Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model

We study numerically and analytically the average length of reduced (primitive) words in so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth modelComment: 29 pages, LaTeX, 7 separated Postscript figures (available on request), submitted to J. Phys. (A): Math. Ge

Multifractality of entangled random walks and non-uniform hyperbolic spaces

Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally nonsymmetric tree, statistical properties of topological invariants, such as drift and return probabilities, have been studied by means of a renormalization group (RG) technique. The comparison of the analytical RG--results with numerical simulations as well as with the rigorous results of P.Gerl and W.Woess demonstrates clearly the validity of our approach. It is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space (which has an exponentially large number of states) has been broken.Comment: 28 pages, 11 eps-figures (enclosed

Native ultrametricity of sparse random ensembles

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the distribution of linear chains is purely exponential. We analyze in detail the spectral density of ensembles of linear subgraphs, discuss its ultrametric nature and show that near the spectrum boundary, the tail of the spectral density exhibits a Lifshitz singularity typical for Anderson localization. We also discuss an intriguing connection of the spectral density to the Dedekind $\eta$-function. We conjecture that ultrametricity is inherit to complex systems with extremal sparse statistics and argue that a number-theoretic ultrametricity emerges in any rare-event statistics.Comment: 24 pages, 9 figure

On physical nanoscale aspects of compatibility of steels with hydrogen and natural gas

The possibilities of effective solutions of relevant technological problems are considered based on the analysis of fundamental physical aspects, elucidation of the nano-structural mechanisms and interrelations of aging and hydrogen embrittlement of materials (steels) in the hydrogen industry and gas-main industries. The adverse effects which these mechanisms and processes have on the service properties and technological lifetime of materials are analyzed. The concomitant fundamental process of formation of carbohydride-like and other segregation nanostructures at dislocations (with the segregation capacity 1 to 1.5 orders of magnitude greater than in the widely used Cottrell 'atmosphere' model) and grain boundaries is discussed in the context of how these nanostructures affect technological processes (aging, hydrogen embrittlement, stress corrosion damage, and failure) and the physicomechanical properties of the metallic materials (including the technological lifetimes of pipeline steels)