Homogenization of the Dyson Brownian Motion P. Bourgade, joint work - PowerPoint PPT Presentation

Homogenization of the Dyson Brownian Motion P. Bourgade, joint work with L. Erd˝ os, J. Yin, H.-T. Yau Cincinnati symposium on probability theory and applications, September 2014

Introduction Universality Log-correlated Gaussian field Homogenization for eigenvector moment flow . . . . . . . . . . . . . . . . . . . A spacially confined quantum mechanical system can only take on certain discrete values of energy. Uranium-238 : Quantum mechanics postulates that these values are eigenvalues of a certain Hermitian matrix (or operator) H , the Hamiltonian of the system. The matrix elements H ij represent quantum transition rates between states labelled by i and j . Wigner’s universality idea (1956). Perhaps I am too courageous when I try to guess the distribution of the dis- tances between successive levels. The situation is quite simple if one attacks the problem in a simpleminded fa- shion. The question is simply what are the distances of the characteristic values of a symmetric matrix with random coefficients.

Introduction Universality Log-correlated Gaussian field Homogenization for eigenvector moment flow . . . . . . . . . . . . . . . . . . . Wigner’s model : the 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 (2, 4 for invariance under unitary or symplectic conjugacy). • Semicircle law as N → ∞ . • Limiting bulk local statistics of GOE/GUE/GSE calculated by Gaudin, Mehta, Dyson.

Introduction Universality Log-correlated Gaussian field Homogenization for eigenvector moment flow . . . . . . . . . . . . . . . . . . . Dyson’s description of the first experiments. All of our struggles were in vain. 82 levels were too few to give a statistically significant test of the model. As a contribution of the understanding of nuclear phy- sics, random matrix theory was a dismal failure. By 1970 we had decided that it was a beautiful piece of work having nothing to do with physics. When N → ∞ and the nu- clei statistics performed over a large sample, the gap probabi- lity agree (resonance levels of 30 sequences of 27 different nu- clei).

Introduction Universality Log-correlated Gaussian field Homogenization for 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 Homogenization for 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 Homogenization for 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 This does not include Jimbo, Miwa, Mori, Sato relations for gaps in Bernoulli matrices, for example. 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 ? Related developments : gaps universality by Erd˝ os Yau (2012). The gaps are much more stable statistics than the fixed energy ones : N 1 ⟨ λ i , λ j ⟩ ∼ 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 Homogenization for eigenvector moment flow . . . . . . . . . . . . . . . . . . . Theorem. Fixed energy universality holds for Wigner matrices from all symmetry classes. Individual eigenvalues fluctuate as a 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) − → step is replaced What makes the Hermitian universality easier ? The by HCIZ formula : correlation functions of � H t are explicit only for β = 2.

Introduction Universality Log-correlated Gaussian field Homogenization for eigenvector moment flow . . . . . . . . . . . . . . . . . . . A few facts about the 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 Homogenization for 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 Homogenization for 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. Rigidity of the eigenvalues, optimal Wegner estimates (for level-repulsion), and the H¨ older regularity of the discrete-space parabolic equation.

Introduction Universality Log-correlated Gaussian field Homogenization for 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, never considered as a second-order Hilbert transform fundamental solution. 3. Explicit kernels can be obtained for all Bernstein-Szego measures, c α,β (1 − x 2 ) 1 / 2 ϱ ( x ) = ( α 2 + (1 − β 2 )) + 2 α (1 + β ) x + 4 βx 2 .