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Fabian Schwarzenberger

Publications and source records attributed to Fabian Schwarzenberger.

10 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↗
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

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↗
arXiv2014

Uniform existence of the integrated density of states on metric Cayley graphs

Given a finitely generated amenable group we consider ergodic random Schrödinger operators on a Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states as the limit can be expressed by a Pastur-Shubin formula. The spectrum supports the corresponding measure and discontinuities correspond to the existence of compactly supported eigenfunctions.

math-ph↗
arXiv2013

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper we study unimodular amenable groups. The first part is devoted to results on the existence of uniform families of epsilon-quasi tilings for these groups. In this context, constructions of Ornstein and Weiss are extended by quantitative estimates for the covering properties of the corresponding decompositions. Afterwards, we apply the developed methods to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable, amenable group into some Banach space. This significantly extends and complements the previous results of Lenz, Müller, Schwarzenberger and Veselić. Further, using the Lindenstrauss ergodic theorem, we describe a link of our results to classical ergodic theory. We conclude with two important applications: the uniform approximation of the integrated density of states on amenable Cayley graphs, as well as the almost-sure convergence of cluster densities in an amenable bond percolation model.

math.SP↗
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↗
arXiv2011

A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.

math-ph↗