Properties of Linearly Sofic Groups

We consider (projectively) linearly sofic groups, i.e. groups which can be approximated using (projective) matrices over arbitrary fields, as a generalization of sofic groups. We generalize known results for sofic groups and groups which can be approximated with complex matrices, including the fact that free products of linearly sofic groups (using a fixed field) are linearly sofic.Comment: 20 page

Banking Supervision in Integrated Financial Markets: Implications for the EU

I analyze the optimal design of banking supervision in the presence of cross-border lending. Cross-border lending could imply that an individual bank failure in one country could trigger negative spillover effects in another country. Such cross-border contagion effects could turn out to be important in the EU because national banking problems could easily spread via the highly integrated interbank market. I show that if benevolent supervisors are accountable only to their own jurisdiction, they will not take cross-border contagion effects into account. Supervisors with such a national mandate fail to implement the optimum from a supranational perspective. In consequence, the probability of a bank failure will be inefficiently high. Against the background of this result, I argue in favor of institutionalizing an EU ”Supervisory Coordination Authority” to which national supervisors are accountable.banking supervision, systemic risk, cross-border contagion

The infinite XXZ quantum spin chain revisited: Structure of low lying spectral bands and gaps

We study the structure of the spectrum of the infinite XXZ quantum spin chain, an anisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves the number of down spins (or particle number), allowing to represent it as a direct sum of N-particle interacting discrete Schr\"odinger-type operators restricted to the fermionic subspace. In the Ising phase of the model we use this representation to give a detailed determination of the band and gap structure of the spectrum at low energy. In particular, we show that at sufficiently strong anisotropy the so-called droplet bands are separated from higher spectral bands uniformly in the particle number. Our presentation of all necessary background is self-contained and can serve as an introduction to the mathematical theory of the Heisenberg and XXZ quantum spin chains.Comment: 32 pages, 3 figure

Positivity of Lyapunov exponents for Anderson-type models on two coupled strings

We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol'dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices) to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies in the first model and for sufficiently large energies in the second model). The methods used allow for singularly distributed random parameters, including Bernoulli distributions.Comment: 19 page

Traces in monoidal categories

The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property

A generalization of Gordon's theorem and applications to quasiperiodic Schr\"odinger operators

We prove a criterion for absence of eigenvalues for one-dimensional Schr\"odinger operators. This criterion can be regarded as an $L^1$-version of Gordon's theorem and it has a broader range of application. Absence of eigenvalues is then established for quasiperiodic potentials generated by Liouville frequencies and various types of functions such as step functions, H\"older continuous functions and functions with power-type singularities. The proof is based on Gronwall-type a priori estimates for solutions of Schr\"odinger equations.Comment: 8 page