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)
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
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
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
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
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
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
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 2... 8
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
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
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.
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
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
( 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
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
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
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
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