Stochastic renormalization group approach to random point processes. A fractal generalization of Poisson statistics

Abstract

A generalization of the Shlesinger–Hughes stochastic renormalization method is suggested for random point processes. A decimation procedure is introduced in terms of two parameters: the probability α that a decimation step takes place and the probability β that during a decimation step a random dot is removed from the process. At each step a random number of dots are removed; by this procedure a chain of point processes is generated. The renormalized point process is a superposition of the intermediate processes attached to the different steps. For a statistical ensemble the fraction Lq of systems for which q decimation steps occur is a power function of the fraction ρq of points which survive q decimation steps Lq = (ρq)1−df where df=1−ln α/ln(1−β) is a fractal exponent smaller than unity, df<1. If the random points are initially independent then a complete analysis is possible. In this case explicit expressions for the renormalized Janossy densities, the joint densities, and the generating functional of the process are derived. Even though the initial process is made up of independent random points the points in the renormalized process are correlated. The probability of the number of points is a superposition of Poissonians corresponding to the different steps; however, it is generally non-Poissonian. It may be considered as a fractal generalization of Poisson statistics. All positive moments of the number of points exist and are finite and thus the corresponding probability does not have a long tail; the fractal features are displayed by the dependence of the probability on the initial average number of points λ: for λ→∞ the probability has an inverse power tail in λ modulated by a periodic function in ln λ with a period −ln(1−β). The new formalism is of interest for describing the lacunary structures corresponding to the final stages of chemical processes in low dimensional systems and for the statistics of rare events.