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We discuss several models in order to shed light on the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments…
Dynamical systems in nature such as fluid flows, heart beat patterns, rainfall variability, stock market price fluctuations, etc. exhibit selfsimilar fractal fluctuations on all scales in space and time. Power spectral analyses of fractal…
Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on…
The standard semiclassical calculation of transmission correlation functions for chaotic systems is severely influenced by unitarity problems. We show that unitarity alone imposes a set of relationships between cross sections correlation…
The maintenance of diversity, the `commonness of rarity', and compositional turnover are ubiquitous features of species-rich communities. Through a minimal model, we consider how these features reflect the interplay between environmental…
A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict…
I review several issues related to statistical description of gravitating systems in both static and expanding backgrounds. After briefly reviewing the results for the static background, I concentrate on gravitational clustering of…
Recent research on the non-stationary nature of the dynamics of complex systems is reviewed through three specific models. The long time dynamics consists of a slow, decelerating but spasmodic release of generalized intrinsic strain. These…
Turbulence exhibits significant velocity fluctuations even if the scale is much larger than the scale of the energy supply. Since any spatial correlation is negligible, these large-scale fluctuations have many degrees of freedom and are…
Physical kinetic roughening processes are well known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available…
Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed the models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the…
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often…
There exist some boundary-driven open systems with diffusive dynamics whose particle current fluctuations exhibit universal features that belong to the Edwards-Wilkinson universality class. We achieve this result by establishing a mapping,…
Scaling analysis reveals striking regularities in earthquake occurrence. The time between any one earthquake and that following it is random, but it is described by the same universal-probability distribution for any spatial region and…
Density fluctuations in traffic current are studied by computer simulations using the deterministic coupled map lattice model on a closed single-lane circuit. By calculating a power spectral density of temporal density fluctuations at a…
The day-to day fluctuations of Dow Jones Index exhibit fractal fluctuations, namely, a zigzag pattern of successive increases followed by decreases on all space-time scales. Self-similar fractal fluctuations are generic to dynamical systems…
We compare systematically several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution…
In this paper we review some general properties of probability distributions which exibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of…
Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex…
The theory of diffusion seeks to describe the motion of particles in a chaotic environment. Classical theory models individual particles as independent random walkers, effectively forgetting that particles evolve together in the same…