Quantitative concentration inequalities for the uniform approximation of the IDS

TL;DR

This paper establishes a quantitative concentration inequality for the empirical integrated density of states (IDS) of a discrete random Schrödinger operator, providing explicit probabilistic bounds for its uniform approximation of the theoretical IDS.

Contribution

It introduces a new concentration inequality that quantifies how closely the empirical IDS approximates the theoretical IDS with high probability.

Findings

Explicit high-probability bounds for IDS approximation

Confidence regions for spectral distribution

Quantitative error estimates for finite-volume approximations

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 Schr\"odinger 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.