ,h E a ( α c α ξ α ) 2 + (h E α d α ξ α ) 2 ξ α = N u α a 2, c α = ε N 2 (λ α E) 2 + ε 2, d α = N(λ α E) N 2 (λ α E) 2 + ε 2 Local semicircle law implies that there are c α ε and d α. Use two such c s and two other such d s: 23
E a ( α c α ξ α ) 2 + (h E α d α ξ α ) 2 sup E E a ) 2 (εξ α + εξ α2 ) 2 + (E ξ β ξ β2 Think of ξ α s as squares of independent complex Gaussians. sup E 4j= e z j 2 dz j dz j ( ε z 2 + ε z 2 2) 2 + ( E z3 2 z 4 2) 2 2 j= e z j 2 dzj dz j ε ( z 2 + z 2 2) Cε Note that the ε 2 singularity is mollified to ε by integrating out the second term coming from the real part: s 3 e s2 ds (εr 2 ) 2 + (E s 2 ) 2 r3 e r2 dr εr 2 r3 e r2 dr The main technical lemma assures that this intuition is correct: 24
LEMMA: Let u,...u N an ONB, set ξ α = b u α where the components of b = (b,... b N ) are i.i.d. with smoothness of order p+3. Let α,... α p, β, β 2 N distinct indices, then for any < r < p + we have E b ( p j= ) 2+ N c j ξ αj (E α= d α ξ α ) 2 r/2 C(p, r) [ p c j]r p min{d β, d β2 } [The estimate is the same as with i.i.d. Gaussian assumption] It is not LLN, not CLT, not LDE but local smoothness. Proof: Change variables, integrate by parts along a vectorfield that is transversal to the singularity (E α d α ξ α ) r and trace the derivative of the Jacobian. Spectral averaging : de(...) E(...) 25
LEVEL REPULSION Theorem 5. Fix k and assume smoothness ˆν(t) ( + Ct 2 ) k2 +5, t Let E 2 κ, I ε = [E 2N ε, E + 2N ε ], then P ( N Iε k ) C k ε k2 uniformly in ε > 0 and N N(k). Remarks: i) Level repulsion for k = 2. [Independent ev s it would be ε k ]. ii) Exponent is optimal, it agrees with the GUE case: p(µ,... µ N ) j<k (µ j µ k ) 2 e j µ2 j iii) For symmetric matrices the exponent is different. 26
Idea of the proof: Induction over k. By the basic estimate and the interlacing property N Iε ε N N j= Im (N(j) h jj z N I ε k ) α ξ(j) λ (j) α z P ( N Iε k ) ε k E λ E a (N () I ε k ) [ ( α c α ξ α ) 2 + ( E h α d α ξ α ) 2 ] k/2 ξ α = N u α a 2, c α = ε N 2 (λ α E) 2 + ε 2, d α = N(λ α E) N 2 (λ α E) 2 + ε 2 Now we have at least k indices with c α ε! Thus, using the basic lemma to perform the E a expectation P ( N Iε k ) Cε 2k P ( N () I ε k ) k... C ε (2j ) = C ε k2 27
SUMMARY All results for general Wigner matrices, no Gaussian formulas Semicircle saw for the DOS on the optimal scale /N Eigenvectors are fully delocalized away from the spectral edges. Tail of the gap distribution decays at least subexponentially Optimal Wegner estimate: averaged density of states is bounded Level repulsion for k eigenvalues with optimal exponent. DREAM: Wigner-Dyson distribution of level spacing for Wigner matrices 28