Similarity and Probability Distribution Functions in Many-body Stochastic Processes with Multiplicative Interactions

Analytical and numerical studies on many-body stochastic processes with multiplicative interactions are reviewed. The method of moment relations is used to investigate effects of asymmetry and randomness in interactions. Probability distribution functions of the processes generally have similarity solutions with power-law tails. Growth rates of the system and power-law exponents of the tails are determined via transcendental equations. Good agreement is achieved between analytical calculations and Monte Carlo simulations.Comment: 8 pages, 4 figures, CN-Kyoto proceeding

Asymptotic analysis of the model for distribution of high-tax payers

The z-transform technique is used to investigate the model for distribution of high-tax payers, which is proposed by two of the authors (K. Y and S. M) and others. Our analysis shows an asymptotic power-law of this model with the exponent -5/2 when a total ``mass'' has a certain critical value. Below the critical value, the system exhibits an ordinary critical behavior, and scaling relations hold. Above the threshold, numerical simulations show that a power-law distribution coexists with a huge ``monopolized'' member. It is argued that these behaviors are observed universally in conserved aggregation processes, by analizing an extended model.Comment: 5pages, 3figure

Effects of Randomness on Power Law Tails in Multiplicatively Interacting Stochastic Processes

Effects of randomness on non-integer power law tails in multiplicatively interacting stochastic processes are investigated theoretically. Generally, randomness causes decrease of the exponent of tails and the growth rate of processes. Explicit calculations are performed for two examples: uniformly distributed and two peaked systems. Significant influence is demonstrated when a bare growth rate is low and coupling is weak. It should be emphasized that even the sign of the growth rate can be changed from positive to negative growth.