The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents

The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, $$, can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions $D > 2$ one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes: low-dimensional systems ($2\leq D \leq 3$), where localized states are always exponentially localized and high-dimensional systems ($D\geq D_c=4$), where states with non-exponential localization are also formed. The value of the upper critical dimension is found to be $D_0=6$ for the Anderson localization problem; this value is also characteristic of a related problem - percolation.Comment: 17 pages, 5 figures, to appear in Eur. Phys.J.

Disorder effects on the static scattering function of star branched polymers

We present an analysis of the impact of structural disorder on the static scattering function of f-armed star branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good solvent in the presence of structural defects, correlated at large distances r according to a power law \sim r^{-a}. In particular, we are interested in the ratio g(f) of the radii of gyration of star and linear polymers of the same molecular weight, which is a universal experimentally measurable quantity. We apply a direct polymer renormalization approach and evaluate the results within the double \varepsilon=4-d, \delta=4-a-expansion. We find an increase of g(f) with an increasing \delta. Therefore, an increase of disorder correlations leads to an increase of the size measure of a star relative to linear polymers of the same molecular weight.Comment: 17 pages, 7 figure

Microscopic analysis of K^+-nucleus elastic scattering based on K^+N phase shifts

We investigate $K^{+}$-nucleus elastic scattering at intermediate energies within a microscopic optical model approach. To this effect we use the current $K^{+}$-nucleon {\it (KN)} phase shifts from the Center for Nuclear Studies of the George Washington University as primary input. First, the {\it KN} phase shifts are used to generate Gel'fand-Levitan-Marchenko real and local inversion potentials. Secondly, these potentials are supplemented with a short range complex separable term in such a way that the corresponding unitary and non-unitary {\it KN} $S$ matrices are exactly reproduced. These {\it KN} potentials allow to calculate all needed on- and off-shell contributions of the $t$ matrix,the driving effective interaction in the full-folding $K^{+}$-nucleus optical model potentials reported here. Elastic scattering of positive kaons from $^{6}$Li, $^{12}$C, $^{28}$Si and $^{40}$Ca are studied at beam momenta in the range 400-1000 MeV/{$c$}, leading to a fair description of most differential and total cross section data. To complete the analysis the full-folding model, three kinds of simpler $t\rho$ calculations are considered and results discussed. We conclude that conventional medium effects, in conjunction with a proper representation of the basic {\it KN} interaction are essential for the description of $K^{+}$-nucleus phenomena.Comment: 11 pages, 1 table, 12 figures, submitted to PR

Scaling in public transport networks

We analyse the statistical properties of public transport networks. These networks are defined by a set of public transport routes (bus lines) and the stations serviced by these. For larger networks these appear to possess a scale-free structure, as it is demonstrated e.g. by the Zipf law distribution of the number of routes servicing a given station or for the distribution of the number of stations which can be visited from the chosen one without changing the means of transport. Moreover, a rather particular feature of the public transport network is that many routes service common subsets of stations. We discuss the possibility of new scaling laws that govern intrinsic features of such subsets.Comment: 9 pages, 4 figure