Power-law behavior in a cascade process with stopping events: a solvable model

Fat tails and black swans: Exact results for multiplicative processes with resets

We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails-ubiquitous in the statistics of social, economic, and ecological systems. Our main goal is to provide a series of exact results on the dynamics and asymptotic behavior of increasingly complex versions of a basic multiplicative process with resets, including discrete and continuous-time variants and several degrees of randomness in the parameters that control the process. In particular, we show how the power-law distributions are built up as time elapses, how their moments behave with time, and how their stationary profiles become quantitatively determined by those parameters. Our discussion emphasizes the connection with financial systems, but these stochastic processes are also expected to be fruitful in modeling a wide variety of social and biological phenomena.

Splash detail due to a single grain incident on a granular bed

Using the discrete element method, we study the splash processes induced by the impact of a grain on a randomly packed bed. Good correspondence is obtained between our numerical results and the findings of previous experiments for the movement of ejected grains. Furthermore, the distributions of the ejection angle and ejection speed for individual grains vary depending on the relative timing at which the grains are ejected after the initial impact. Obvious differences are observed between the distributions of grains ejected during the earlier and later splash periods: the form of the vertical ejection-speed distribution varies from a power-law form to a lognormal form with time; this difference may determine grain trajectory after ejection.

Equipartitions and a distribution for numbers: A statistical model for Benford's law

A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to the distribution of fragments. The resulting power law directly leads to Benford's law for the first digits of the parts.

Minimal fragmentation of regular polygonal plates

Minimal fragmentation models intend to unveil the statistical properties of large ensembles of identical objects, each one segmented in two parts only. Contrary to what happens in the multifragmentation of a single body, minimally fragmented ensembles are often amenable to analytical treatments, while keeping key features of multifragmentation. In this work we present a study on the minimal fragmentation of regular polygonal plates with up to 100 sides. We observe in our model the typical statistical behavior of a solid torn apart by a strong impact, for example. We obtain a robust power law, valid for several decades, in the small mass limit. In the present case we were able to analytically determine the exponent of the cumulative probability distribution to be ½. Less usual, but also reported in a number of experimental and numerical references on impact fragmentation, is the presence of a sharp crossover to a second power-law regime, whose exponent we found to be between ⅓ and 1 depending on the way anisotropy is introduced in the model.

Stochastic model of Zipf's law and the universality of the power-law exponent

We propose a stochastic model of Zipf's law, namely a power-law relation between rank and size, and clarify as to why a specific value of its power-law exponent is quite universal. We focus on the successive total of a multiplicative stochastic process. By employing properties of a well-known stochastic process, we concisely show that the successive total follows a stationary power-law distribution, which is directly related to Zipf's law. The formula of the power-law exponent is also derived. Finally, we conclude that the universality of the rank-size exponent is brought about by symmetry between an increase and a decrease in the random growth rate.