Random Matrix Approach to Fluctuations and Scaling in Complex Systems

Abstract

The study of fluctuations, self-similarity and scaling in physical and socioeconomic sciences in the last several years has brought in new insights and new ideas for modelling them. For instance, one of the important empirical results of the market dynamics is that the probability distribution of price returns r in a typical market displays a power-law, i.e, (P|r| > x) ∼ r −α , where α ∼ 3.0 [1]. In fact, this “inverse cube law” is known to hold good for volume of stocks traded in stock exchanges, though the exponent in this case is α ∼ 1.5 [1]. Similar power laws appear for the cumulative frequency distribution of earth quake magnitudes, often called the Gutenberg-Richter relation [2]. Infect, anything from size distribution of cities and wealth distributions, display power law. These apparently universal power laws pertain to the distribution of actual values taken by some quantity of interest, say, a stock market index and these distributions reveal scaling with certain parameters.