arXiv subjects
Publications and source records attributed to Werner Kirsch.
The Penrose-Banzhaf index and the Shapley-Shubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in particular those coalitions a voter is decisive for. We present a new combinatorial formula how to calculate both indices solely using the set of minimal winning coalitions.
We investigate the voting rules in the Council of the European Union. It is known that the current system, according to the Treaty of Nice, and the voting system proposed in the Lisbon treaty both strongly deviate from the square root law by Penrose. This is known to be the ideal voting rule under certain assumptions. In 2004 Slomczynski and Zyczkowski designed a voting system, now known as the Jagiellonian Compromise. It satisfies the square root law with very high accuracy. Each member state in this system obtains a voting weight proportional to the square root of the population. Additionally the quota is fixed in such a way that the voting power of each country is also proportional to the square root of the population. In this paper we investigate to which extent a change of the quota in the Treaty of Nice and the Treaty of Lisbon may bring the voting power closer to the ideal square root distribution. Our computations show that even with optimal quota both systems are way off the ideal power distribution.
This review is an extended version of my mini course at the Etats de la recherche: Operateurs de Schroedinger aleatoires at the Universite Paris 13 in June 2002, a summer school organized by Frederic Klopp. These lecture notes try to give some of the basics of random Schroedinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization.
We prove a Wegner estimate for a large class of multiparticle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper.
In certain bodies, like the Council of the EU, the member states have a voting weight which depends on the population of the re- spective state. In this article we ask the question which voting weight guarantees a `fair' representation of the citizens in the union. The tra- ditional answer, the square-root law by Penrose, is that the weight of a state (more precisely: the voting power) should be proportional to the square-root of the population of this state. The square root law is based on the assumption that the voters in every state cast their vote inde- pendently of each other. In this paper we concentrate on cases where the independence assumption is not valid.
We survey some aspects of the theory of the integrated density of states (IDS) of random Schroedinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current research.
We consider the discrete Laplace operator $Δ^{(N)}$ on Erdős--Rényi random graphs with $N$ vertices and edge probability $p/N$. We are interested in the limiting spectral properties of $Δ^{(N)}$ as $N\to\infty$ in the subcritical regime $0<p<1$ where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of $Δ^{(N)}$ exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.
Bond-percolation graphs are random subgraphs of the d-dimensional integer lattice generated by a standard bond-percolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, self-adjoint, ergodic random operators with off-diagonal disorder. They possess almost surely the non-random spectrum [0,4d] and a self-averaging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the non-percolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge, where the exponent equals 1/2.
We consider Schrödinger operators on $L^2(R^d)$ with a random potential concentrated near the surface $R^{d_1}\times\{0\}\subset R^d $. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.
This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered : fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably "reduced" to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.
We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials $f: \mathbbm{R}^d \to [0, \infty[ $ decay at infinity in an anisotropic way, for example, $f(x_1,x_2)\sim (|x_1|^{α_1}+|x_2|^{α_2})^{-1}$ as $ |(x_1,x_2)| \to \infty $. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent $ d/2 $ if both $α_1$ and $α_2$ are big enough, and there is a so-called classical regime with Lifshits exponent depending on $α_1$ and $α_2$ if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on $ α_1$ and $α_2$ simultaneously.
We determine the leading low-energy fall-off of the integrated density of states of a magnetic Schroedinger operator with repulsive Poissonian random potential in case its single impurity potential has a slow anisotropic decay at infinity. This so-called magnetic Lifshits tail is shown to coincide with the one of the corresponding classical integrated density of states.
We study spectral properties of Schrödinger operators with random potentials of alloy type on $L^2(\RR)$ and their restrictions to finite intervals. A Wegner estimate for non-negative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the density of states. Our estimate is valid for all bounded energy intervals. Wegner estimates play a key role in an existence proof of pure point spectrum.
We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.