1 / 25

Spectra and Density in Sparse Systems - From Fibonacci Sequences to Anderson Localization

This study explores the spectra of sparse matrices and random operators, ranging from Fibonacci sequences to Anderson localization. It examines the density of linear subgraphs (chains) at the percolation point and analyzes the spectral density of ensembles of linear chains. Additionally, it investigates the asymptotic behavior of the Dedekind function and its relationship to the spectrum tail for q approaching 1. The research includes direct numeric simulations and explores topics such as isometric embedding, hyperbolic geometry, and relief construction using the Dedekind h-function.

pauln
Download Presentation

Spectra and Density in Sparse Systems - From Fibonacci Sequences to Anderson Localization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Native ultrametricity in random sparse systems Sergei Nechaev LPTMS, Orsay, France

  2. Spectra of sparse matrices and random operators: from Fibonacci sequences to Anderson localization…and more Sergei Nechaev LPTMS, Orsay, France In collaboration with V. Avetisov (Moscow), P. Krapivsky (Boston)

  3. Density of linear subgraphs (chains) at percolation point: Total cluster density: 95% subgraphs at percolation point are linear chains

  4. Linear bi-diagonal operators

  5. Spectral density of ensembles of linear chains

  6. Principle series of peaks Generic expression of the principle series

  7. Degeneracy of series of peaks

  8. Degeneracy of a principle series of peaks Visibility diagram

  9. Spectrum tail for q < 1 Lifshitz tail of 1D Anderson localization

  10. Spectrum tail for q  1

  11. Asymptotic behavior of the Dedekind function

  12. Conjecture about asymptotic behavior of the Dedekind function and spectrum tail for q  1 Direct numeric simulations Limiting eigenvalue density Log of Dedekind h

  13. Thomae (Dirichlet) function Static and Dynamical Phyllotaxis in Magnetic Cactus C. Nisoli et al: ArXiv: cond-mat/0702335

  14. Attempts to squeeze the surface in the 3D space Jupe a godets

  15. Isometric embedding of an uniform open Cayley tree Lobachevsky plane (space)

  16. Explicit construction by conformal maps is realized by the Dedekind eta-function

  17. Hyperbolic geometry in Nature Relief on the basis of Dedekind h-function coral

  18. Surface cut near maxima of the relief

More Related