Long-time fluctuations in a dynamical model of stock market indices.

Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. Recent empirical studies of stock market indices examined whether the distribution P(r) of returns r(tau) after some time tau can be described by a (truncated) Lévy-stable distribution L(alpha)(r) with some index 02, namely, beyond the range of Lévy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results.

Evidence for universality within the classes of deterministic and stochastic sandpile models.

Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.

Pattern formation and a clustering transition in power-law sequential adsorption.

We present a model that describes adsorption and clustering of particles on a surface. A clustering transition is found that separates between a phase of weakly correlated particle distributions and a phase of strongly correlated distributions in which the particles form localized fractal clusters. The order parameter of the transition is identified and the fractal nature of both phases is examined. The model is relevant to a large class of clustering phenomena such as aggregation and growth on surfaces, population distribution in cities, and plant and bacterial colonies, as well as gravitational clustering.

Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements.

A generic model of stochastic autocatalytic dynamics with many degrees of freedom w(i,) i=1, em leader,N, is studied using computer simulations. The time evolution of the w(i)'s combines a random multiplicative dynamics w(i)(t+1)=lambdaw(i)(t) at the individual level with a global coupling through a constraint which does not allow the w(i)'s to fall below a lower cutoff given by cw, where w is their momentary average and 0infinity and the limit of decoupled free multiplicative random walks c-->0 do not commute: alpha(0,N)=0 for any finite N while alpha(c,infinity)>or=1 (which is the common range in empirical systems) for any positive c. The time evolution of w(t) exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution L(alpha)(r) with the same index alpha. This nontrivial relation between the distribution of the wi's at a given time and the temporal fluctuations of their average is examined, and its relevance to empirical systems is discussed.