Diffusion Profile and Delocalization for Random Band Matrices

Transcription

1 Diffusion Profile and Delocalization for Random Band Matrices Antti Knowles Courant Institute, New York University Newton Institute 19 September 2012 Joint work with László Erdős, Horng-Tzer Yau, and Jun Yin

2 Diffusion: a zoo of models Model Classical Quantum Stochastic dynamics (no memory) Hamiltonian particle in heat bath Hamiltonian particle in random environment Random Walk [Wiener] Einstein s kinetic model [Dürr-Goldstein-Lebowitz] Lorentz gas (random scatterers) [Kesten-Papanicolau, Komorowsky-Ryzhik] Random kick model with no time correlation [Pillet, Schenker-Kang] Particle in phonon bath [Erdős, Erdős-Adami, Spohn- Lukkarinen, De Roeck-Fröhlich, De Roeck-Kupiainen] Anderson model, band matrix [Spohn, Erdős-Yau, Erdős-Salmhofer-Yau, Disertori- Spencer-Zirnbauer, Erdős-K] Periodic Lorentz gas Billiards [Bunimovich-Sinai] Ballistic (trivial) [Bloch] Many-body Hamiltonian Boltzmann equation [Landford: short times] Quantum boltzmann [unsolved] Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 2

3 Two standard models of quantum disorder Models on {1,..., N}. (For simplicity, take one dimension.) Wigner random matrix. The entries of H are i.i.d. up to the constraint H = H. This is a mean-field model with no spatial structure. Eigenvectors are delocalized, local spectral statistics are given by random matrix theory (RMT). Anderson Model. On-site randomness + short-range hopping. v 1 1 H = + 1 v 2 1 v x = x... vn v N Eigenvectors are localized, local spectral statistics are Poisson. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 3

4 Universality conjecture for disordered quantum systems Wigner: The statistics of any disordered quantum system (described by a large random matrix) falls into one of the following categories. Weak disorder: eigenvectors are delocalized; local spectral statistics are given by RMT. Strong disorder: eigenvectors are localized; local spectral statistics are Poisson. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 4

5 Band matrices A family of interpolating models. Allow a precise statement of Wigner s conjecture. Let H = (H xy ) be an N N matrix with mean-zero entries independent up to contraint H = H. Let f be a probability density on R (the band profile) and suppose that Here W [1, N] is the band width. E H xy 2 = s xy := 1 W f ( x y W ). Summary: If W = O(1) then H Anderson model. If W = O(N) then H Wigner matrix. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 5

6 Anderson transition for band matrices W = O(1) = eigenvectors are localized. W = O(N) = eigenvectors are delocalized. Varying 1 W N provides a means to test the Anderson transition. Conjecture (Fyodorov-Mirlin [1991]) The Anderson transition occurs at W N 1/2. Let l denote the typical localization length of the eigenvectors of H. Then the conjecture means that l W 2. Previous results: l/w W 7 (Schenker [2010]). l/w W 1/6 (Erdős, K [2010]). This talk: l/w W 1/4. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 6

7 Quantum diffusion for the propagator Define the expected quantum transition probability from 0 to x in time t through ϱ(t, x) := E (e ith ) x0 2. Consider the diffusive regime t = ζt and x = ζ 1/2 W X with ζ. Diffusion cannot hold for x W 2 = choose ζ = W κ for 0 < κ < 2. Theorem (Erdős-K [2010]) Fix 0 < κ < 1/3. Then for all T 0 ϱ ( W κ T, W 1+κ/2 X ) 1 0 dλ 4 π λ 2 G(λT, X) 1 λ 2 weakly in X, where 1 G(T, X) := e 1 2T D X2, D := 1 X 2 f(x) dx. 2πT D 2 Proof: Diagrammatic expansion and perturbative renormalization. Corollary: eigenvectors have localization length at least W 1+κ/2. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 7

8 The resolvent approach to quantum diffusion Instead of studying the unitary time evolution e ith, consider the resolvent G(z) := (H z) 1 with Im z 0. These are equivalent by the identities G(z) = i e itz e ith dt, e ith = i e itz G(z). 2π Γ 0 Challenge is to control: e ith for large t G(E + iη) for small η. These two problems are comparable for η t 1. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 8

9 Instead of (e ith ) xy 2 use (H z) 1 2 = G xy 2. Set S := (s xy ), the step distribution of a random walk. xy Theorem (Spencer [2010]) For fixed η > 0 we have where is the diffusion profile and E G xy 2 = Θ xy ( 1 + O(W 1 ) ), Θ := m := 1 2π m 2 S 1 m 2 S ξ 2 dξ ξ z is the Stieltjes transform of Wigner s semicircle law. Proof: perturbative renormalization. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 9

10 Interpretation of Θ Fourier transform Ŝ(p) := x e ipx s x0 f(w p) = 1 D(W p) 2 +. An elementary computation yields (recall z = E + iη) m 2 = 1 γ 1 η + O(η 2 ), where γ := 4 E 2. 2 In Fourier space Θ therefore reads Recall the interpretation t η 1. m 2 Ŝ(p) 1 m 2Ŝ(p) γ η + Dγ(W p) 2, This is the resolvent of the classical diffusion operator Dγ(W p) 2. The diffusion constant D is renormalized by the factor γ 1. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 10

11 Going back to x-space we get ( 1 W Θ xy η exp ) η x y W 1 Nη if η ( ) W 2 N if η ( ) W 2 N. Plot of ηθ x0 : Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 11

12 Quantum diffusion (η 0) in the resolvent language We consider versions of G xy 2 with slight averaging: T xy := u s xu G uy 2 or Y xy := E x G xy 2, etc. All of these behave in the same way, i.e. G xy 2 is self-averaging. Focus on T xy : Theorem (Erdős-K-Yau-Yin, [2012]) Suppose that N W 5/4 and η W 1/2. Then with high probability ( ) 1 T xy = Θ xy + O. Nη Corollary: For N W 5/4 the eigenvectors are delocalized with high probability. Variants: Higher dimensions. Heavy-tailed band shape: f(x) x 1 β as x, where β (0, 2). Yields superdiffusion with symbol W p β. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 12

13 Sketch of proof (1) Derive a self-consistent matrix equation T = m 2 ST + m 2 S + E (1) with a random error term E. (Note: setting E = 0 would give T = Θ.) (2) Error term E is controlled using Fluctuation Averaging Theorem for polynomials in entries of G and G. The size is controlled using Λ := max Gxy δ xy m. x,y (3) Solve (1) for T : control (1 m 2 S) 1 E. (4) T controls Λ with high probability. (5) Self-improving scheme Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 13

14 Fluctuation Averaging Theorem [Erdős-K-Yau, 2012] Main work: control E. It consists of a zoo of terms such as s xa G ya G az, s xa (1 E a ) G ay 2, s xa s ab G ba G ay G yb, etc. a a Deals with arbitrary averages of arbitrary polynomials of entries of G and G. In particular used to estimate E. Example: R = a s xa (1 E a ) G ay 2. Each entry of G has a natural small size Λ: naive size of R is Λ 2. Since ER = 0, use cancellations to get R Λ 4. (Similar to CLT.) Problem: G ay 2 and G ay 2 are strongly correlated for small η. (CLT-type arguments fail.) Proof uses an intricate hierarchical expansion (indexed by graphs) to a very high order. a,b Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 14

15 The idea of using averaging of resolvent entries to improve bounds first appeared in Erdős-Yau-Yin [2010], Erdős-K-Yau-Yin [2011] for the estimate 1 N (1 E a ) 1 Λ 1+1 G a used in proof of universality of random matrices. Here: Arbitrary polynomials in {G xy, G xy}. a More gain from averaging (exploiting several novel mechanisms). Examples: a,b s xa (1 E a ) G ay 2 Λ 2+2, a s xa s ab G za G ab G bw Λ 3+2, a,b 1 N s xb(1 E b ) ( G ya G ab G ) bz Λ 3+3. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 15

16 The self-consistent equation Set G (i) := (H (i) z) 1 where H (i) is the minor of H without the i-th row and column. Key identities (A) G ij = G (k) ij (B) G ij = G ii + G ikg kj G kk, k i h ik G (i) kj = G jj Use G uu = m + and (B) to get G uy 2 = m 2 δ uy + m 2 (1 δ uy ) Take partial expectation and use (A): E u G uy 2 a,b u = m 2 δ uy + m 2 a u k j G (j) ik h kj (i j). h ua G (u) ay G (u) yb h bu +. s ua G (u) 2 ay + = m 2 δ uy + m 2 a u s ua G ay 2 +. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 16

17 Average over u: T xy = u s xu G uy 2 = u s xu E u G uy 2 + u s xu (1 E u ) G uy 2 = m 2 s xy + m 2 u s xu T uy + E xy. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 17

18 Conclusion Diffusion profile with strong control for N W 5/4 and η W 1/2. Eigenvectors are delocalized for N W 5/4. Proof relies on a self-consistent matrix equation combined with a general Fluctuation Averaging Theorem. Major open questions: Improve N W 5/4 to N W 2. Control resolvent for η W 1. Spectral universality of random band matrices for N W 2. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 18

19 Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 19

20 Interpretation of λ The quantum particle spends a macroscopic time λt moving according to a random walk, with jump rate O(1) in time t and transition kernel p(y x) = E H xy 2. The remaining fraction (1 λ)t is the time the particle wastes in backtracking. Probability density of λ: 4 λ 2 π 1 λ Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 20

21 Naive (and doomed) attempt: power series expansion of e ith/2 The moment method (Wigner 1955: semicircle law,... ) involves computing E Tr H n = x 1,...,x n E H x1x 2 H x2x 3 H xnx 1 for large n. Because of EH xy = 0, nonzero terms have a complete pairing (or a higher-order lumping). Graphical representation: path x 1, x 2,..., x n, x 1. The path is nonbacktracking if x i x i+2 for all i. The path is fully backtracking if it can be obtained from x 1 by successive replacements of the form a aba. (This generates double-edged trees.) Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 21

22 A fully backtracking path is paired by construction; its contribution is 1. Proof. Sum over all vertices, starting from the leaves. Each summation yields a factor y E H xy 2 = 1. Fully backtracking paths give the leading order contribution as W. Wigner s original derivation of the semicircle law involved counting the number of fully backtracking paths. Applying this strategy to ϱ leads to trouble: the expansion ϱ(t, x) = n,n 0 i n n t n+n 2 n+n n!n! EHn 0xH n x0 is unstable as t. The main contribution comes from fully backtracking graphs, whose number is of the order 4 n+n. The main contribution to the sum over n, n comes from n + n t (Poisson), diverges like e 4t as t. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 22

23 Getting rid of the trees: perturbative renormalization Simple example: Let z = E + iη with η > 0 and compute EG xx (z) = E(H z) 1 xx. Assuming that the semicircle law holds, we know what to expect: EG xx (z) = E 1 G yy (z) = E 1 1 N N λ α z = Em N(z) m(z) y where m N (z) is the Stieltjes transform of the empirical eigenvalue density N 1 α δ λ α, and 1 4 x 2 m(z) := dx x z 2π is the Stieltjes transform of the semicircle law. Note: m(z) is uniquely characterized by α z + m(z) + 1 m(z) = 0, m(z) < 1 (z / [ 2, 2]). Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 23

24 Choose µ µ(z) C and expand G(z) around µ 1 : Thus we get 1 H z = 1 µ + H µ z = 1 µ + 1 µ ( µ + z H ) n. µ µ n=1 EG xx = 1 µ + 1 µ 2 (µ + z) + 1 ( (µ + z) 2 µ 3 + EHxx 2 ) 1 + µ 4 ( ) +. Choose µ so that red terms cancel: 1 µ 2 (µ + z) = 1 µ 3 EH2 ii = 1 µ 3 z + µ + 1 µ = 0. We need µ > 1 for convergence: choose µ = m 1. Using a graphical expansion, one can check that this choice of µ leads to a systematic cancellation of leading-order pairings (trees) up to all orders. What remains are higher-order corrections of size O(W 1 ). In particular, EG xx = m + o(1). This is essentially two-legged subdiagram renormalization in perturbative field theory. Works also for more complicated objects like E G xy 2. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 24

25 Renormalization using Chebyshev polynomials A more systematic and powerful renormalization: use a beautiful algebraic identity due to Bai, Yin, Feldheim, Sodin,.... Define the n-th nonbacktracking power of H through H (n) x 0x n := Assume from now on that x 1,...,x n 1 H x0x 1 H xn 1x n H xy = n 2 i=0 1(1 x y W ) 2W 1 Unif(S 1 ). We shall see later how to relax this condition. 1(x i x i+2 ). Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 25

26 Lemma [Bai, Yin] H (n) = HH (n 1) H (n 2) Proof. Introduce 1 = 1(x 0 x 2 ) + 1(x 0 = x 2 ) into (HH (n 1) ) x0x n. Feldheim and Sodin inferred that H (n) = Ũn(H), where Ũn(ξ) = U n (ξ/2) and U n is the standard Chebyshev polynomial of the second kind. Indeed, we have Ũ n (ξ) = ξũn 1(ξ) Ũn 2(ξ). Thus, we expand the propagator e ith/2 in terms of Chebyshev polynomials: e itξ = n 0 α n (t)u n (ξ). We can compute the coefficients α n (t) = 2 π 1 1 e itξ U n (ξ) 1 ξ 2 dξ = 2( i) n n + 1 J n+1 (t), t where J n is the n-th Bessel function of the first kind. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 26

27 The graphical representation The Chebyshev expansion yields ϱ(t, x) = Represent matrix multiplication by loop; upper edge represents H (n) 0x and lower edge H (n ) x0. Taking the expectation yields a lumping (or partition) Γ of the edges: EH (n) 0x H(n ) x0 = Γ G n,n V x (Γ). Each lump γ Γ contains at least two edges. The most important lumpings are the pairings; their contribution estimates the contribution of all other lumpings (see later). n,n 0 α n (t)α n (t) EH (n) 0x H(n ) x0. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 27

28 The leading order contribution to ϱ is given by the ladder pairings L 0, L 1, L 2,... : ϱ ladder (t, x) = n 0 The ladder pairings α n (t) 2 V x (L n ) The family of weights { α n (t) 2 } n=0 is a t-dependent probability distribution on N (since the family {U n } n=0 is orthonormal). The number α n (t) 2 is the probability that the particle performs n steps of a random walk during the time t. The steps of the random walk have the transition kernel p(y x) = E H 2 xy = 1 W f ( x y W ). The distribution of the number of jumps does not concentrate at n t because of possible delays due to backtracking The function λ t α [λt] (t) 2 for t = 150 (brown), and its weak limit 4 λ2 (blue) as t. π 1 λ2 Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 28

29 The non-ladder lumpings We have to prove that the sum of the contributions of all non-ladder lumpings vanishes. This is the main work! First, we estimate the contribution of all non-pairings in terms of the contribution of all non-ladder pairings. = It is enough to show that the contribution of all non-ladder pairings vanishes as W. Problem: The summation labels are associated with vertices, but edges are paired. We need to extract conditions on the vertex labels from a pairing of the edges. Basic philosophy: The more complicated a pairing, the more constraints it induces on the vertex labels, and therefore the smaller its contribution. This fights against the larger number of complicated pairings. = We need a means of quantifying the combinatorial complexity of a pairing. Observation: A group of parallel lines has a large contribution, but a trivial combinatorics = parallel lines should not contribute to the combinatorial complexity of a pairing. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 29

30 Step 1. Collapse the parallel lines. We assign to each pairing Γ its skeleton pairing S(Γ) obtained from Γ by collapsing all parallel lines of Γ. The size 2m of S(Γ) is the correct notion of complexity for Γ. We recover Γ by replacing each line σ of S(Γ) with a number l σ of parallel lines. The contribution of l σ parallel lines is given by a random walk with l σ steps = use heat kernel bounds on each line of S(Γ) (local CLT). Step 2. Estimate the number of free labels in a skeleton: the 2/3 rule. Since parallel lines are forbidden, each vertex label must appear at least three times. Thus, the number L of free labels satisfies 3L 2m, i.e. L 2m/3. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 30

31 Step 3. Encode the skeleton as a multigraph and sum everything up using heat kernel bounds. Each edge σ of the multigraph carries a random walk of l σ steps. To sum up the graph, choose a spanning tree of the multigraph. Apply heat kernel bounds: l 1 -bound for each tree edge ( factor 1) l -bound for each nontree edge ( factor l 1/2 σ W 1 ). The 2/3 rule implies that the number of nontree edges is at least m/3. = Contribution of skeletons of size m is (roughly) bounded by ( ) n m! (W 1 ) m/3. m This is summable for n t W 1/3. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 31

32 The necessity of the condition κ < 1/3 The following family Σ 1, Σ 2, Σ 3,... of skeletons is critical; Σ k is defined as The bound in the 2/3 rule is saturated: each vertex label occurs exactly three times (6k edges and 4k free labels). It is not hard to see that the contribution of all such skeletons diverges as κ 1/3. = going beyond κ = 1/3 requires (i) resummations of terms with different n, n, or (ii) a more refined use of heat kernel bounds. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 32

33 General band matrices So far we assumed that H xy = W 1/2 1(1 x y W ) Unif(S 1 ), which was necessary for the algebraic identity to hold. H (n) = HH (n 1) H (n 2) (2) If the entries of H are general, the algebraic relation (2) is no longer exact; the RHS of (2) receives the error terms Φ 2 H (n 2) Φ 3 H (n 3), where ( (Φ 2 ) xy := δ xy Hxz 2 E H xz 2), (Φ 3 ) xy := H xy 2 H xy. z Strategy: Φ 3 is small by power counting (easy), Φ 2 has zero expectation (hard). The rigorous treatment requires a considerably more complicated class of graphs; out of the simple loop grow backtracking branches. The cancellation of backtracking paths is no longer complete. The organization of the expansion of the backtracking branches is quite involved. The threshold t W 1/3 is again necessary. Antti Knowles Diffusion Profile and Delocalization for Random Band Matrices 33