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Quantitative concentration inequalities for the uniform approximation of the IDS

Max Kämper, Christoph Schumacher, Fabian Schwarzenberger, Ivan Veselic

Abstract

The integrated density of states (IDS) is a fundamental spectral quantity for quantum Hamiltonians modeling condensed matter systems, describing how densely energy levels are distributed. It can be interpreted as a volume-averaged spectral distribution. Hence, there are two equivalent definitions of the IDS related by the Pastur-Shubin formula: an operator-theoretic trace formula and a limit of normalized eigenvalue counting functions on finite volumes. We study a discrete random Schrödinger operator with bounded random potentials of finite-range correlations and prove a quantitative concentration inequality ensuring, with explicit high probability, that the empirical IDS (normalized eigenvalue counting function) uniformly approximates the abstract IDS trace formula within a prescribed error, thereby implying confidence regions for the IDS.

Quantitative concentration inequalities for the uniform approximation of the IDS

Abstract

The integrated density of states (IDS) is a fundamental spectral quantity for quantum Hamiltonians modeling condensed matter systems, describing how densely energy levels are distributed. It can be interpreted as a volume-averaged spectral distribution. Hence, there are two equivalent definitions of the IDS related by the Pastur-Shubin formula: an operator-theoretic trace formula and a limit of normalized eigenvalue counting functions on finite volumes. We study a discrete random Schrödinger operator with bounded random potentials of finite-range correlations and prove a quantitative concentration inequality ensuring, with explicit high probability, that the empirical IDS (normalized eigenvalue counting function) uniformly approximates the abstract IDS trace formula within a prescribed error, thereby implying confidence regions for the IDS.
Paper Structure (11 sections, 12 theorems, 142 equations, 2 figures)

This paper contains 11 sections, 12 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation, model assumptions, and some basic properties
  3. Main results: Concentration inequalites for the evcf
  4. Discussion of achievements and limitations of our new findings
  5. Strategy of the proofs and outline for the remainder of the paper
  6. Geometric approximation estimates
  7. Probabilistic approximation estimates
  8. Bracket coverings for eigenvalue counting functions
  9. Concentration inequalities
  10. Proofs of the main results
  11. Extensions to other settings

Key Result

Theorem 1

For $d \geq 3$ and all $\alpha, \beta \in (0,1)$ provided where Here the supremum is taken over all Borel probability measures $\mu$ on $\mathbb{R}$ with bounded support.

Figures (2)

  • Figure 1: Illustration of some lattice subsets for $d=2$.
  • Figure 2: Illustration of the choice of $x_j (\varepsilon)$ (left) and a nested monotone bracketing cover consisting of evcf (reight).

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Theorem 2
  • Remark 3
  • Remark 4: What is different for dimensions one and two?
  • Theorem 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 17 more