Dyson's spike for random Schroedinger operators and Novikov-Shubin invariants of groups
| Autor: | Bálint Virág, Marcin Kotowski |
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| Rok vydání: | 2016 |
| Předmět: |
FOS: Physical sciences
Group Theory (math.GR) Poisson distribution 01 natural sciences Measure (mathematics) Combinatorics symbols.namesake 0103 physical sciences FOS: Mathematics 0101 mathematics Independence (probability theory) Mathematical Physics Mathematics 010102 general mathematics Probability (math.PR) Zero (complex analysis) Lie group Statistical and Nonlinear Physics Mathematical Physics (math-ph) symbols Novikov self-consistency principle 010307 mathematical physics Random matrix Mathematics - Group Theory Schrödinger's cat Mathematics - Probability |
| DOI: | 10.48550/arxiv.1602.06626 |
| Popis: | We study one dimensional Schroedinger operators with random edge weights and their expected spectral measures \({\mu_H}\) near zero. We prove that the measure exhibits a spike of the form \({\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{\mid{{\rm log}\varepsilon}\mid^2}}\) (first observed by Dyson), without assuming independence or any regularity of edge weights. We also identify the limiting local eigenvalue distribution, which is different from Poisson and the usual random matrix statistics. We then use the result to compute Novikov–Shubin invariants for various groups, including lamplighter groups and lattices in the Lie group Sol. |
| Databáze: | OpenAIRE |
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