We review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager, and discuss its quantitative description using the theory of random walks. We illustrate Lévy flight foraging by comparison to Brownian random walks and...

Recently it has been shown that the most efficient strategy for searching randomly located objects, when the sites are randomly distributed and can be revisited any number of times, leads to a power law distribution P(ℓ)=ℓ−μ of the flights ℓ, with μ=2. We show analytically that the...

There has been growing interest in the study of Lévy flights observed in the movements of biological organisms performing random walks while searching for other organisms. Here, we approach the problem of what is the best statistical strategy for optimizing the encounter rate between...

We discuss the possible utility of statistical physics in elucidating some of the puzzling phenomena that seem to occur in the brains of patients affected with Alzheimer’s disease. Further, we report three specific results from this approach: (i) The size distribution of senile plaques appears...

We present a scaling Ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for (i) the effect of the finite size of the system, and (ii) the dependence of this distribution on the site occupancy probability p. We present...

Inspired by work of both Widom and Mandelbrot, we analyze the Computstat database comprising all publicly traded United States manufacturing companies in the years 1974–1993. We find that the distribution of company size remains stable for the 20 years we study, i.e., the mean value and...

Financial networks are dynamic. To assess their systemic importance to the world-wide economic network and avert losses we need models that take the time variations of the links and nodes into account. Using the methodology of classical mechanics and Laplacian determinism we develop a model that...

The presence of significant cross-correlations between the synchronous time evolution of a pair of equity returns is a well-known empirical fact. The Pearson correlation is commonly used to indicate the level of similarity in the price changes for a given pair of stocks, but it does not measure...

In the current era of worldwide stock market interdependencies, the global financial village has become increasingly vulnerable to systemic collapse. The recent global financial crisis has highlighted the necessity of understanding and quantifying interdependencies among the world's economies,...

This brief overview is designed to introduce some of the advances that have occurred in our understanding of percolation phenomena. We organize our presentation around three simple questions: (i) What are percolation phenomena? (ii) Why do we care? (iii) What do we actually do? To answer the...