Explicit concentration inequalities for eigenvalue-counting functions in the Anderson model
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Date
2024
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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.
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Thermodynamic limit, Ergodic theorem, Random Schrödinger operator, Concentration inequalities, Anderson model, integrated density of states, Eigenvalue-counting function