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 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.
Keywords
Cite
@article{arxiv.2602.18214,
title = {Quantitative concentration inequalities for the uniform approximation of the IDS},
author = {Max Kämper and Christoph Schumacher and Fabian Schwarzenberger and Ivan Veselic},
journal= {arXiv preprint arXiv:2602.18214},
year = {2026}
}