Universality of Random Matrices and Dyson Brownian Motion

Transcription

1 Universality of Random Matrices and Dyson Brownian Motion Horng-Tzer Yau Harvard University Joint work with L. Erdős, and B. Schlein, J. Yin 1

2 Lecture 1: History of Universality and Main Results Basic question: 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 of the matrix? Gaussian unitary ensemble: H =(h jk ) 1 j,k hermitian with h jk = 1 (x jk + iy jk ) and h jj = 2 x jj and where x jk, y jk, j > k, and x jj are independent centered Gaussian random variables with variance 1/2. Classical ensembles: Gaussian unitary ensemble (GUE), Gaussian orthogonal ensemble (GOE), Gaussian symplectic ensemble(gse), sample covariance ensembles 2

3 Probability density of eigenvalues (w.r.t. Lebesgue measure) p (x 1,..., x ) Correlation function for two eigenvalues: p (2) (x 1,x 2 )= Density of states: ρ (x) = R 2 p (x 1,x 2,..., x )dx 3...dx R 1 p (x, x 2,..., x )dx 2...dx 3

4 . 1 ρ (x) = 4 x 2 2π 2 2 Eigenvalues: λ 1 λ λ, λ i+1 λ i 1/. Global statistics: Density of state ρ(x) follows the Wigner semicircle law for GUE, GOE and GSE Sample covariance ensembles, matrix of the form A + A: Marchenko- Pastur law 4

5 Gaudin, Mehta, Wigner, Dyson: level correlation for local statistics ( lim p(2) E + a 1,E+ a ) 2 = det { S(a i a j ) } 2 i,j=1, πa S(a) =sin πa (for GUE), E < 2 Gap distribution E # of e.v. pairs with gap less than s near E # of eigenvalues near E s 0 d 2 dα 2 det(1 K α)dα K α (x, y) = sin π(x y) π(x y) as operator on L 2 ((0,α)) 5

6 Edge Universality λ : the largest eigenvalue of a random matrix. The probability distribution functions of λ for the classical Gaussian ensembles are identified by Tracy-Widom to be lim P( 2/3 (λ 2) t) =F β (t) where F β can be computed in terms of Painleve equations and β = 1, 2, 4 corresponds to the standard classical ensembles. Edge universality is less universal than the bulk. TW distribution appears in random interfaces, random corners and random processes. 6

7 History, Applications and Conjectures uclear Physics: the excitation spectra of heavy nuclei are expected to have the same local statistical properties as the eigenvalues of GOE (Wigner, 1955). Quantum Chaos Conjecture (Bohigas-Giannoni-Schmit 1984, Berry-Tabor, 1977 ) Riemann ζ-function: Gap distribution of zeros of ζ function is given by GUE (Montgomery, 1973). Anderson Model (1958): V ω random potential. random Schrödinger operator: H Λ = + λv ω the eigenvalue distributions of H in the delocalization regime (λ small) are given by GOE. Frohlich-Spencer, Disertori-Spencer-Zirnbauer, Supersymmetry etc 7

8 KOW RESULTS FOR UITARY ESEMBLES Unitary ensemble: Hermitian matrices with density P(H)dH e Tr V (H) dh Invariant under H UHU 1 for any unitary U (GUE) Joint density function of the eigenvalues is explicitly known p(x 1,..., x ) = const. (x i x j ) β e j V (x j) i<j classical ensembles β =1, 2, 4 GOE, GUE, GSE. large asymptotic of orthogonal polynomials Mehta-Gaudin (1960- ), Dyson ( ), via Hermite polynomials and general cases by Deift-Its-Zhou (1997), Pastur- Schcherbina (1997), Bleher-Its (1999), Deift-et al ( ), Lubinsky (2008)... 8

9 Universality of Generalized Wigner Ensembles H =(h kj ) 1 k,j, h ji = h ij independent Eh ij =0, E h ij 2 = σ 2 ij, i σ 2 ij =1,σ ij 1/2. δ C inf inf,i,j {σ2 ij } sup{σij 2 } C sup δ 1,i,j (A) sub-exponential decay: Ee h ij ε < (B) If h ij are i.i.d. then it is called Wigner ensembles. Wigner ensembles are completely different from the invariant ensembles due to the lack of explicit formula for the eigenvalue distribution. Only Gaussian are Wigner and invariant. (A) rules out the band and sparse matrices, but some estimates of the proof are valid for these two ensembles. 9

10 Theorem [Erdoes-Schlein-Y-Yin, ] The bulk universality holds for generalized Wigner ensembles satisfying (A) and (B), i.e., for 2 < E < 2 lim ( p (k) F, p(k) µ, )( E + b 1,..., E + b ) k 0 F µ generalized symmetric matrices GOE generalized hermitian GUE generalized self-dual quaternion GSE real covariance real Gaussian Wishart complex covariance complex Gaussian Wishart Edge universality: Two generalized Wigner ensembles have the same edge distributions if the first two moments of the two ensembles are the same. Earlier results for edge universality: Soshnikov, Sodin (moment methods and generalization). Tao-Vu: Wigner ensembles with vanishing third moments. 10

11 Three Steps to the Universality of Random Matrices Step 1. Local Semicircle Law, LSC (valid for more general class of ensembles) Step 2. Universality of Gaussian divisible ensembles H = H 0 + tv, t > 0, H 0 is Wigner V is GUE i.e., the matrix entries have a substantial Gaussian component. 2a. Asymptotic analysis of explicit formulas (Johansson, Brezin- Hikami) for correlation functions for eigenvalues: valid only for Hermitian matrices 2b. Local ergodicity of Dyson Brownian motion Gaussian convolution matrix H = e t/2 H 0 + (1 e t ) 1/2 V H 0 + tv 11

12 Step 3. Approximation by Gaussian divisible ensembles (A density argument via perturbation expansion) 3a. Reverse heat flow (eed smoothness) 3b. Green function comparison theorem Approach of this lecture: LSC + DBM + Green fn comparison on the bulk for generalized Wigner ensembles, all symmetry classes. LSC + Green fn comparison on the edges (Surprising fact: Green function methods are good both on the bulk and on the edges!) Tao-Vu s result: four moment theorem. (1): LSC+ Johansson+ 4-moments = universality for Wigner Hermitian ensembles for distribution whose support contains at least three points (Bernoulli excluded). (2) LSC+ 4-moments = universality for Wigner ensembles with the first four moments matched (three moments on the edges). o Johansson result for symmetric or quaternion case. 12

13 Outline of lectures 1. Statement of uniform local semicircle law and applications to rigidity of eigenvalues, eigenfunction delocalization estimates. Idea of proof of uniform local semicircle law. 2. Local ergodicity of DBM 3. Green function comparison theorem and universality. 4. Dyson s conjecture on local relaxation time of DBM. Pseudo-equilibrium measures, local relaxation flow, logarithmic Sobolev inequality, entropy estimates. 13

14 Step 1: Strong Local semicircle law Theorem [Erdös-Schlein-Y-Yin, 2010] Suppose assumptions (A) and (B) hold. Then for any z = E+iη with E 5, 1 η 10 we have ( P max G ii (z) m sc (z) 1 ) i η ( P max G ij (z) 1 ) ij η ( P m(z) m sc (z) 1 ) η where G ij = 1 H z (i, j) and m(z) = 1 TrG = 1 i G ii. ϱ(x)dx Stieltjes transform: m(z) = x z These estimates are optimal and are first results on matrix elements of Green functions. 14

15 Corollary [Rigidity of Eigenvalues] Let γ j be the classical location of the j-th eigenvalue, i.e. γj ϱ sc(x)dx = j Then [ P ( j : λ j γ j (log ) L min ( j, j+1 )] 1/3 ) 2/3 All estimates are optimal (up to log corrections) Tao-Vu: For Wigner matrices with vanishing third moment: (E [ λ j γ j 2]) 1/2 [ min ( j, j +1 )] 1/3 1/6 ε 0 with some small positive ε 0 15

16 From Stieltjes transform to counting function Helffer-Sjöstrand formula: Let f C 1 (R). Let χ(y) be a cutoff function in [ 1, 1]. Define the quasianalytic extension of f as f(x + iy) =(f(x)+iyf (x))χ(y), then = 1 2π f(λ) = 1 2π R 2 z f(x + iy) λ x iy dxdy R 2 iyf (x)χ(y)+i(f(x)+iyf (x))χ (y) λ x iy dxdy. 16

17 Detour: Similar results without Assumption (A) Theorem [Erdös-Yau-Yin, 2010] Let Eh ij = 0, E h ij 2 = σij 2 and Assumption (B) holds. Define M := (max ij σij 2 ) 1. Then for any z = E + iη with Mη 1 and ε E 2 κ 0 > 0: ( P max G ii (z) m sc (z) 1 ) i Mη ( P max G ij (z) 1 ) ij Mη P ( m(z) m sc (z) ε ) Mη Eigenfn. delocalization estimate Denote the eigenfunction by v α = (v α (1),..., v α ()). Then sup j v α (j) 2 C log /M with very high probability. Disertori Pinson-Spencer: Density of states for Gaussian band matrices. Related work on supersymmetric models 17

18 Local semicircle law H (i) denotes the ( 1) ( 1) 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 = 1 H z, 1 G(i) = H (i) z Then by inverting 2 2 matrix we have G ii = ( 1 H z ) ii = 1 h ii z a i G (i) a i H = ( h a a B ), h C, a C 1, B C ( 1) ( 1) (H z) 1 (1, 1) = 1 h z a (B z) 1 a 18

19 Crudest local semicircle law (for σ 2 ij = 1 ) G ii = 1 h ii z E i a i G (i) a i Z i Z i := a i G (i) a i E i a i G (i) a i simple computation: E i a i G (i) a i = 1 m(i) By the interlacing property of the eigenvalues of H and H (i), = G ii = m(z) = m(z) m (i) (z) 1 η, 1 z m(z)+ω i, 1 z m(z) + 1 The main error term is in terms of i Z i! i η = Im z ( ) 1 Ω i = h ii Z i + O η 1 (z + m(z)) 2Ω i + 19

20 G ii = 1 z m(z)+ω i, ( ) 1 Ω i = h ii Z i + O η Expand and sum up, with Ω = max i Ω i 1 1 m(z) = z + m(z) + O(Ω) Large deviation (CLT) estimate: Z i Imm(i) η Imm η ( ) 1 Imm m(z) = z + m(z) + O η By the stability analysis of this equation, and m sc = 1 z+m sc, m m sc C 1 η, E 2 > 0 20

21 Formulas relating Green functions: G ij = G jj G (j) ii K ij, K ij = h ij zδ ij a i G (ij) a j G ii = G (j) ii + G ijg ji, G ij = G (k) G ij jj Rule of thumb: G ii O(1), G ij is small. + G ikg kj G kk i, j, k, different Lemma For any even number p and away from the edges, we have E 1 p Z i i=1 C p (η) p for sufficiently large, i.e., with high probability: 1 i=1 Z i (η) 1 21

22 Step 2: Local Ergodicity of Dyson Brownian Motion Gaussian convolution matrix H = e t/2 H 0 + (1 e t ) 1/2 V where V is a GUE matrix can be obtained by evolving matrix elements by an OU process: t u t (x) =Lu t (x), L = 1 2dx 2 x d 2dx Denote the probability density of the eigenvalues by f t (x)µ(dx). GUE : µ = Ce H(x) dx, H(x) = t f t = L f t, L = 1 2 fl fdµ = 1 2 i=1 2 i + i=1 d 2 x 2 i /2 1 log x j x i i=1 i j f 2 dµ, 1 2 x i + 1 gradient flow of GUE j i 1 i, x i x j BM of matrix elements = DBM on eigenvalues. Johansson (essentially): local equilibrium is reached t = O(1). 23

23 (SDE) dx i = 1 db i + ( 1 2 x i + 1 j i 1 x i x j ) dt Derivation of DBM (for BM case, h ij = 1/2 B ij for simplicity): λ α h ik =ū α (i)u α (k), u α (i) h lk = β α 1 λ α λ β ū β (l)u α (j)u β (i) 2 λ α = h ik h lj β α 1 λ α λ β u β (l)ū α (j)ū β (i)u α (k)+c.c. dλ α = 1 ik ū α (i)u α (k)db ik + 1 ik β α 1 λ α λ β u β (i)ū α (k)ū β (i)u α (k)dt = 1 db α + 1 β α 1 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. 24

24 Assumption III. There exists an ε> 0 such that sup t 2a 1 j=1 with a constant C uniformly in. (x j γ j ) 2 f t (dx)µ(dx) C 1 2a The typical spacing between neighboring points is of order 1/ away from the spectral edges. Assumption III guarantees that typically the random points x j remain in the 1/2 a vicinity of their classical location. (The notation of eigenvalues are changed to x j ). 25

25 Theorem [Erdös-Schlein-Y-Yin, 2009] Suppose that Assumption III hold for the solution of f t = L f t. Let E < 2 and b>0. Then δ > 0, k 2 we have lim lim b 0 sup t 2a+δ E+b E b ( p (k) t, p(k) µ, de 2b )( E + α 1 ϱ(e),..., E + where a > 0 is the exponent from Assumption III. α ) k ϱ(e) 0 Explicit error estimates were obtained. Strong local semicircle law = a = 1/2 ε = local relaxation time to DBM is 1+ε (Dyson s conjecture). t 1 then the eigenvalue does not move on the scale of 1. 26

26 Once Step 2 is completed, the matrices fall into the universality class are in some sense dense.thus we need a density argument. Step 3 Approximation by Gaussian divisible ensembles: Reverse heat flow: Given a smooth distribution u(x)dx, choose then g t = (1 t + ( t )2 2! e t g t +... ) u e t u will have a Gaussian component and will be very close to u since t ε is small. 27

27 3c [Erdös-Y-Yin 10] (Green Function Comparison theorem) Suppose that the first four moments of v ij and w ij for two random matrices almost match. Let q j = E j + iη, j =1, 2 with E j < 2 and η 1 ε. Then we have ( ) ( ) ( ) 1 1 E v E w 0 q 1 H q 2 H ( = p (2) )( v, p(2) w, E + α 1,E+ α ) 2 0. Bulk universality: For any matrix ensemble H, find another matrix ensembles H 0 such that the first four moments of H and H 0 + tv are the same. ij kl 28

28 Proof E 1 Tr 1 H (v) z E 1 Tr 1 = 2 γ=1 [ H (w) z E 1 Tr 1 H γ z E 1 Tr 1 H γ 1 z H γ 1 with H γ differ only in the (i, j) and (j, i) matrix elements: H γ 1 = Q + 1 V, V := v ij e i e j + v ji e j e i H γ = Q + 1 W, W := w ij e i e j + w ji e j e i, Q ij = Q ji =0 ]. 29

29 resolvent identity: 1 z Q A = 1 z Q + 1 z Q A 1 z Q + 1 z Q A 1 z Q A 1 z Q +... Power counting: A h ij 1/2. So four moments matching implies a factor of 2 δ provided that all green functions in the resolvent expansions are finite. This is guaranteed by the LSC. Without LSC up to η 1, this expansion could be badly divergent. Edge universality: Power counting changes due to the density on the edge is 2/3. 30

30 3b 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. Inputs: 1. Local semicircle law (ESY, no smoothness asumption) 2. eigenfunction estimates (ESY, no smoothness assumption) 3. Level repulsion (ESY proof needs smoothness, TV s estimate is much weaker but sufficient for their purpose and they removed the smoothness assumption.) Main Difficulty: Overlaps of singularities in the perturbation of eigenvalues and eigenfunctions. 31

31 logarithmic Sobolev inequality and convexity Recall entropy of a probability density f w.r.t a measure µ: S(fµ µ) =S(f) = f log fdµ, µ = e H /Z Suppose t f t = L f t and µ is invariant, i.e., L f t dµ = 0. t S(f t )= t D( f t )= 1 2 = 1 2 (L f t ) log f t dµ + ( ft ) 2 f t L f t f t dµ f t dµ = 4D(h t ), h t := ( h)( 2 H) h 1 2 ( ij f t ij h ( ih)( j h) h Bakry-Emery: 2 H τ 1 = t D µ ( f t ) τ 1 D µ ( f t ) = LSI: S(fµ µ) τd µ ( f), S(f t µ µ) e Ct/τ S(f 0 µ µ). ) 2, 32

32 For DBM: 2 H 1(from x 2 j ) relaxation time is O(1) Global equilibrium is reached in time scale of O(1). Local equilibrium is believed to be reached in O( 1 ). F. Dyson on approach to equilibrium of DBM (1962): The picture of the gas coming into equilibrium in two wellseparated stages, with microscopic and macroscopic time scales, is suggested with the help of physical intuition. A rigorous proof that this picture is accurate would require a much deeper mathematical analysis. 33

33 ow we compute t D( f t ). Let h := f for simplicity, then t h t = 1 2h t h 2 t = 1 2h t L h 2 t = L h t + 1 2h t ( h t ) 2. In the last step we used that L h 2 =( h) 2 +2hL h that can be seen directly from L = ( H). 34

34 We compute (dropping the t subscript for brevity and = dµ) t D( f t )= 1 2 t = = = 1 2 ( h)( L h)+ 1 2 ( h)[, L ]h = 1 2 ( h) 2 dµ ( h)( 2 H) h 1 2 ij [ 2( j h)( i h) ij h h ( h)( 2 H) h 1 2 ( h) ( h)2 h ( h)l ( h)+ 1 2 ij ( i j h) 2 ij ( jh) 2 ( i h) 2 ( where we used the commutator ij h 2 [ 2( j h) i j h i h h ] ij h ( ih)( j h) h [, L ]= 1 2 ( 2 H). ) 2, ( jh) 2 ] i h h 2 35

35 Approach to equilibrium: 2 H τ 1 t S(fµ µ) = D µ ( f t ), t D µ ( f t ) τ 1 D µ ( = S(f t µ µ) e Ct/τ S(f 0 µ µ). τ 1 for DBM and there is no room to improve. f t ) Key Idea: Pseudo equilibrium measure ω = ψµ is almost an equilibrium. Then t S(f t µ ω) = t S(g t ω ω) = D ω ( g t )+Ω t, g t = f t /ψ where Ω t is the error term due to that ω is non-equilibrium. 36

36 Choose ω so that (1) ω is more convex than µ so that the LSI w.r.t. ω improves. (2) the error Ω t is still not too large. (3) ω and µ have almost identical local equilibrium. This implies that D ω ( g t ) is small provided Ω t is small. Thus f t ω and we can deduce the local statistics of f t from ω. ote: We never compared two dynamics! We estimate the Dirichlet form of f t w.r.t. the pseudo equilibrium measure ω with f t given by DBM. 37

37 Local relaxation flow pseudo equilibrium measure, ω e H, H = H + W, W = j=1 Local relaxation flow: W j (x j ), W j (x j )= 1 2τ (x j γ j ) 2, t u t = L u t, f L fdω = 1 2 (density of system at time t) =u t ω f 2 dω := D ω (f) gradient flow w.r.t. ω L = L j b j j, b j = W j (x j)=τ 1 (x j γ j ) 2 H τ 1 = relaxation time of L is τ. 38

38 Let ω = ψµ and define f t = g t ψ so that f t µ = g t ω. We will call ω pseudo equilibrium measure. ψ is time independent. Theorem: [Relative Entropy w.r.t pseudo equilibrium measure] Let g t = f t /ψ. Assume that S ω (g 0 ) C m with some fixed m. Then we have t S ω (g t ) D ω ( g t )+CQτ 2 Cτ 1 S ω (g t )+CQτ 2 Hence we obtain the estimates: S ω (g τ ) Cτ 1 Q, D ω ( g τ ) Cτ 2 Q. where Q := sup t 0 j (x j γ j ) 2 f t dµ. 39

39 General relative entropy calculation t hf t dµ = (L h)f t dµ = (L h)g t dω = hl ωg t dω d dt S ω(g t )= = = 1 2 g t t = L ωg t, (L ωg t )(log g t ) dω (g t ψ t ) L (log g t ) dµ (g t ψ t ) (log g t ) dµ = 1 2[ ψ t ( g t ) 2 /g t dµ + ( g t )( ψ t ) dµ ] 40

40 t S ω (g t )= 2 j ( j gt ) 2 dω+ j From the Schwarz inequality t S ω (g t ) D ω ( g t )+C j Together with LSI, we have b j j g t dω, b j = W j = x j γ j τ b 2 j g t dω D ω ( g t )+CQτ 2 t S ω (g t ) Cτ 1 S ω (g t )+CQτ 2 which, after integrating it from t = 0 to τ (actually slightly bigger than τ so that the initial entropy disappears) proves the bound on S ω (g τ ). The estimate for D follows from integration from τ to 2τ. 41

41 Theorem [Relaxation of the local relaxation flow] Fact: v, 2 H(x)v 1 τ v Consider the forward equation i<j (v i v j ) 2 t q t = L q t, t 0, (x i x j ) 2, v R. with initial condition q 0 = q and with reversible measure ω. Then t D ω ( q t ) 1 τ D ω( q t ) ds i,j=1 i,j=1 ( i qs j qs ) 2 ( i qt j qt ) 2 (x i x j ) 2 dω, (x i x j ) 2 dω D ω ( q) and the logarithmic Sobolev inequality holds S ω (q) CτD ω ( q) Thus the time to equilibrium is of order τ: S ω (q t ) e Ct/τ S ω (q 0 ). 42

42 Theorem [ Gap distribution-dirichlet form estimate ] Let q be a probability density w.r.t. ω and G is bounded smooth function with compact support. Then we have 1 G((x i x i+1 )) [ q 1 ] ( Dω dω C ( q)τ i where τ is the time to equilibrium for L. (The estimate (*) is time independent!) ) 1/2 (*) As a comparison we show what entropy control alone can reach: Theorem: We have, for any E i < 2 1 i G ( (x i E i ) )[ q 1 ] dω τs C ω (q) 43

43 Recall f t = L f t, L = 1 2 D ω ( g τ ) Cτ 2 Q, Q := sup t 0 i j [ i i H i ] (1) (x j γ j ) 2 f t dµ 2a (2) 1 G((x i x i+1 )) [ q 1 ] ( Dω dω C ( q)τ i Choose q = g τ = f τ /ψ. Then 1 G((x i x i+1 )) [ g τ 1 ] ( Dω dω C ( g τ )τ i C Q τ C 2a τ 0, if 2a τ ) 1/2 (3) ) 1/2 1 G((x i x i+1 ))f τ dµ = i 1 G(...)dω, i indep. of initial data 44

44 Proof. 1 i G((x i x i+1 )) [ q 1 ] dω = i 1 G(...) [ (q q t )+(q t 1) ] dω For the second term use entropy, 1 G(...) q t 1 dω C S ω (q t ) C S ω (q)e ct/τ For the first term, integrate its derivative 1 t 1 G(...)(q q t )dω = ds 0 = = = t i t 0 ds 1 t 0 ds 1 0 ds 1 i j i ( j q s ) j [ 1 i i i G(...) s q s dω ] G((x i x i+1 )) ( i q s i+1 q s ) G ((x i x i+1 )) ( i qs i+1 qs ) q s G ((x i x i+1 )) 45

45 After a Schwarz 1 G((x i x i+1 )) [ ] q q t dω i = t 0 ds 1 i t ds 0 i ( i qs i+1 qs ) q s G ((x i x i+1 )) [ i qs i+1 qs ] 2 2 (x i x i+1 ) 2 dω 1/2 t ds 0 i G ((x i x i+1 )) 2 (x i x i+1 ) 2 q s dω 1/2 t ds 0 i,j [ i qs j qs ] 2 2 (x i x j ) 2 dω 1/2 t The first term is estimated by D ω ( q) from the additional term in entropy dissipation term. 46

46 Three Steps to the Universality of Random Matrices Step 1. Local Semicircle Law Step 2. Strong local ergodicity of Dyson Brownian motion Step 3. Approximation via the Green function comparison thm Universality comes from the local ergodicity of DBM model independent and involves no explicit formula. Universality holds for all classical ensembles; the variances are allowed to be different (but bounded). The only condition on the distribution is subexponential decay (which can be weakened). Step 2 and 3 are now very simple and completely general. Only Step 1 may be model dependent. 47

47 Main Open Problems Universality for band matrices, random Schrödinger, non-classical β-ensembles. Random band matrices: H is symmetric with independent but not identically distributed entries with mean zero and variance E W h kl 2 = e k l /W arrow band, W = Broad band, W = localization, Poisson statistics delocalization, GOE statistics Even the Gaussian case is open. 48

48 Lemma [Large Deviation Estimates] Let a i be (1 i ) independent random complex variables with mean zero, variance σ 2 and are subexponential decay. Let A i, B ij C. Then P i=1 a i B ii a i P i=1 i j σ 2 B ii a i B ij a j ( σ2 i=1 ( σ2 i j B ii 2 ) 1/2 B ij 2 ) 1/2 C, C, From the Lemma: EZ 2 i 1 2 k,j i G (i) kj 2, G (i) = 1 H (i) z identity: G 2 = 1 η ImG, G 2 = GG 1 2 G (i) kl 2 1 η Imm(i), kl i Since m (i) m O(1), we have Ω Z i C η 49