| Title: | Diagonally dominant random matrices: Physical questions, Mathematical challenges |
| Speaker: | Rajinder Mavi Postdoctoral Researcher Institute of Mathematical Physics and Department of Mathematics Michigan State University East Lansing, MI |
| Abstract: | A remarkable phenomenon in quantum physics is that impurities in solid state materials will inhibit mobile quantities, such as electrons, spin orientations, and even information. The consequences range from the familiar to the remarkable: copper wires with impurities of aluminum or silicon have higher resistivity, inhibiting the wire's conductance, a more surprising effect is that, left to their own devices, strongly disordered materials do not reach their thermodynamic equilibrium! This phenomenon is known as Anderson localization and it is a fundamental part of the theory of solid state physics. In the future, we might find applications for disordered materials exhibiting such behavior playing an important role in the construction of quantum computer microchips. A simple mathematical model exhibiting a physically relevant approximation to a disordered material is a diagonally dominant random matrix. A diagonally dominant random matrix is a random diagonal matrix perturbed by a symmetric, non-random, sparse matrix. In quantum mechanics, one is typically interested in the properties of the eigenbasis, i.e. the eigenvectors and eigenvalues of the matrix. If the system is one dimensional, or if the perturbation is small, the eigenbasis is similar to the unperturbed matrix. That is to say, most of the `mass' of most eigenvectors is at a single entry of the vector. Although this may seem unremarkable, the difficulty is showing this is true for fixed perturbation strength with probability one, regardless of the size of the matrix. We will also compare eigenbases of diagonally dominant random matrices to to eigenbases of `traditional' random matrices which have i.i.d. random variables at all entries of the matrix. In the later case, the mass of each eigenvector is more or less equally distributed over all entries of the vector. We will then examine an interesting interpolation between diagonally dominant random matrices and traditional random matrices. Finally, we will discuss some recent results and current questions in the field of diagonally dominant random matrices today. |
| Location: | Palenske 227 |
| Date: | 2/8/2018 |
| Time: | 3:30 PM |
@abstract{MCS:Colloquium:RajinderMavi:2018:2:8,
author = "{Rajinder Mavi}",
title = "{Diagonally dominant random matrices: Physical questions, Mathematical challenges}",
address = "{Albion College Mathematics and Computer Science Colloquium}",
month = "{8 February}",
year = "{2018}"
}