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Christoph Schumacher

Publications and source records attributed to Christoph Schumacher.

15 recordsLinked to original sources
arXiv2026

Quantitative concentration inequalities for the uniform approximation of the IDS

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.

math.ST↗
arXiv2024

Twice Fourier transformable measures and diffraction theory

Mathematical diffraction theory has been developed since about 1995. Hof's initial approach relied on tempered distributions in euclidean space. Nowadays often the Fourier theory by Argabright and Gil de Lamadrid is used, which applies to appropriate measures on locally compact abelian groups. We review diffraction theory using Wiener amalgams as test function spaces. For translation bounded measures, this unifies and simplifies the former two approaches. We treat weighted versions of Meyer's model sets as examples.

math-ph↗
arXiv2021

Lifshitz asymptotics and localization for random breather models

We prove Lifshitz behavior at the bottom of the spectrum for non--negative random potentials, i.\,e.\ show that the IDS is exponentially small at low energies. The theory is developed for the breather potential and generalized to all non--negative random potentials in a second step. Since our models need not be ergodic, we need to identify the minimum of the spectrum. We deduce an initial length scale estimate from the Lifshitz bound for the breather model and combine it with a recent Wegner estimate to establishes Anderson localization via multi-scale analysis. Finally, for ergodic models, we complement the Lifshitz behavior with a lower bound. We provide detailed proofs accessible to non-experts.

math.SP↗
arXiv2020

The Statistics of Noisy One-Stage Group Testing in Outbreaks

In one-stage or non-adaptive group testing, instead of testing every sample unit individually, they are split, bundled in pools, and simultaneously tested. The results are then decoded to infer the states of the individual items. This combines advantages of adaptive pooled testing, i. e. saving resources and higher throughput, with those of individual testing, e. g. short detection time and lean laboratory organisation, and might be suitable for screening during outbreaks. We study the COMP and NCOMP decoding algorithms for non-adaptive pooling strategies based on maximally disjunct pooling matrices with constant row and column sums in the linear prevalence regime and in the presence of noisy measurements motivated by PCR tests. We calculate sensitivity, specificity, the probabilities of Type I and II errors, and the expected number of items with a positive result as well as the expected number of false positives and false negatives. We further provide estimates on the variance of the number of positive and false positive results. We conduct a thorough discussion of the calculations and bounds derived. Altogether, the article provides blueprints for screening strategies and tools to help decision makers to appropriately tune them in an outbreak.

stat.AP↗
arXiv2020

On sampling and interpolation by model sets

We refine a result of Matei and Meyer on stable sampling and stable interpolation for simple model sets. Our setting is model sets in locally compact abelian groups and Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid. This leads to a refined version of the underlying model set duality between sampling and interpolation. For rather general model sets, our methods also yield an elementary proof of stable sampling and stable interpolation sufficiently far away from the critical density, which is based on the Poisson Summation Formula.

math.FA↗
arXiv2019

Uniform existence of the IDS on lattices and groups

We present a general framework for thermodynamic limits and its applications to a variety of models. In particular we will identify criteria such that the limits are uniform in a parameter. All results are illustrated with the example of eigenvalue counting functions converging to the integrated density of states. In this case, the convergence is uniform in the energy.

math.PR↗
arXiv2017

Lifshitz tails for Schrödinger operators with random breather potential

We prove a Lifshitz tail bound on the integrated density of states of random breather Schrödinger operators. The potential is composed of translated single site potentials. The single site potential is an indicator function of set $tA$ where $t$ is from the unit interval and $A$ is a measurable set contained in the unit cell. The challenges of this model are: Since $A$ is not assumed to be star-shaped the dependence of the potential on the parameter $t$ is not monotone. It is also non-linear and not differentiable.

math-ph↗
arXiv2017

Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups

We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost additive, with finite range dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem, more precisely, that averages along a Foelner sequence converge uniformly to a limiting function. Moreover we give explicit error estimates for the approximation in the sup norm.

math-ph↗
arXiv2016

A Glivenko-Cantelli Theorem for almost additive functions

We develop a Glivenko--Cantelli theory for monotone, almost additive functions of i.\,i.\,d.\ sequences of random variables indexed by~$\Z^d$. Under certain conditions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We apply the results to yield uniform convergence for several quantities arising naturally in statistical physics.

math.PR↗
arXiv2015

Graph Laplacians do not generate strongly continuous semigroups

We show that for graph Laplacians $Δ_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $α\,\mathrm{Id}+βΔ_G, α,β\in\mathbb{K},β\ne 0$, generate a strongly continuous semigroup on $\mathbb{K}^V$ when the latter is equipped with the product topology.

math.FA↗
arXiv2013

EURETILE 2010-2012 summary: first three years of activity of the European Reference Tiled Experiment

This is the summary of first three years of activity of the EURETILE FP7 project 247846. EURETILE investigates and implements brain-inspired and fault-tolerant foundational innovations to the system architecture of massively parallel tiled computer architectures and the corresponding programming paradigm. The execution targets are a many-tile HW platform, and a many-tile simulator. A set of SW process - HW tile mapping candidates is generated by the holistic SW tool-chain using a combination of analytic and bio-inspired methods. The Hardware dependent Software is then generated, providing OS services with maximum efficiency/minimal overhead. The many-tile simulator collects profiling data, closing the loop of the SW tool chain. Fine-grain parallelism inside processes is exploited by optimized intra-tile compilation techniques, but the project focus is above the level of the elementary tile. The elementary HW tile is a multi-processor, which includes a fault tolerant Distributed Network Processor (for inter-tile communication) and ASIP accelerators. Furthermore, EURETILE investigates and implements the innovations for equipping the elementary HW tile with high-bandwidth, low-latency brain-like inter-tile communication emulating 3 levels of connection hierarchy, namely neural columns, cortical areas and cortex, and develops a dedicated cortical simulation benchmark: DPSNN-STDP (Distributed Polychronous Spiking Neural Net with synaptic Spiking Time Dependent Plasticity). EURETILE leverages on the multi-tile HW paradigm and SW tool-chain developed by the FET-ACA SHAPES Integrated Project (2006-2009).

cs.DC↗
arXiv2012

Approximation of the integrated density of states on sofic groups

In this paper we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows to treat applications from long-range percolation and the Anderson model. Our results apply to operators on Z^d, amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without any ergodic theorem at hand.

math.SP↗
arXiv2012

Classical Motion in Random Potentials

We consider the motion of a classical particle under the influence of a random potential on R^d, in particular the distribution of asymptotic velocities and the question of ergodicity of time evolution.

math-ph↗
arXiv2011

The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schrödinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its the deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.

math.FA↗