From the mesoscopic to microscopic scale in random matrix theory P. - PowerPoint PPT Presentation

From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Wigner’s universality idea (1956). Perhaps I am too courageous when I try to guess the distribution of the distances between successive levels (of energies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fashion. The question is simply what are the distances of the characteristic values of a symmetric matrix with random coefficients. Gaussian Orthogonal Ensemble : (a) Invariance by H �→ U ∗ HU , U ∈ O( N ). (b) Independence of the H i,j ’s, i ≤ j . The entries are Gaussian and the spectral density is ∏ 1 | λ i − λ j | β e − β N i λ 2 ∑ 4 i Z N i<j with β = 1. Semicircle law as N → ∞ . Eigenstates Haar-distributed.

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Fundamental belief in universality : the macroscopic statistics (like the equilibrium measure) depend on the models, but the microscopic statistics are independent of the details of the systems except the symmetries. • GOE : Hamiltonians of systems with time reversal invariance • GUE : no time reversal symmetry (e.g. application of a magnetic field) • GSE : time reversal but no rotational symmetry Correlation functions. For a point process χ = ∑ δ λ i : ε → 0 ε − k P ( χ ( x i , x i + ε ) = 1 , 1 ≤ i ≤ k ) . ρ ( N ) ( x 1 , . . . , x k ) = lim k For deterministic systems, P is an averaging over the energy level in the semiclassical limit. Gaudin, Dyson, Mehta : for any E ∈ ( − 2 , 2) then ( β = 2 for example) ( ) u 1 u k sin( π ( u i − u j )) ρ ( N ) E + Nϱ ( x ) , . . . , E + N →∞ det − → . k Nϱ ( x ) π ( u i − u j ) k × k

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Wigner matrix : symmetric, Hermitian (or symplectic), entries have variance 1 /N , some large moment is finite. The Wigner-Dyson-Mehta conjecture. Correlation functions of symmetric Wigner matrices (resp. Hermitian, symplectic) converge to the limiting GOE (resp. GUE, GSE). Recently universality was proved under various forms. Fixed (averaged) energy universality. For any k ≥ 1, smooth F : R k → R , for arbitrarily small ε and s = N − 1+ ε , ∫ E + s ∫ ( ) ∫ 1 d x v d v F ( v ) ρ ( N ) d v F ( v ) ρ ( GOE ) lim x + d v = ( v ) k k ϱ ( E ) k s Nϱ ( E ) N →∞ E

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Johansson (2001) Hermitian class, fixed E , Gaussian divisible entries Erd˝ os Schlein P´ ech´ e Ramirez Yau (2009) Hermitian class, fixed E Entries with density Tao Vu (2009) Hermitian class, fixed E Entries with 3rd moment=0 Erd˝ os Schlein Yau (2010) Any class, averaged E Key input for all recent results : rigidity of eigenvalues (Erd˝ os Schlein Yau) : | λ k − γ k | ≤ N − 1+ ε in the bulk. Optimal rigidity ? Jimbo, Miwa, Mori, Sato & condition number of ± 1 matrices ? Related developments : gaps universality by Erd˝ os Yau (2012). The gaps are much more stable statistics than the fixed energy ones : N − 2 N ⟨ λ i , λ j ⟩ ∼ N − 2 log 1 + | i − j | , almost crystal . ⟨ λ i +1 − λ i λ j +1 − λ j ⟩ ∼ 1 + | i − j | 2

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Theorem (B., Erd˝ os, Yau, Yin, 2014). (i) Fixed energy universality : for Wigner matrices from all symmetry classes. (ii) Optimal fluctuations : Log-correlated Gaussian field. The Dyson Brownian Motion (DBM, d H t = d B t N − 1 2 H t d t ) is an essential √ interpolation tool, as in the Erd˝ os Schlein Yau approach to universality, summarized as : H 0 ↕ (DBM) � � H 0 − → H t (DBM) → : for t = N − 1+ ε , the eigenvaues of � − H t satisfy averaged universality. ↕ : Density argument. For any t ≪ 1, there exists � H 0 s.t. the resolvents of H 0 and � H t have the same statistics on the microscopic scale. (DBM) What makes the Hermitian universality easier ? The − → step is replaced by HCIZ formula : correlation functions of � H t are explicit only for β = 2.

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow A few facts about the general proof of fixed energy universality. (i) A game coupling 3 Dyson Brownian Motions. (ii) Homogenization allows to obtain microscopic statistics from mesoscopic ones. (iii) Need of a second order type of Hilbert transform. Emergence of new explicit kernels for any Bernstein-Szeg˝ o measure. These include Wigner, Marchenko-Pastur, Kesten-McKay. (iv) The relaxing time of DBM depends on the Fourier support of the test (DBM) function : the step − → becomes the following. N ∑ ( ) � F ( λ , ∆) = F { N ( λ i j − E ) + ∆ , 1 ≤ j ≤ k } i 1 ,...,i k =1 F ⊂ B(0 , 1 / √ τ ), then for t = N − τ , Theorem. If supp ˆ E � F ( λ t , 0) = E � F ( λ (GOE) , 0) .

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow First step : coupling 3 DBM. Let x (0) be the eigenvalues of � H 0 and y (0) , z (0) those of two indepndent GOE.   √ ∑ N d B i ( t ) + 1 2 1 − 1  d t d x i / d y i / d z i = 2 x i /y i /z i N x i /y i /z i − x j /y j /z j j ̸ = i Let δ ℓ ( t ) = e t/ 2 ( x ℓ ( t ) − y ℓ ( t )). Then we get the parabolic equation ∑ B kℓ ( t ) ( δ k ( t ) − δ ℓ ( t )) , ∂ t δ ℓ ( t ) = k ̸ = ℓ 1 where B kℓ ( t ) = N ( x k ( t ) − x ℓ ( t ))( y k ( t ) − y ℓ ( t )) > 0. By the de Giorgi-Nash-Moser method (+Caffarelli-Chan-Vasseur+Erd˝ os-Yau), this PDE is older-continuous for t > N − 1+ ε , i.e. δ ℓ ( t ) = δ ℓ +1 ( t ) + O( N − 1 − ε ), i.e. gap H¨ universality. This is not enough for fixed energy universality.

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Second step : homogenization. The continuum-space analogue of our parabolic equation is ∫ 2 f t ( y ) − f t ( x ) ∂ t f t ( x ) = ( K f t )( x ) := ϱ ( y )d y. ( x − y ) 2 − 2 K is some type of second order Hilbert transform. Theorem. Let f 0 be a smooth continuous-space extension of δ (0) : f 0 ( γ ℓ ) = δ ℓ (0). Then for any small τ > 0 ( t = N − τ ) thre exists ε > 0 such that ( ) e t K f 0 ℓ + O( N − 1 − ε ) . δ ℓ ( t ) = Proof. Key inputs are the rigidity of the eigenvalues and optimal Wegner estimates (for level-repulsion).

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Third step : the continuous-space kernel. 1. For the translation invariant equation ∫ g t ( y ) − g t ( x ) ∂ t g t ( x ) = d y, ( x − y ) 2 R c t the fundamental solution is the Poisson kernel p t ( x, y ) = t +( x − y ) 2 . 2. For us, t will be close to 1, so the edge curvture cannot be neglected. Fortunately, K can be fully diagonalized and ( x = 2 cos θ , y = 2 cos ϕ ) c t k t ( x, y ) = | e i( θ + ϕ ) − e − t/ 2 | 2 | e i( θ − ϕ ) − e − t/ 2 | 2 . Called the Mehler kernel by Biane in free probability context. Here it appears as a second-order Hilbert transform fundamental solution. 3. Explicit kernels can be obtained for all Bernstein-Szeg˝ o measures, c α,β (1 − x 2 ) 1 / 2 ϱ ( x ) = ( α 2 + (1 − β 2 )) + 2 α (1 + β ) x + 4 βx 2 .

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Fourth step : microscopic from mesoscopic. Homogenization yields ∫ k t ( x, y ) f 0 ( y ) ϱ ( y )d y + O( N − 1 − ε ) δ ℓ ( t ) = The LHS is microscopic-type of statistics, the RHS is mesoscopic. This yields, up to negligible error, Nx ℓ ( t ) = Ny ℓ ( t ) − Ψ t ( y 0 ) + Ψ t ( x 0 ) , where Ψ t ( x 0 ) = ∑ h ( N τ ( x i (0) − E )) for some smooth h . We wanted to prove E � F ( x t , 0) = E � F ( z t , 0) + o(1) . We reduced it to E � F ( y t , − Ψ t ( y 0 ) + Ψ t ( x 0 )) = E � F ( y t , Ψ t ( y 0 ) + Ψ t ( z 0 )) + o(1) . where Ψ t ( y 0 ), Ψ t ( x 0 ) and Ψ t ( z 0 ) are mesoscopic observables and independent.

Introduction Universality Log-correlated Gaussian field The eigenvector moment flow Fifth step and conclusion : CLT for GOE beyond the natural scale. Do Ψ t ( x 0 ) and Ψ t ( y 0 ) have the same distribution ? No, their variance depend on their fourth moment. A stronger result holds : E � F ( y t , − Ψ t ( y 0 ) + c ) does not depend on the constant c . We know that E � F ( y t , − Ψ t ( y 0 ) + Ψ t ( z 0 ) + c ) = E � F ( y t , − Ψ t ( y 0 ) + Ψ t ( z 0 )) for all c (why ?). Exercise : let X be a random variable. If E g ( X + c ) = 0 for all c , is it true that g ≡ 0 ? Not always. But true if X is Gaussian (by Fourier). ( e i λ Ψ t ( z (0)) ) = e − λ 2 2 τ log N + O( N − 1 / 100 ) . Lemma. E The proof uses algebraic ideas of Johansson and rigidity of β -ensembles. F has size 1 / √ τ . This is why DBM By Parseval, proof when the support of ˆ needs to be run till time almost 1.