Tutte polynomial of a small-world farey graph

In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs.Comment: 6 page

Ambiguity in the Determination of the Free Energy for a Model of the Circle Map

We consider a simple model to describe the widths of the mode locked intervals for the critical circle map. Using two different partitions of the rational numbers, based on Farey series and Farey tree levels respectively, we calculate the free energy analytically at selected points for each partition. It is found that the result of the calculation depends on the method of partition. An implication of this is that the generalized dimensions $D_q$ are different for each partition except when $q=0$, i.e. only the Hausdorff dimension is the same in each case.Comment: 14 page

A quantum-geometrical description of fracton statistics

We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$$$<$$$$h$$$$<$$$$2$, a fractal distribution function associated with a fractal von Neumann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational values of spin. This formulation can be considered in the context of the fractional quantum Hall effect-FQHE and number theory.Comment: Typos corrected, latex, 8 pages, Talk given at the 2nd International Londrina Winter School: Mathematical Methods in Physics, August, 26-30 (2002), Universidade Estadual de Londrina, Paran\'a, Brazil. Version to be published in Int. J. Mod. Phys. {\bf A}, (2003

Relative blocking in posets

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.Comment: 29 pages. Corrected version of original publication which is available at http://www.springerlink.com, see Corrigendu

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

Thirty Years of Turnstiles and Transport

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio