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Register a new userThe Eigenvector Moment Flow and local Quantum Unique Ergodicity
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We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.
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Consider $Ntimes N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W geq N^{3/4+varepsilon}$ for any $varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+varepsilon}$ for any $varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${rm L}^infty$ norms are all simultaneously bounded by $N^{-frac{1}{2}+varepsilon}$ (after normalization in ${rm L}^2$) with overwhelming probability, for any $varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method cite{BouErdYauYin2017}, which required $W=Omega(N)$, to the current setting $W ge N^{3/4+varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W geq N^{3/4+varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation cite{BouYau2017} under the matrix Brownian motions.
A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coud`ene, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coud`enes theorem to other kinds of foliations.
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ e}$ for non-Hermitian random matrices at any point $z in C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $ |z|-1=oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $ |z|-1=oo(1)$ up to scale $N^{-1/4+ e}$.
We prove an analogue of Shnirelman, Zelditch and Colin de Verdieres Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.
We consider a repeated quantum interaction model describing a small system $Hh_S$ in interaction with each one of the identical copies of the chain $bigotimes_{N^*}C^{n+1}$, modeling a heat bath, one after another during the same short time intervals $[0,h]$. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order $h$. After giving the GNS representation, we establish the relation between the time scale $h$ and the classical low density limit. We introduce a chemical potential $mu$ related to the time $h$ as follows: $h^2=e^{betamu}$. We further prove that the solution of the associated discrete evolution equation converges strongly, when $h$ tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.
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