Bulk Universality for Random Matrices via the local relaxation flow

Transcription

1 Bulk Universality for Random Matrices via the local relaxation flow László Erdős Ludwig-Maximilians-Universität, Munich, Germany Disentis, Jul 26 30, 200 Joint with B. Schlein, H.T. Yau, and J. Yin

2 INTRODUCTION Basic question [Wigner]: Consider a large matrix whose elements are random variables with a given probability law. What can be said about the statistical properties of the eigenvalues? Do some universal patterns emerge and what determines them? Gaussian Unitary Ensemble (GUE): H = (h jk ) j,k N hermitian N N matrix with h jk = h kj = N (x jk + iy jk ) and h kk = 2 N x kk where x jk, y jk (for j < k) and x kk are independent, centered Gaussian random variables with variance /2. Classical ensembles: GUE, GOE, GSE, Wishart (H = A A) 2

3 ρ (x) = 4 x 2 2π 2 2 Eigenvalues: x x x N Typical eigenvalue spacing is x i+ x i N (in the bulk) Observations: i) Semicircle density. ii) Level repulsion. Holds for GUE, GOE, GSE. For Wishart: Marchenko-Pastur law 3

4 SINE KERNEL Probability density of the eigenvalues (wrt. Lebesgue) p(x, x 2,..., x N ) The k-point marginal density (= correlation) is given by p (k) N (x, x 2,..., x k ) = R N k p(x,... x k, x k+,..., x N )dx k+...dx N Special case: k = (density ) N(x) := p () N (x) = R N p(x, x 2,..., x N )dx 2...dx N It allows to compute expectation of observables with one eigenvalue: E N N i= O(x i ) = O(x) N(x)dx 2π O(x) 4 x 2 dx 4

5 Higher k is similar, for example, k = 2 gives E N(N ) N i j O(x i, x j ) = O(x, y)p (2) N (x, y)dxdy Correlation functions answer to the questions like: How many eigenvalues are in an interval I? What is the prob. that there are 5 eigenvalues in I and 3 in J? They do not (directly) answer to questions like; What is the distribution of the 200-th eigenvalue? What is the distribution of the gap between neighboring evalues? Nevertheless, the second question can be answered by exclusioninclusion formulas. 5

6 Local level correlation statistics [Wigner, Dyson, Mehta] lim [ρ N(E)] 2 p (2) N N ( E + x Nρ N (E), E + x 2 Nρ N (E) ) = det { S(x i x j ) } 2 i,j=, sin πx S(x) = πx, E < 2 = ( ) sin π(x x 2 ) 2 = Level repulsion π(x x 2 ) k-point correlation functions are given by k k determinants: lim [ρ N(E)] k p (k) N N ( E + x Nρ N (E), E + x 2 Nρ N (E),..., E + = det { S(x i x j ) } k i,j= x k Nρ N (E) Note that the limit is independent of E as long as E is in the bulk spectrum, i.e. E < 2. 6 )

7 Gap distribution E # of e.v. pairs with gap less than s near E # of eigenvalues near E s 0 p(α)dα, p(α) := d2 dα 2 det( K α)dα K α (x, y) = sin π(x y) π(x y) as operator on L 2 ((0, α)) Wigner surmise (not exact but very good approximation) p(α) πα 2 e πα2 /4 for the symmetric case p(α) 32π 2 α 2 e 4α2 /π for the hermitian case 7

8 More precisely: Fix an energy E with E < 2 and for any s > 0 and for some N-dependent parameter t with /N t let Λ(s) := 2Nt sc (E) # { j N : λ j+ λ j s N sc (E), λ j E t i.e., the proportion of rescaled eigenvalue differences below a threshold s in a large but still microscopic vicinity of an energy E. } By the exclusion-inclusion formula, change of variables and the universal limit of the correlation functions, EΛ(s) = 2Nt m=2 m=2 where = sc (E). ( ) m( N m ) t t dv... t t dv m { max v i v j p (m) N (E + v, E + v 2,..., E + v m ) ( ) m s ( s sin π(ai (m )! da 2... da a j ) m det 0 0 π(a i a j ) ) m i,j= s N, } 8

9 History, Applications and Conjectures Nuclear Physics: the excitation spectra of heavy nuclei are expected to have the same local statistical properties as the eigenvalues of GOE (Wigner, 955). Quantum Chaos Conjecture (vague) classical dynamics with potential V chaotic integrable e.v. gap of + V GOE statistics Poisson statistics Geodesic flow: Bohigas-Giannoni-Schmit (984), Berry-Tabor (977) 9

10 Riemann ζ-function: Gap distribution of zeros of ζ function is given by GUE (Montgomery, 973). Anderson Model (958): V ω random potential on R d or Z d random Schrödinger operator: H = + λv ω the eigenvalue distribution of H in the delocalization regime (λ small) is given by GOE. In the localization regime (λ large or near the spectral edge) Poisson statistics is proven. Universality conjecture (vague): Statistics of eigenvalues depends on symmetry but is otherwise independent of the details of the ensemble. 0

11 KNOWN RESULTS FOR UNITARY ENSEMBLES Unitary ensemble: Hermitian matrices with density P(H)dH e Tr V (H) dh Invariant under H UHU for any unitary U (GUE) Joint density function of the eigenvalues is explicitly known p(λ,..., λ N ) = const. (λ i λ j ) β e j V (λ j) i<j classical ensembles β =,2,4 (orthogonal, unitary, symplectic symmetry classes; GOU, GUE, GSE for Gaussian case) Correlation functions can be explicit computed via orthogonal polynomials due to the Vandermonde determinant structure. large N asymptotic of orthogonal polynomials = local eigenvalue statistics indep of V. But density of e.v. depends on V.

12 Many previous results: Dyson (962-76), Mehta (960- ) classical Gaussian ensembles via Hermite polynomials General case by Deift etc. (999), Pastur-Schcherbina (2008), Bleher-Its (999), Deift etc (2000-, GOE and GSE), Lubinsky (2008)... All results are limited to invariant ensembles. 2

13 PREVIOUS METHODS FOR RANDOM MATRICES Moment method: compute ETrH k. Semicircle law on macro scale [Wigner, 955]. Suitable on the spectral edges [Sinai, Soshnikov 999] for the Tracy-Widom law. Green Function Method: compute ETr z H. Resolvent is unstable inside the spectrum, randomness stabilizes. Explicit formulas + Asymptotic analysis. E.g: orthogonal polynomial methods or Brezin-Hikami and Johansson s formulae for hermitian case. Problems: Need explicit formulas. Universality??? 3

14 (HERMITIAN) WIGNER ENSEMBLE H = H = (h jk ) j,k N is a hermitian N N matrix, N. h kj = N ( xkj + iy kj ) h kk = 2 N x kk where x kj, y kj ( k < j N) and x kk ( k N) are i.i.d. with E x jk = 0 and E x 2 jk = 2 ( E h jk 2 = N ) No explicit formula for the eigenvalue distribution. Not unitary invariant, unless V (x) = x 2 Gaussian. Symmetric Wigner ensemble is defined similarly. Our final result: Universality for (generalized) Wigner matrices 4

15 SEMICIRCLE LAW (WIGNER 955) N[I] : = number of eigenvalues in interval I sc (E) : = (4 E 2 ) + 2π For any δ > 0 and E 2 ( lim lim P N[E η 2, E + η 2 ] η 0 N Nη ) sc (E) δ = 0 Remark : Semicircle is independent of the distribution of entries Remark 2: Wigner s result concerns the macroscopic density, i.e. density in intervals containing order N eigenvalues. 5

16 UNIVERSALITY IN THE BULK Theorem [E-Yau-Yin, 200] Suppose that H is a hermitian or symmetric Wigner matrix with Eh ij = 0, and σ 2 ij = E h ij 2, with and normalization (cn ) j σ 2 ij σ 2 ij =, CN and subexponential decay. Then the corr. functions of the local statistics are given by the GUE/GOE in the bulk, i.e. for E < 2, k for any nice test fn O : R k R we have (# is GUE or GOE): lim lim b 0 N 2b E+b E b de ( sc (E) k p (k) N i R k dα...dα k O(α,..., α k ) p(k) #,N )( E + α N sc (E),..., E + α ) k N sc (E) = 0. [Earlier result for Wigner with σ 2 ij = /N: E-Schlein-Yau, 2009] 6

17 DYSON BROWNIAN MOTION Gaussian convolution matrix (after rescaling) H t = H 0 + tv interpolates (Johansson [200] for t > 0, hermitian case). It can also be obtained by evolving the matrix elements with an OU process: t u t = Lu t, L = 2dx 2 x d 2dx, H t = e t/2 H 0 + ( e t ) /2 V. Denote the prob density of the eigenvalues by f t (x)µ(dx). d 2 GUE : µ = Ce NH(x) dx, H(x) = (PDE) t f t = Lf t, N i= flfdµ = 2N x 2 i 2 N i j log x j x i f 2 dµ,=: D µ (f) L = 2N N i= 2 i + N i= ( 2 x i + N j i x i x j ) i, 7

18 (SDE) dx i = N db i + ( 2 x i + N j i x i x j ) dt Derivation of DBM (for the BM case, h ij = N /2 B ij for simplicity): λ α h ik = ū α (i)u α (k), u α (i) h lk = β α λ α λ β ū β (l)u α (j)u β (i) 2 λ α = h ik h lj β α λ α λ β u β (l)ū α (j)ū β (i)u α (k) + c.c. dλ α = N ik ū α (i)u α (k)db ik + N ik β α λ α λ β u β (i)ū α (k)ū β (i)u α (k)dt = N db α + N β α λ α λ β dt where one can check that ik ū α (i)u α (k)db ik are independent white noises for different α s. 8

19 (SDE) dx i = N db i + ( 2 x i + N j i x i x j ) dt Idea: Equilibrium is the invariant ensemble (GUE, etc.) with known local statistics. For local statistics, only local equilibrium needs to be achieved which is faster. Bakry-Emery method: Speed of convergence to a measure µ = e H /Z is given by the lower bound on the Hessian matrix H : H K = S µ (f t ) e tk, D µ ( and LSI holds: KS µ (f t ) D µ ( f t ). f t ) e tk In our case, K = O(), so time to global equilibrium is O(). [Morally, Johansson s work is t O(), but with different method and for different reason] 9

20 KEY STEPS Step. Good local semicircle law including a control near the edge. Method: System of self-consistent equations for the Green function, control the error by large deviation methods. Step 2. Universality for Gaussian divisible matrices with a small ( N ε ) Gaussian component. Method: Modify DBM to speed up its local relaxation, then show that the modification is irrelevant for statistics involving differences of eigenvalues. Step 3. Universality for arbitrary Wigner matrices. Method: Remove the small Gaussian component in Step 2 by resolvent perturbation theory and moment matching. 20

21 DYSON CONJECTURE Theorem [Strong local ergodicity of DBM] [E-Yau-Yin, 200] Suppose that H is a hermitian or symmetric Wigner matrix with Eh ij = 0, and σ 2 ij = E h ij 2, with (cn ) σ 2 ij CN, j σ 2 ij =, and subexponential decay. Let E [2 c,2 + c] with some c > 0. Then for any ε > 0 and 0 < b < c/2, any integer n and for any nice test function O : R n R we have lim sup N t N +ε where p (n) t,n E+b E b de 2b ( p (n) t,n p(n) µ,n i dα R n...dα n O(α,..., α n ) (E) n )( E + α N (E),..., E + α n N (E) ) = 0 are the corr. functions of the evalues of the DBM flow. Dyson s conjecture [962]: Local equilibrium is already attained after time t N +ε. 2

22 HISTORICAL DETOUR We have been subsequently improving the local semicircle law in several papers focusing on shrinking the window. The following prototype result is optimal away from the edge: Local Semicircle Law. [E-Schlein-Yau, ] Let N[I] := #{λ n I} be no. of evalues in I R. Then P N[E K 2N, E + K 2N ] K sc (E) δ Ce cδ K for all E < 2 and K > 0 uniformly in N N 0 (E, δ) and the eigenvectors are completely delocalized. Method: Green function + Large deviation Local semicircle law is always the starting point of any analysis. 22

23 Extended states: eigenvector delocalization No concept of abs. continuous spectrum unlike in random Schr. Let v = (v, v 2,... v N ) C N be l 2 -normalized. Complete localization: a few large component, e.g. v = (,0,0,...0) = v p = for all 2 p Complete delocalization: components have the same size, e.g. v = ( N /2, N /2,... ) = v p = N p 2 for all 2 p Theorem 2. [E-Schlein-Yau, 2008] Assume E e x ij ε <. Fix E < 2, K > 0 and 2 < p <, then for large M P v, v 2 =, Hv = xv, x E K N, v p MN p 2 Ce c M 23

24 Level repulsion and Wegner estimate Theorem 3. [E-Schlein-Yau, 2008] Fix k and assume that the distribution ν is sufficiently smooth and decays. Let E < 2, then P ( N[E ε 2N, E + ε 2N ] k) C k ε k2 uniformly in ε, N i) Level repulsion for k = 2. [For Poisson ev s it would be ε k ]. ii) Exponent is optimal, it agrees with the GUE case: p(x,... x N ) (x j x k ) 2 e j x2 j P(N ε k) ε k2 j<k (Exponent in symmetric case is different, but also optimal) Method: Green function + local semicircle + integration by parts in probability space 24

25 Universality for hermitian Wigner matrices Johansson [200], Ben Arous-Peche [2005] proved sine-kernel for hermitian Wigner matrices with Gaussian component of the form H = H 0 + tv C N N, H 0 is Wigner, V is GUE, t > 0. Key explicit formula: [Brézin-Hikami 996, Johansson 200] The two-point corr. fn. of H in terms of p 0 (e.v. distribition of H 0 ) is with p (2) (x, x 2 ) = K t,n ( E, E + τ N ;y ) p 0 (y)dy = γ dz det ( ) 2 K t,n (x i, x j ;y) i,j= Γ dw g N(z, w; τ)e N(f N(w) f N (z)) f N (z) = 2t (z2 2Ez) + N j log(z y j ), g N =... Remark: Available only for the hermitian case. 25

26 To identify the saddles at energy E, one needs to solve for z 0 = f N (z) = t (z E) + N j z y j where y j s are the eigenvalues of the Wigner matrix. Approximately 0 = t (z E) + N and here Im z t. j 0 = sc z y j t (z E) + (y)dy z y ( ) Thus (*) can be justified if semicircle law is valid on scale O(t). Choosing t = N +ε, the sine-kernel is proved for matrices H = H 0 + tv, t = N +ε 26

27 Theorem [E-Peche-Ramirez-Schlein-Yau, 2009] Sine kernel holds for hermitian Wigner ensemble if the distribution of the matrix elements has subexp decay and some smoothness (no Gaussian component). Method: Extend Johansson s result to a tiny Gaussian component t = N +ε using local semicircle law in the saddle point analysis of the contour integral. To remove this tiny Gaussian component: use reverse heat flow: Given a smooth distribution u(x)dx, choose then g t = ( t + ( t )2 2! e t g t +... ) u e t u will have a Gaussian component and will be close to u. 27

28 Method of the reverse heat flow Let u 0 be the density of the matrix elements of H 0, then the matrix elements of H = H 0 + tv have density d 2 u t = e tl u 0, L = 2dx 2 The probability density of all matrix elements, F 0 = u N2 0 at t = 0 and F t = (e tl u 0 ) N2 satisfy F t F 0 N 2 e tl u 0 u 0 CN 2 t, t N +ε Key observation Compare F with e tl G for some G that satisfied the local semicircle law; G may even depend on t. Simple way to find a better G: For smooth u 0, choose G = g N2 t, g t = ( tl)u 0 then e tl g t u 0 = O(t 2 ) 28

29 Related results Four-Moment Theorem [Tao-Vu 2009]: if the first four moments of the single entry distribution of two Wigner matrices match, then the local e.v. statistics coincide. Idea: Expand the function x i (x jk, j =,..., N; k =,..., N) in Taylor series of x jk (recall N /2 x jk is the matrix element distribution). Input: Local semicircle law + eigenvector delocalization With Johansson s result, it yields universality for hermitian Wigner matrix under a third moment and a support condition. Universality for hermitian case with no condition apart from subexp decay [ERSTVY, 2009] (Combine the two methods) All results used explicit hermitian formulas; method is not robust. End of the historical detour: back to new method. 29

30 KEY STEPS (recall) Step. Good local semicircle law including a control near the edge. Method: System of self-consistent equations for the Green function, control the error by large deviation methods. Step 2. Universality for Gaussian divisible matrices with a small ( N ε ) Gaussian component. Method: Modify DBM to speed up its local relaxation, then show that the modification is irrelevant for statistics involving differences of eigenvalues. Step 3. Universality for arbitrary Wigner matrices. Method: Remove the small Gaussian component in Step 2 by resolvent perturbation theory and moment matching. 30

31 Step : Strong semicircle law for general Wigner matrices Theorem [E-Yau-Yin, 200] Let Eh ij = 0, E h ij 2 = σij 2, where B := (σij 2) is double stochastic, c Nσ2 ij C. Assume ) P ( h ij xσ ij ϑ e xϑ, x > 0 Then for any z = E + iη with E 5, N η 0 we have ( P max G ii (z) m sc (z) ) N i Nη ( P max G ij (z) ) ij Nη ( P m(z) m sc (z) ) Nη N N where G ij = H z (i, j) and m(z) = N TrG = N (Here means log N corrections) i G ii. 3

32 Corollary. [Counting functions] then for any E n(e) := N #{λ j E}, n sc (E) := ( P n(e) n sc (E) ) N E N sc(x)dx Corollary 2. [Eigenvalue locations] Let γ j be the classical location of the j-th eigenvalue, i.e. γj Then for any j N/2 P ( λ j γ j sc(x)dx = j N N 2/3 j /3 ) N All these estimates are optimal (up to log corrections) 32

33 From Stieltjes transform to counting function Let (λ)dλ be a signed measure on a compact interval in R, say [ 3,3] and suppose its Stieltjes transform satisfies Im m(x + iy) U, y > 0, x + y 0 Ny Then for any E, E 2 [ 3,3] we have CU log η f E,E 2,η(λ) (λ)dλ N where f E,E 2,η is a smoothed char. function of [E, E 2 ] on scale η. Helffer-Sjöstrand formula: Let f C (R). Let χ(y) be a cutoff function in [,]. Define the quasianalytic extension of f as f(x + iy) = (f(x) + iyf (x))χ(y), then f(λ) = 2π R 2 z f(x + iy) λ x iy dxdy = 2π R 2 iyf (x)χ(y) + i(f(x) + iyf (x))χ (y) λ x iy d 33

34 Detour: similar results for universal matrices Theorem [E-Yau-Yin, 200] Let Eh ij = 0, E h ij 2 = σij 2, where B := (σij 2) is double stochastic. Let M := (max ij σij 2). Assume ) P ( h ij xσ ij ϑ e xϑ, x > 0 Then for any z = E + iη with Mη and ε, K ( P max G ii (z) m sc (z) ) i Mη ( P max G ij (z) ) ij Mη P ( m(z) m sc (z) Nε ) Mη E 2 κ 0 > 0: N N K N For E away from the edge, P ( n(e) n sc (E) Nε M ) N K 34

35 Step 2: Local ergodicity of Dyson Brownian Motion Assumption I. [Convexity] The Hamiltonian of the invariant measure µ e NH(x) on R N has the representation H(x) = β N i= U(x i ) N i<j log x j x i f t = Lf t, L = 2N i i H Assumption II [Macroscopic limit of density] There exists a continuous, compactly supported density function (x) 0, R =, such that for any fixed a, b R lim sup N t 0 N N j= (x j [a, b])f t (x)dµ(x) b (x)dx = 0. a 35

36 Assumption III [Eigenvalues are close to their classical location] Let γ j denote the classical location of the j-th eigenvalue (given by ). Suppose there is ε > 0 such that sup N t N 2ε N j= (x j γ j ) 2 f t (dx)µ(dx) CN 2ε I.e., typical spacing between the random points x j and their classical location is less than N /2 ε. Assumption IV. [Upper bound on local density] For any I R, let N I := N i= (x i I) denote the number of points in I. For any compact subinterval I {E : (E) > 0}, I N +σ sup τ N 2ε { N I KN I } f τ dµ C n K n, n =,2,... 36

37 Theorem [E-Schlein-Yau-Yin, 2009] Suppose Assumption I [convexity] holds. Suppose at time t 0 = N 2ε, the density f t0 satisfy a bounded entropy condition, i.e., for some m S µ (f t0 ) := f t0 log f t0 dµ CN m and suppose that Assumptions II, III and IV hold for the solution of f t = Lf t. Let E R and b > 0 such that min{ (x) : x [E b, E + b]} > 0. Then δ > 0, k and for any nice test fn O : R k R we have lim N sup t N 2ε+δ 2b (E) k E+b E b de ( p (k) t,n p(k) µ,n dα R k...dα k O(α,..., α k ) )( E + α N (E),..., E + where ε > 0 is the exponent from Assumption III. α ) k N (E) = 0 The method gives an effective error bound with trade-off between ε and b, in particular b = N power is allowed. 37

38 Local relaxation flow We modify the dynamics by adding a fictitious term to H: H = H + W, W = j (x j γ j ) 2 where R and γ j are the classical location of ev s, defined by γj sc(x)dx = j N New equilibrium measure: ω e N H. R 2 We then have v, 2 H(x)v R 2 v 2 + N i<j (v i v j ) 2 (x i x j ) 2, v RN. i.e. the relaxation time is t R 2 (Bakry-Emery). We also keep the second term from log x i x j. 38

39 Theorem. [Relaxation of the modified dynamics] Suppose v, 2 H(x)v R 2 v 2 + N Consider the forward equation i<j (v i v j ) 2 t q t = Lq t, t 0, (x i x j ) 2, v RN. with initial condition q 0 = q and with reversible measure ω. Then t D ω ( q t ) R 2D ω( q t ) 2N 2 2N 2 0 ds N i,j= N i,j= ( i qs j qs ) 2 and the logarithmic Sobolev inequality holds ( i qt j qt ) 2 (x i x j ) 2 dω, (x i x j ) 2 dω D ω ( q) S ω (q) CR 2 D ω ( q) Thus the time to equilibrium is of order R 2 : S ω (q t ) e Ct/R2 S ω (q 0 ). 39

40 Theorem 2. [Gap distribution estimated by Dir. form] Let q L be a density, qdω =. Then for any τ > 0 we have G(N(x i x i+ ))qdω G(N(x i x i+ ))dω N N i ( Dω ( ) q)τ /2 C + C S ω (q)e cτ/r2. N Similar result holds for G i,m (x). As a comparison we show what entropy control alone can reach Theorem: We have, for any τ > 0, and E < 2 N i G ( N(x i E) )[ q ] dω C S ω (q) [τ + e cτ/r2] i = CR S ω (q) after optimizing for τ in the last step. N problem! /2 40

41 We will use this for q = g τ, where g t dω = f t dµ, or, g t = f t /ψ. Comparison of dynamics: t f t = Lf t f t µ µ slow [true DBM] t q t = Lq t q t ω ω fast [approx] Initial state f 0 = q 0 ψ. Write f t = g t ψ, compare f t µ = g t ω with q t ω. Theorem 3: [Comparison of dynamics] Set τ = R 2 N ε. Let g t = f t /ψ. Assume that S ω (g 0 ) CN m with some fixed m. Then the entropy and the Dirichlet form satisfy the estimates: S ω (g τ/2 ) CNR 2 Q, D ω ( g τ ) CNR 4 Q. Recall: Q := sup t 0 j (x j γ j ) 2 f t dµ. Note: g t does NOT evolve according to the L dynamics (then the estimate would be exp. small). 4

42 Still, differentiating entropy (similar to Boltzmann s t S = D) t S ω (g t ) = 2 N j ( j gt ) 2 dω+ j From the Schwarz inequality t S ω (g t ) D ω ( g t )+CN j Together with LSI, we have b j j g t dω, b j = W j = x j γ j R 2 b 2 j g tdω D ω ( g t )+CNQR 4 t S ω (g t ) CR 2 S ω (g t ) + CNQR 4 which, after integrating from t = t 0 to τ/2, proves the bound on S ω (g τ ), using that τ = R 2 N ε and S ω (g 0 ) = S µ (f t0 ψ) = S µ (f t0 ) log(z/ Z) + N In fact, we have S ω (g s ) CR 2 NQR 4 = CNQR 2 Wf t0 dµ CN m. for any s τ/2. The estimate for D is integration of (*) from τ/2 to τ. 42 ( )

43 Step 3: Green function comparison theorem Theorem: Let H (v) and H (w) be two generalized N N Wigner matrices, with matrix elements h ij given by the random variables N /2 v ij and N /2 w ij, respectively, with v ij and w ij satisfying the uniform subexponential decay condition. Define interpolating matrices, H 0 = H (v), H,... H N 2 = H (w), i.e. order the elements in some way and let H γ such that its matrix elements h ij follow the v-distribution up to γ and then they follow the w-distribution; Assume that for any y N +ε and E < 2, ( Assume moment matching H γ E iy ) kk Evij k Ev = Ewk ij, k =,2,3, 4 ij Ewij 4 N δ 43

44 Let F be a nice function and G (v) (z) = (H (v) z) be the resolvent. Then EF k N Tr G (v) (zj ),..., k n N Tr G (v) (zj n ) EF ( G (v) G (w)) j= j= for any collection of spectral parameters, zj m N ε, Ej m 2 2κ. = E m j ± iη, with η Similar statement holds for matrix elements. Moral of the statement: if four moments (almost) match, local statistics are the same. 44

45 The four moment condition first appeared in Tao-Vu s Four momemt theorem, stating that observables of the form ( ) Nλi, Nλ i2,..., Nλ ik i <i 2 <...<i k F have the same expectation under the two ensembles, if the first four moments of the single entry distribution match. Their proof also used local semicircle law and it was quite involved due to resonances [they used level repulsion]. Our proof is a simple resolvent perturbation, exploiting the stability of the resolvent; G(z) η (never blows up). 45

46 Matching lemma We have proved so far universality for Wigner matrices with a single entry distribution having an N ε Gaussian convolution. We also proved that local stat are stable if four moments (almost) match. To prove the universality for any single entry distribution, we need to approximate it by another one with a small Gaussian components so that the first four moments almost match. This is and elementary lemma. Let m k (ξ) = Eξ k be the k-th moment of a random variable ξ. 46

47 Matching lemma: Let m 3 and m 4 be two numbers such that m 4 m 2 3 0, m 4 C 2 Let ξ G be a Gaussian random variable with mean 0 and variance. Then for any sufficient small γ > 0 there exists a real random variable ξ γ with subexponential decay and independent of ξ G, such that the first four moments of ξ = ( γ) /2 ξ γ + γ /2 ξ G are m (ξ ) = 0, m 2 (ξ ) =, m 3 (ξ ) = m 3 and m 4 (ξ ), and m 4 (ξ ) m 4 Cγ Note that m 4 m holds for the moments of any random variable with m = 0, m 2 =. 47

48 SUMMARY: KEY STEPS Step. Good local semicircle law including a control near the edge. Method: System of self-consistent equations for the Green function, control the error by large deviation methods. Step 2. Universality for Gaussian divisible matrices with a small ( N ε ) Gaussian component. Method: Modify DBM to speed up its local relaxation, then show that the modification is irrelevant for statistics involving differences of eigenvalues. Step 3. Universality for arbitrary Wigner matrices. Method: Remove the small Gaussian component in Step 2. by perturbation theory and moment matching. 48

49 OPEN PROBLEMS. Extend universality to band matrices for W N 2. Prove delocalization for band matrices, W N 3. Extend the method to β-ensembles. Random band matrices: H is symmetric with independent but not identically distributed entries with mean zero and variance Conjecture Narrow band, W N = Broad band, W N = E N h kl 2 = e k l /W Even the Gaussian case is completely open. localization, Poisson statistics delocalization, GOE statistics 49

50 Some proofs for Step : Local semicircle law H (i) denotes the (N ) (N ) minor of H after removing the i-th column/row and similarly for H (ij) etc. Let a i be the i-th column. The Green functions are: G = H z, G(i) = H (i) z Then from Schur decomposition we have G ii = ( H z ) ii = h ii z a i G (i) a i With the notation K ij = h ij zδ ij a i G (ij) a j we also have G ij = G jj G (j) ii K ij and a different set of formulas (i, j, k different) G ii = G (j) ii + G ijg ji, G ij = G (k) G ij jj Rule of thumb: G ii O(), G ij is small. + G ikg kj G kk 50

51 Crudest local semicircle law (for σ 2 ij = N ) with G ii = h ii z E i a i G (i) a i Z i By a simple computation Z i := a i G (i) a i E i a i G (i) a i E i a i G (i) a i = N N m(i), and by the interlacing property of the eigenvalues of H and H (i), Putting all together G ii = m(z) m (i) (z) Nη, z m(z) + Ω i, η = Im z ( ) Ω i = h ii Z i + O Nη 5

52 G ii = z m(z) + Ω i, ( ) Ω i = h ii Z i + O Nη Expand and sum up, with Ω = max i Ω i m(z) = z + m(z) + O(Ω) By the stability analysis of this equation, and m sc = z+m sc, m m sc CΩ Ω + κ, κ := E 2 By the large deviation bound on Z i, we have Ω (Nη) /2, thus ( ) m m sc min, Nηκ (Nη) /4 It can be improved in various directions. 52

53 Some large deviation results Let a i ( i N) be independent complex random variables with mean zero, variance σ 2 and having a uniform subexponential decay P( a i xσ) ϑ exp ( x ϑ), x, with some ϑ > 0. Let A i, B ij C ( i, j N). Then for any ξ > there exists a constant L 0 depending on ϑ and ξ such that we have P N i= P a i B ii a i P N i= N i= i j a i A i (log N)L 0σ σ 2 B ii (log N)L 0σ 2 a i B ij a j (log N)L 0σ 2 i i i j A i 2 C exp [ (log N) ξ], B ii 2 C exp [ (log N) ξ], B ij 2 C exp [ (log N) ξ]. 53

54 For example since using j,k i σ ij G (i) and similarly jk σ ki k Z i 2 j,k i N 2 j σ ij G (i) jk σ ki 2 Nη ( G (i) 2) jj = Nη N G jk 2 = G jk G kj = (GG ) jj = η Im G jj Z ij = a i G (ij) a j Nη j Im G (i) jj C Nη 54

55 Improved local semicircle law. I: Bound on matrix elements Self-consistent equation for the diagonal elements: G ii = z j σij 2 G jj + Y i Y i := h ii + A i Z i A i := σ 2 ii G ii + j σ 2 ij G ji G ij G ii Z i := a i G (i) a i E i a i G (i) a i Further key quantities Λ d := max k G kk m sc, Λ o := max j k G jk, Λ := m m sc (they all depend on z = E + iη, but suppressed in the notation) 55

56 Dichotomy lemma Away from the spectral edge (for simplicity), we have that IF Λ o +Λ d (log N) 2, THEN Λ o+λ d Nη with high prob Proof. For the offdiagonal part, use G ij = G ii G (i) jj K ij C( h ij + Z ij ) Nη For the diagonal part, we will use the self-consistent equation and Y i h ii + A i + Z i N + ( N + C Nη ) + Nη Nη 56

57 Expansion of the self-consistent equation With the notation v i := G ii m sc, we have v i = ( ) m sc z m sc j σij 2 v j Y i We know that z + m sc c and by the assumption (...), so we can expand ( ) ( ) 2 v i = m 2 sc σij 2 v j Y i + O σij 2 v j Y i j j using m sc + z+m sc = 0. (More precise expansion is possible this is the key to further improvements). Introduce v = N i v i, Ȳ := N i Y i, Y := max Y i, ζ := m 2 sc v = ζ v ζȳ + O(Λ 2 d + Y 2 ) 57

58 v = ζ ( ) ζ ζȳ + O ζ (Λ2 d + Y 2 ) ( ) ζ = O ζ (Λ2 d + Y ) where the second bound is crude (another key to improvement). Let B ij = σij 2 be the covariance matrix, and let Q = I e e be the projection onto the orth. complement of the trivial eigenvector. v i v = ζ j B ij (v j v) ζ(y i Ȳ ) + O(Λ 2 d + Y 2 ) The error sums up to 0, so we can invert I ζb on the range of Q. v i v = ( ) ζq ζq (Y j Ȳ ) + O(Λ 2 d j ζb ij ζb + Y 2 ) ζq = O(Λ 2 d ζb + Y ) (crude) v i ζq ζb + ζ ζ O(Λ 2 d + Y ) 58

59 v i ζq ζb + ζ ζ O(Λ 2 d + Y ) Here use that Spec(B) [ + δ, δ + ] {}, δ ± > 0, thus ζq ζb k n ζ k+ QB k + N k>n ζ k+ QB k 2 2 n + N ( δ ± ) k k>n C log N, by choosing n log N δ ± δ ± Neglect the edge-deterioration (i.e. that ζ = m 2 sc κ + η), Quadratic dichotomy: Λ d C(Λ 2 d + Y )(log N) If Λ d (log N), then Λ d C(log N)Y Nη 59

60 Continuity argument How to guarantee Λ d (log N)? R(z) := (log N) 2, S(z) := (log N)C Nη, S(z) < R(z) We have proved that (with high prob) Λ o (z) + Λ d (z) R(z) = Λ o (z) + Λ d (z) S(z) ( ) For η = Imz = 0 (say), one has Λ o (z)+λ d (z) 3/0, so expansion is possible ( z + m 0 0 > 3/0) and one gets that the LHS of (*) holds for η = 0. Now fix E and reduce η, note that η Λ o (z) + Λ d (z) is cont. so Λ o (z) + Λ d (z) S(z) as long as S(z) < R(z). 60

61 Improved local semicircle law II. The key to improve the previous argument is to exploit a CLT-like cancellation in Ȳ = Y i, i.e., mainly in Z i N N Recall that i Z i = a i G (i) a i E i a i G (i) a i Here Z i and Z j are not independent, but almost. Lemma E N i Z i p p Cp E [ Λ 2p o + N p] or, after plugging in the size of Λ o (Nη) /2 E N i Z i p p Cp (Nη) p = Z (Nη) i 6

62 Sketch of the sketch of the proof: p = 2 E N i Z i 2 = N 2 i j EZ i Z j + N 2 i E Z i 2 We look only at the first term, this is EZ Z 2 (by symmetry) ] EZ Z 2 = E [a G () a E a G () a ][a 2 G (2) a 2 E a 2 G (2) a 2 Make the main terms independent by G () kl = G (2) kl + G() k2 G() 2l G () 22 =: P (2) kl + P () kl (superscripts indicate independence of the givens rows/columns) EZ Z 2 = E[ E a P (2) a + where ][ E a P () a E2 a 2 P (2) a 2 + E i := I E i, i.e. acting on a r.v. X, we have ] E a 2 P (2) a 2 2 E i X = X E i X. 62

63 EZ Z 2 = E[ E a P (2) a + with E i := I E i. ][ E a P () a E2 a 2 P (2) a 2 + ] E a 2 P (2) a 2 2 Note that if A (i) is indep of i and Y is arbitrary, then in particular, E i A (i) = 0, and E i [ Y A (i) ] = A (i) Ei Y E ( E X (2))( E2 X ()) = E E [X (2) ] E2 X () = 0 (with Y := X (2), A () := E X () ), since E Ei = 0. Thus 2 EZ Z 2 = E [ E a P () a ][ E2 a 2 P (2) a 2] and E[a P () a ] 2 kl = P () kl 2 σ 2 k σ2 l Λ4 o. 63

64 Some proofs for Step 2: Bakry-Emery argument Simple calculation shows t S(f t ) = using Lf t dµ = 0. (Lf t )log f t dµ+ f t Lf t f t dµ = 2 ( ft ) 2 f t dµ = 4D( f t ), () Now we compute t D( f t ). Let h := f for simplicity, then t h t = 2h t h 2 t = 2h t Lh 2 t = Lh t + 2h t ( h t ) 2. In the last step we used that Lh 2 = ( h) 2 + 2hLh that can be seen directly from L = 2 2 ( H). 64

65 We compute (dropping the t subscript for brevity and = dµ) t D( f t ) = 2 t = = = 2 ( h)( Lh) + 2 ( h)[, L]h = 2 ( h) 2 dµ h = ( h)( 2 H) h 2 ij [ 2( j h)( i h) ij h h ( h)( 2 H) h 2 ( h) ( h)2 h ( h)l( h) + 2 ij ( i j h) 2 ij ( jh) 2 ( i h) 2 ( where we used the commutator ij h 2 f [ 2( j h) i j h i h h ] ij h ( ih)( j h) h [, L] = 2 ( 2 H). ( jh) 2 ] i h h 2 ) 2, (2) 65

66 Local relaxation flow We modify the dynamics by adding a fictitious term to H: H = H + W, W = j (x j γ j ) 2 where R and γ j are the classical location of ev s, defined by γj sc(x)dx = j N New equilibrium measure: ω e N H. R 2 We then have v, 2 H(x)v R 2 v 2 + N i<j (v i v j ) 2 (x i x j ) 2, v RN. i.e. the relaxation time is t R 2 (Bakry-Emery). We also keep the second term from log x i x j. 66

67 Theorem. [Relaxation of the modified dynamics] Suppose v, 2 H(x)v R 2 v 2 + N Consider the forward equation i<j (v i v j ) 2 t q t = Lq t, t 0, (x i x j ) 2, v RN. with initial condition q 0 = q and with reversible measure ω. Then t D ω ( q t ) R 2D ω( q t ) 2N 2 2N 2 0 ds N i,j= N i,j= ( i qs j qs ) 2 and the logarithmic Sobolev inequality holds ( i qt j qt ) 2 (x i x j ) 2 dω, (x i x j ) 2 dω D ω ( q) S ω (q) CR 2 D ω ( q) Thus the time to equilibrium is of order R 2 : S ω (q t ) e Ct/R2 S ω (q 0 ). 67

68 Theorem [Reformulation] Q := sup t t 0 j (x j γ j ) 2 f t dµ, t 0 = N 2ε and assume that Q CN m with some exponent m. Fix n and an array of positive integers, m = (m, m 2,..., m n ) N n +. Let G : R n R be a nice fn and define G i,m (x) := G ( N(x i x i+m ), N(x i+m x i+m2 ),..., N(x i+mn x i+mn ) Then for any sufficiently small ε > 0, and for τ 3t 0 we have G i,m (x)f τ dµ G i,m (x)dµ N N CNε Qτ + Ce cnε. i i ). Since Q N 2ε in applications, the rhs is small if τ N 2ε. From this theorem and the upper bound one can easily get the limit of correlation functions. Note that the observable G depends on differences. 68

69 Theorem 2. [Gap distribution estimated by Dir. form] Let q L be a density, qdω =. Then for any τ > 0 we have G(N(x i x i+ ))qdω G(N(x i x i+ ))dω N N i ( Dω ( ) q)τ /2 C + C S ω (q)e cτ/r2. N Similar result holds for G i,m (x). As a comparison we show what entropy control alone can reach Theorem: We have, for any τ > 0, and E < 2 N i G ( N(x i E) )[ q ] dω C S ω (q) [τ + e cτ/r2] i = CR S ω (q) after optimizing for τ in the last step. N problem! /2 69

70 Proof. Run the L flow with q as initial condition. Split G(N(x i x i+ )) [ q ] dω = G(...) [ (q q τ )+(q τ ) ] dω N N i For the second term use entropy, G(...) q τ dω C S ω (q τ ) C S ω (q)e cτ/r2 N i For the first term, integrate its derivative τ G(...)(q q τ )dω = N ds 0 N = i τ 0 ds N j ( j q s ) j [ N i i i G(...) s q s dω G(N(x i x i+ )) ] = τ 0 ds N i ( i q s i+ q s ) G (N(x i x i+ )) = τ 0 ds N i ( i qs i+ qs ) q s G (N(x i x i+ )) 70

71 After a Schwarz G(N(x i x i+ )) [ ] q q τ dω N i = τ 0 ds N i τ ds 0 i ( i qs i+ qs ) q s G (N(x i x i+ )) [ i qs i+ qs ] 2 N 2 (x i x i+ ) 2 dω /2 τ ds 0 i G (N(x i x i+ )) 2 (x i x i+ ) 2 q s dω /2 τ ds 0 i,j [ i qs j qs ] 2 N 2 (x i x j ) 2 dω /2 τ N The first term is estimated by D ω ( q) from the additional term in Bakry-Emery. 7

72 We will use this for q = g τ, where g t dω = f t dµ, or, g t = f t /ψ. Comparison of dynamics: t f t = Lf t f t µ µ slow [true DBM] t q t = Lq t q t ω ω fast [approx] Compare f t µ = g t ω with ω for t 3t 0. Theorem 3: [Comparison of dynamics] Set τ = R 2 N ε 3t 0. Let g t = f t /ψ. Assume that S µ (f t0 ) CN m with some fixed m. Then the entropy and the Dirichlet form satisfy the estimates: S ω (g τ/2 ) CNR 2 Q, D ω ( g τ ) CNR 4 Q. Recall: Q := sup t t 0 j (x j γ j ) 2 f t dµ. Note: g t does NOT evolve according to the L dynamics (then the estimate would be exp. small). 72

73 Still, differentiating entropy (similar to Boltzmann s t S = D) t S ω (g t ) = 2 N j ( j gt ) 2 dω+ j From the Schwarz inequality t S ω (g t ) D ω ( g t )+CN j Together with LSI, we have b j j g t dω, b j = W j = x j γ j R 2 b 2 j g tdω D ω ( g t )+CNQR 4 t S ω (g t ) CR 2 S ω (g t ) + CNQR 4 which, after integrating from t = t 0 to τ/2, proves the bound on S ω (g τ ), using that τ = R 2 N ε and S ω (g 0 ) = S µ (f t0 ψ) = S µ (f t0 ) log(z/ Z) + N In fact, we have S ω (g s ) CR 2 NQR 4 = CNQR 2 Wf t0 dµ CN m. for any s τ/2. The estimate for D is integration of (*) from τ/2 to τ. 73 ( )

74 Theorem [Recall] Completion of the proof of Step 2. Q := sup t t 0 j (x j γ j ) 2 f t dµ, t 0 = N 2ε ( ) G i,m (x) := G N(x i x i+m ), N(x i+m x i+m2 ),..., N(x i+mn x i+mn ). Then for any sufficiently small ε > 0, and for τ 3t 0 we have G i,m (x)f τ dµ G i,m (x)dµ N N CNε Qτ + Ce cnε. i i Proof: Recall that t 0 = N 2ε, τ = R 2 N ε. Choose q = g τ = f τ /ψ in Theorem 2. Then Thm 2 and 3 give: G i,m (x)f τ dµ G i,m (x)dω N N CNε Qτ + Ce cnε. i i Use the same for f 0 = (so f t = ) to compare f τ µ with µ. 74

75 Step 3: Green function comparison theorem Theorem Let H (v) and H (w) be two generalized N N Wigner matrices, with matrix elements h ij given by the random variables N /2 v ij and N /2 w ij, respectively, with v ij and w ij satisfying the uniform subexponential decay condition. Define interpolating matrices, H 0 = H (v), H,... H N 2 = H (w), i.e. order the elements in some way and let H γ such that its matrix elements h ij follow the v-distribution up to γ and then they follow the w-distribution; Assume that for any y N +ε and E < 2, ( Assume moment matching H γ E iy ) kk Evij k Ev = Ewk ij, k =,2,3, 4 ij Ewij 4 N δ 75

76 Let F be a nice function and G (v) (z) = (H (v) z) be the resolvent. Then EF k N Tr G (v) (zj ),..., k n N Tr G (v) (zj n ) EF ( G (v) G (w)) j= j= for any collection of spectral parameters, zj m N ε, Ej m 2 2κ. = E m j ± iη, with η Similar statement holds for matrix elements. Moral of the statement: if four moments (almost) match, local statistics are the same. 76

77 Sketch of the proof of the comparison theorem E F ( ) N Tr H (v) E F z = γ(n) γ= ( ) N Tr H (w) z ( E F N Tr H γ z [ ) E F ( )] N Tr. H γ z H γ with H γ differ only in the (i, j) and (j, i) matrix elements: H γ = Q + N V, V := v ij e i e j + v ji e j e i H γ = Q + N W, W := w ij e i e j + w ji e j e i, Q ij = Q ji = 0 Define the Green functions R = Q z, S = H γ z. 77

78 Step : We have S kk = ( H γ z ) kk by resolvent perturbation: = R kk = ( Q z ) kk R = S + N /2SV S N 9/5(SV )9 S + N 5(SV )0 R. Note that offdiagonal elements can be estimated by diagonal, morally G kl max G kk V has only at most two nonzero element, so (SV S) kl = S ki v ij S jl + S kj v ji S il. In general, (SV SV...V S) kk is a finite sum of products of S kl and v ij. For the last term, use the trivial bound R ij η. 78

79 Step 2. Compare the resolvents of H γ and H γ with the resolvent R = (Q z) : S = R N RV R+ N (RV )2 R N 3/2(RV )3 R+ N 2(RV )4 R N 5/2(RV )5 S, so we can write with N TrS = R + ξ, ξ = 4 m= N m/2 R (m) + N 5/2Ω R = N TrR, R (m) = ( ) m N Tr(RV )m R, Ω = N Tr(RV )5 S. Each of them is a sum of a few terms, e.g. R (2) = [ R ki v ij R jj v ji R ik +R ki v ij R ji v ij R ik +R kj v ji R ii v ij R jk +R kj v ji R ij v ji R ik N k ((i, j) fixed!) and similar formulas hold for the other terms. 79

80 Taylor expand: ( EF N Tr H γ z ) =EF ( R + ξ ) =E [ F( R) + F ( R)ξ + F ( R)ξ F (5) ( R + ξ )ξ 5] = 5 m=0 N m/2ea(m), where ξ [0, ξ]; the A (m) s are defined as A (0) = F( R), A () = F ( R) R (), A (2) = F ( R)( R () ) 2 +F ( R) R (2), etc. The expectation values of the terms A (m), m 4, with respect to v ij are determined by the first four moments of v ij, for example E A (2) = F ( R) [ N k R ki R jj R ik +... ] E v ij 2 +F ( R) [ ] N 2 R ki R jk R lj R il +... E v ij 2. k,l 80

81 The same expansion holds for the resolvent of H (γ ) but w ij replaces v ij. If the four moments match, then the first four (fully expanded) terms in ( ) EF N Tr 5 = H γ z m=0 N m/2ea(m) coincide. The difference is given by the fifth (error) term, which is of order N 5/2 : A (5) = F ( R)Ω + F (5) ( R + ξ )( R () ) , Ω = N Tr(RV )5 S. It involves S, that is correlated with V, but Schwarz it out. If the fourth moments match only up to N δ, then ( ) ( ) E F N Tr E F H γ z N Tr CN 5/2 + CN 2 δ. H γ z Sum up the telescopic sum with N 2 terms, we get ( ) ( ) E F N Tr H (v) E F z N Tr H (w) CN /2 + CN δ. z 8

82 SUMMARY All results for general Wigner matrices, no explicit formula used Local semicircle law for the DOS on the optimal scale /N Eigenvectors are fully delocalized away from the spectral edges. Level repulsion for k eigenvalues with optimal exponent. Universality of local eigenvalue statistics for general Wigner matrices with subexponentially decaying matrix elements. OPEN: Universality for band matrices and random Schrödinger. 82