Semicircle law on short scales and delocalization for Wigner random matrices

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1 Semicircle law on short scales and delocalization for Wigner random matrices László Erdős University of Munich Weizmann Institute, December 2007 Joint work with H.T. Yau (Harvard), B. Schlein (Munich)

2 WIGNER ENSEMBLE H = (h jk ) 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 GUE. Normalization ensures that Spec(H) = [ 2, 2] + o() Typical eigenvalue spacing is /N. 2

3 MAIN QUESTIONS ) Density of states (DOS) Wigner semicircle law. 2) Eigenvalue spacing distribution (Wigner-Dyson statistics and level repulsion); 3) (De)localization properties of eigenvectors. RELATIONS: 2) is finer than ) [bulk vs. individual ev.] Level repulsion Delocalization??? [Big open conjecture] Motivation in background: Random Schrödinger operators in the extended states regime. 3

4 DENSITY OF STATES N I := #{µ n I} number of evalues µ n of H in I R. Smoothed density of states around E with window size η: ϱ η (E) = Nπ ImTr H E iη = Nπ α η (µ α E) 2 + η 2 ϱ η (E) and N I with I = [E η 2, E + η 2 ] are closely related. For any fixed I R, lim N EN I N = WIGNER SEMICIRCLE LAW ϱ sc(x)dx, ϱ sc (x) = 4 x 2 ( x 2) I 2π Similar statement for ϱ η (E), window size η = O() fixed. Fluctations and almost sure convergence are also known. 4

5 The Wigner-Dyson statistics (universal distribution of eigenvalue spacing) requires info on individual evalues on a scale η /N. It is believed to hold for general Wigner matrices, but proven only for Gaussian and related models and the proofs use explicit formulas for the joint ev. distribution. [Dyson, Deift, Johansson] GOAL: Understand the DOS for small windows η N α. Semicircle Law is expected to hold for any scales η N. Theorem: [Upper bound]. Assume g <. Let I log N N P{N I KN I } e ckn I for large K. Similar result holds for P{ϱ η (E) K}., then 5

6 Lower bound is much harder! Two steps: Theorem: [Fluctuation]. Let ϱ = ϱ η (E) and η N 2/3. Assume spectral gap for dν, then E ϱ Eϱ 2 N 2 η 3. If, additionally, dν satisfies log. Sobolev ineq. then P{ ϱ Eϱ ε} e cnηεmin{,nη2 ε} Previously known result is up to η N /2 [Guionnet-Zeitouni] Theorem: [Identification of the expectation]. Assume log. Sobolev { } lim sup Eϱ η (E) ϱ sc (E) : E 2 κ, η N 2/3 = 0 N Previously known result is up to η N /2 [Bai, Miao, Tsay] COROLLARY: Wigner Semicircle Law holds down to energy scales η N 2/3 with exponential precision. 6

7 Proof of the upper bound: 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 and define ξ α := N a u α 2, Eξ α = For the (,) matrix element of G z = (H z), z = E + iη: G z (,) = G z (,) h z a (B z) a = Im [ ] η + N α h z N ξ α (λ α E) 2 + η 2 for any interval I = [E η, E + η]. Recall that N I Cη k G z (k, k) N α= ξ α λ α z Nη α : λ α I ξ α so to get an upper bound on N I, we need a lower bound on ξ α. 7

8 Repeating the above construction for each k, Suppose then we had k= N I CNη 2 N α : λ (k) α I α : λ (k) α I ξ α (k) ξ (k) α cn I (E ξ = ) N I N2 η 2 N I = N I Nη We also used that in the decomposition ( ) h a H =, a B the eigenvalues µ α of H and λ α of B are interlaced: µ λ µ 2 λ

9 Lower bound on α ξ α : Recall ξ α = N a u α 2. Note that a is indep of λ α,u α. The ξ α s are not independent, but have a strong concentration if g < : Lemma: Let g <, then P α A ξ α δ A e c A Note α A ξ α = N α A a u α 2 = N P A a 2, P A = proj Lemma: Let z = (z,... z N ), z j = x j +iy j, x j, y j e g(x) dx with g <. Let P be a projection of rank m in C N. Then E e c(pz,pz) e c E(Pz,Pz) = e c m Proof: Brascamp-Lieb 9

10 Proof of the Thm on the fluctuation: Recall ϱ = N α f(µ α ), f(µ) = Spectral gap = E ϱ Eϱ 2 E ij η (µ E) 2 + η 2 ϱ Nh ij 2 = N E ij α f (µ α ) µ α h ij 2 From first order perturbation theory, µ α h ij = v α (i)v α (j) + c.c. E ϱ Eϱ 2 E N 3 f (µ α )f (µ β ) v α (i)v α (j)v β (i)v β (j) α,β ij = N 3E f (µ α ) 2 α N 3 η4e#{ev s in [E η, E + η]} N 3 η 4 Nη = N 2 η 3 0

11 Proof of the expectation: Eϱ(E) ϱ sc (E) Consider the Stieltjes transform ϱ(x)dx m(z) = x z The Stieltjes tr. of the semicircle law satisfies m sc (z) + m sc (z) + z = 0 This fixed point equation is stable away from the spectral edge.

12 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] p < 2 Absence of localized evectors (support is large) p > 2 Lack of concentration. 2

13 Theorem. Assume g < and e δx2 dν <. (i) [Absence of localization] Let p < 2 P { v ev. such that v p N p 2 } e cn (ii) [Lack of concentration] Let p > 2. Then only a negligible fraction of eigenvectors {v β } can be concentrated: P N { β : vβ p N p } 2 = o() = o() [The opposite inequalities are trivial by Schwarz] Case (i) is easier, even a stronger version is true. Def: v is (L, η)-localized if A, A = L, s.t. j A v j 2 η. Theorem: Assume the above conditions. Let η, κ = N/L small P { v ev. exhibiting (L, η) localization } e cn 3

14 ( h a Proof of (ii): Decompose as before H = a B Let Hv = µv and v = (v,w), w C N. Then hv +a w = µv, av +Bw = µw = w = (µ B) av ), Recall λ α,u α are the ev s of B and let A = { α : λ α µ = q N } q N From the normalization, = w 2 + v 2, we have v 2 = + N α ξ α (µ λ α ) 2 N, (ξ α := N a u α 2 ) (q/n) 2 α A ξ α If we knew that A, then ξ α > 0 would imply v 2 q 2 /N. Use the interlacing property and elementary combinatorics to show that most eigenvalues cannot be isolated. This proves v N /2, then interpolate for 2 p <. 4

15 Small fraction of the evectors are not controlled b/c no control on the microscopic ev. distribution on scale η /N. There may be a few isolated evalues and their evectors may be localized. There is no good control on the big gaps: Open question: P { α : µ α µ α q } e cq N We even do not know that this probability vanishes as q Note that Wigner surmise predicts even Gaussian tail! Using the control on η N 2/3, we have Corollary: Assume g <, Gaussian moment and log-sobolev. Then, with o() probability, v N /3 for all eigenvectors away from the spectral edges. [The optimal bound should be N /2 ] 5

16 EXTENDED STATES: GREEN FUNCTION ESTIMATES Matrix element of the boundary value of the Green function: G(j, j) = G E (j, j) = u α (j) 2 α λ α E i0, j fixed α u α (j) 2 =. If uniform, u α (j) 2 = N, then G(j, j) O() apart from a Lebesgue-small set of E s. True also for the square: Theorem: Assume g < and e δx2 dν <. ( P Meas E : N j G E (j, j) 2 (log N) 2 ) C log N e c(log N)2 Follows from the v bound and dyadic decomposition in the energy space. 6

17 SUMMARY All results for general Wigner matrices, no Gaussian formulas Upper bound on the DOS on scale log N N DOS on scale N 2/3 converges to semicircle Most eigenvectors have no concentration No eigenvector is localized. Control the second moment of the Green fn. OPEN QUESTION: Lower bound on DOS on scale /N. Why is there no gap??? Wigner-Dyson distribution of level spacing. [DREAM...] 7

18 More precise version in the special case: p =. P N { β : vβ 2 q2 (log N) 2 } C N q = e c(log N)2, = Density of evectors with loc. length L Nq 2 is at most C/q. = v N /2 for most eigenvectors (log corrections) 8