Abstract
This article discusses some mathematical results and conjectures about random band matrix ensembles (RBM) and sparse matrix ensembles. Spectral problems of RBM and sparse matrices can be expressed in terms of supersymmetric (SUSY) statistical mechanics that provides a dual representation for disordered quantum systems. This representation offers important insights into nonperturbative aspects of the spectrum and eigenfunctions of RBM. The article first presents the definition of RBM ensembles before considering the density of states, the behaviour of eigenvectors, and eigenvalue statistics for RBM and sparse random matrices. In particular, it highlights the relations with random Schrödinger (RS) and the role of the dimension of the lattice. It also describes the connection between RBM and statistical mechanics, the spectral theory of large random sparse matrices, conjectures and theorems about eigenvectors and local spacing statistics, and the RS operator on the Cayley tree or Bethe lattice.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
