Explicit concentration inequalities for eigenvalue-counting functions in the Anderson model
| dc.contributor.advisor | Veselic, Ivan | |
| dc.contributor.author | Kämper, Max | |
| dc.contributor.referee | Pogorzelski, Felix | |
| dc.date.accepted | 2024-09-24 | |
| dc.date.accessioned | 2024-12-16T11:59:55Z | |
| dc.date.available | 2024-12-16T11:59:55Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | The Anderson model is a discrete approximation of the Hamiltonian describing the quantum mechanical behaviour of electrons in crystals or metals, featuring a Schrödinger operator on a graph consisting of the sum of the discrete Laplace operator Δ and a random multiplication operator V_ω. Compressing this operator to the subspace of functions with support in a finite set Λ yields a hermitian matrix with real eigenvalues. This thesis provides the first explicit quantification of the uniform and almost-sure convergence of the normalized eigenvalue-counting functions N_ω^Λ of these matrix operators to the integrated density of states N of the original Anderson model. Results are given as concentration inequalities both for the lattice Z^d as well as general Cayley graphs of finitely generated amenable groups. Central to these results is a uniform law of large numbers, which is quantified by a bound on the Orlicz norm of a supremum over an empirical process. For Z^d, d ≥3 and suitable random fields V_ω there exists a universal constant K<1186 and sets Ω(n) such that |(|N_ω^(Λ_n )-N|)|_∞≤c(n) for all ω∈Ω(n) with explicit c(n)∼1/√n and P(Ω(n))≥1-2 exp(-√(⌊n\/⌊√n⌋⌋^d )/⌊√n⌋K), where Λ_n=[0,n)^d∩Z^d. Analogous results for d=1,2 as well as approximations along monotiling Følner sequences are also given. For finitely generated amenable groups analogous bounds are shown along Følner sequences based on the Ornstein-Weiss concept of ε-quasitilings. Treatment for operators with unbound hopping range is also achieved in the case of the Laplace operator on long-range percolation graphs on Z^d. | en |
| dc.identifier.uri | http://hdl.handle.net/2003/43283 | |
| dc.identifier.uri | http://dx.doi.org/10.17877/DE290R-25115 | |
| dc.language.iso | en | |
| dc.subject | Thermodynamic limit | en |
| dc.subject | Ergodic theorem | en |
| dc.subject | Random Schrödinger operator | en |
| dc.subject | Concentration inequalities | en |
| dc.subject | Anderson model | en |
| dc.subject | integrated density of states | en |
| dc.subject | Eigenvalue-counting function | en |
| dc.subject.ddc | 510 | |
| dc.subject.rswk | Anderson-Modell | de |
| dc.subject.rswk | Eigenwert | de |
| dc.subject.rswk | Konvergenz von Funktionen | de |
| dc.subject.rswk | Konzentration <Wahrscheinlichkeitsverteilung> | de |
| dc.title | Explicit concentration inequalities for eigenvalue-counting functions in the Anderson model | de |
| dc.type | Text | |
| dc.type.publicationtype | PhDThesis | |
| dcterms.accessRights | open access | |
| eldorado.secondarypublication | false |