Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.Comment: 10 pages, 13 figure

Modeling temporal fluctuations in avalanching systems

We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean field approach to sandpile dynamics by formulating it as a generalization of Itoh's SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters, and a drift term reflecting the tendency for small avalanches to grow and large avalanches to be constricted by the finite system size. If one defines avalanching to take place when the toppling activity exceeds a certain threshold the stochastic model allows us to compute the avalanche exponents in the continum limit as functions of the Hurst exponent of the noise. The results are found to agree well with numerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also provides a method for computing the probability density functions of the fluctuations in the toppling activity itself. We show that the sandpiles do not belong to the class of phenomena giving rise to universal non-Gaussian probability density functions for the global activity. Moreover, we demonstrate essential differences between the fluctuations of total kinetic energy in a two-dimensional turbulence simulation and the toppling activity in sandpiles.Comment: 14 pages, 11 figure

Statistical-thermodynamical foundations of anomalous diffusion

It is shown that Tsallis' generalized statistics provides a natural frame for the statistical-thermodynamical description of anomalous diffusion. Within this generalized theory, a maximum-entropy formalism makes it possible to derive a mathematical formulation for the mechanisms that underly Levy-like superdiffusion, and for solving the nonlinear Fokker-Planck equation.Comment: 13 pages, 8 figures; to appear in special issue of Braz. J. Phys. as invited revie

Geometry of River Networks II: Distributions of Component Size and Number

The structure of a river network may be seen as a discrete set of nested sub-networks built out of individual stream segments. These network components are assigned an integral stream order via a hierarchical and discrete ordering method. Exponential relationships, known as Horton's laws, between stream order and ensemble-averaged quantities pertaining to network components are observed. We extend these observations to incorporate fluctuations and all higher moments by developing functional relationships between distributions. The relationships determined are drawn from a combination of theoretical analysis, analysis of real river networks including the Mississippi, Amazon and Nile, and numerical simulations on a model of directed, random networks. Underlying distributions of stream segment lengths are identified as exponential. Combinations of these distributions form single-humped distributions with exponential tails, the sums of which are in turn shown to give power law distributions of stream lengths. Distributions of basin area and stream segment frequency are also addressed. The calculations identify a single length-scale as a measure of size fluctuations in network components. This article is the second in a series of three addressing the geometry of river networks.Comment: 16 pages, 13 figures, 4 tables, Revtex4, submitted to PR

On the extended Kolmogorov-Nagumo information-entropy theory, the q -> 1/q duality and its possible implications for a non-extensive two dimensional Ising model

The aim of this paper is to investigate the q -> 1/q duality in an information-entropy theory of all q-generalized entropy functionals (Tsallis, Renyi and Sharma-Mittal measures) in the light of a representation based on generalized exponential and logarithm functions subjected to Kolmogorov's and Nagumo's averaging. We show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality. The generalized partition function also results to be a scalar invariant under the q -> 1/q transformation which can be interpreted as a non-extensive two dimensional Ising model duality between systems governed by two different power law long-range interactions and temperatures. This does not hold only for Tsallis statistics, but is a characteristic feature of all stationary distributions described by q-exponential Boltzmann factors.Comment: 13 pages, accepted for publication in Physica