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.Comment: 19 pages; presentation slightly improved, some comments and references added; to appear in Mathematische Zeitschrif

The capabilities of the functional-analytic and of the functional-integral approach for the construction of the Hamiltonian as a self-adjoint operator on Hilbert space are compared in the context of non-relativistic quantum mechanics. Differences are worked out by taking the one-dimensional hydrogen atom as an example, that is, a point mass on the Euclidean line subjected to the inverse-distance potential. This particular choice is made with the intent to clarify a long-lasting discussion about its spectral properties. In fact, for the four-parameter family of possible Hamiltonians the corresponding energydependent Green functions are derived in closed form. The multiplicity of Hamiltonians should be kept in mind when modelling certain experimental situations as, for instance, in quantum wires.

We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the acconductivity is bounded from above by Cν 2 (log 1 ν ) d+2 at small frequencies ν. This is to be compared to Mott's formula, which predicts the leading term to be Cν 2 (log 1 ν ) d+1 . IntroductionThe occurrence of localized electronic states in disordered systems was first noted by Anderson in 1958 [An], who argued that for a simple Schrödinger operator in a disordered medium,"at sufficiently low densities transport does not take place; the exact wave functions are localized in a small region of space." This phenomenon was then studied by Mott, who wrote in 1968 [Mo1]: "The idea that one can have a continuous range of energy values, in which all the wave functions are localized, is surprising and does not seem to have gained universal acceptance." This led Mott to examine Anderson's result in terms of the Kubo-Greenwood formula for σ EF (ν), the electrical alternating current (ac) conductivity at Fermi energy E F and zero temperature, with ν being the frequency. Mott used its value at ν = 0 to reformulate localization: If a range of values of the Fermi energy E F exists in which σ EF (0) = 0, the states with these energies are said to be localized; if σ EF (0) = 0, the states are nonlocalized.Mott then argued that the direct current (dc) conductivity σ EF (0) indeed vanishes in the localized regime. In the context of Anderson's model, he studied the behavior of Re σ EF (ν) as ν → 0 at Fermi energies E F in the localization region (note Im σ EF (0) = 0). The result was the well-known Mott's formula for the ac-conductivity at zero temperature [Mo1], [Mo2], which we state as in

Shear relaxation and dynamic density fluctuations are studied within a Rouse model, generalized to include the effects of permanent random crosslinks. We derive an exact correspondence between the static shear viscosity and the resistance of a random resistor network. This relation allows us to compute the static shear viscosity exactly for uncorrelated crosslinks. For more general percolation models, which are amenable to a scaling description, it yields the scaling relation k = φ − β for the critical exponent of the shear viscosity. Here β is the thermal exponent for the gel fraction and φ is the crossover exponent of the resistor network. The results on the shear viscosity are also used in deriving upper and lower bounds on the incoherent scattering function in the long-time limit, thereby corroborating previous results. 61.25.Hq, 64.60.Ht, 61.20.Lc

Abstract. -We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation k = φ − β between the critical exponent k of the viscosity, the thermal exponent β associated with the gel fraction and the crossover exponent φ of a random resistor network.

We provide an ergodic theorem for certain Banach-space valued functions on structures over Z d , which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated discrete finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.

Abstract. We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterise ergodicity geometrically in terms of pattern frequencies. The general framework allows to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs.

We present event-driven simulations of a granular gas of inelastic hard disks with incomplete normal restitution in two dimensions between vibrating walls (without gravity). We measure hydrodynamic quantities such as the stress tensor, density and temperature profiles, as well as velocity distributions. Relating the local pressure to the local temperature and local density, we construct a local constitutive equation. For strong inelasticities the local constitutive relation depends on global system parameters, like the volume fraction and the aspect ratio. For moderate inelasticities the constitutive relation is approximately independent of the system parameters and can hence be regarded as a local equation of state, even though the system is highly inhomogeneous with heterogeneous temperature and density profiles arising as a consequence of energy injection. With respect to local velocity distributions we find that they do not scale with the square root of the local granular temperature. Moreover the high-velocity tails are different for the distribution of the x and the y components of the velocity, and even depend on the position in the sample, the global volume fraction, and the coefficient of restitution.