Random Block Operators

We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.Comment: 22 pages, 3 figure

Lifshitz tails for spectra of Erd\H{o}s--R\'{e}nyi random graphs

We consider the discrete Laplace operator $\Delta^{(N)}$ on Erd\H{o}s--R\'{e}nyi random graphs with $N$ vertices and edge probability $p/N$. We are interested in the limiting spectral properties of $\Delta^{(N)}$ 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 $\Delta^{(N)}$ exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.Comment: Published at http://dx.doi.org/10.1214/1050516000000719 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nuclear shock waves in heavy-ion collisions

It is shown that nuclear matter is compressed during the encounter of heavy ions. If the relative velocity of the nuclei is larger than the velocity of first sound in nuclear matter (compression sound for isospin T=0), nuclear shock waves occur. They lead to densities which are 3-5 times higher than the nuclear equilibrium density &#961;0, depending on the energy of the nuclei. The implications of this phenomenon are discussed

Interfaces between highly incompatible polymers of different stiffness: Monte Carlo simulations and self-consistent field calculations

We investigate interfacial properties between two highly incompatible polymers of different stiffness. The extensive Monte Carlo simulations of the binary polymer melt yield detailed interfacial profiles and the interfacial tension via an analysis of capillary fluctuations. We extract an effective Flory-Huggins parameter from the simulations, which is used in self-consistent field calculations. These take due account of the chain architecture via a partial enumeration of the single chain partition function, using chain conformations obtained by Monte Carlo simulations of the pure phases. The agreement between the simulations and self-consistent field calculations is almost quantitative, however we find deviations from the predictions of the Gaussian chain model for high incompatibilities or large stiffness. The interfacial width at very high incompatibilities is smaller than the prediction of the Gaussian chain model, and decreases upon increasing the statistical segment length of the semi-flexible component.Comment: to appear in J.Chem.Phy

Privacy sets for constrained space-filling

The paper provides typology for space filling into what we call "soft" and "hard" methods along with introducing the central notion of privacy sets for dealing with the latter. A heuristic algorithm based on this notion is presented and we compare its performance on some well-known examples

Monte Carlo simulations of copolymers at homopolymer interfaces: Interfacial structure as a function of the copolymer density

By means of extensive Monte Carlo simulations of the bond fluctuation model, we study the effect of adding AB diblock copolymers on the properties of an interface between demixed homopolymer phases. The parameters are chosen such that the homopolymers are strongly segregated, and the whole range of copolymer concentrations in the two phase coexistence region is scanned. We compare the ``mushroom'' regime, in which copolymers are diluted and do not interact with each other, with the ``wet brush'' regime, where copolymers overlap and stretch, but are still swollen by the homopolymers. A ``dry brush'' regime is never entered for our choice of chain lengths. ``Intrinsic'' profiles are calculated using a block analysis method introduced by us in earlier work. We discuss density profiles, orientational profiles and contact number profiles. In general, the features of the profiles are similar at all copolymer concentrations, however, the profiles in the concentrated regime are much broader than in the dilute regime. The results compare well with self-consistent field calculations.Comment: to appear in J. Chem. Phy