STOCHASTIC RENORMALIZATION-GROUP APPROACH TO RANDOM POINT-PROCESSES -A FRACTAL GENERALIZATION OF POISSON STATISTICS

A generalization of the Shlesinger-Hughes stochastic renormalization m ethod is suggested for random point processes. A decimation procedure is introduced in terms of two parameters: the probability alpha that a decimation step takes place and the probability beta that during a de cimation 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 superposi tion of the intermediate processes attached to the different steps. Fo r a statistical ensemble the fraction L(q) of systems for which q deci mation steps occur is a power function of the fraction rho(q) of point s which survive q decimation steps L(q) = (rho(q))1-df where d(f) = 1- ln alpha/ln(1 - beta) is a fractal exponent smaller than unity, df<1. If the random points are initially independent then a complete analysi s 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 a re correlated. The probability of the number of points is a superposit ion of Poissonians corresponding to the different steps; however, it i s generally non-Poissonian. It may be considered as a fractal generali zation of Poisson statistics. All positive moments of the number of po ints exist and are finite and thus the corresponding probability does not have a long tail; the fractal features are displayed by the depend ence of the probability on the initial average number of points lambda : for lambda --> infinity the probability has an inverse power tail in lambda modulated by a periodic function in ln lambda with a period - In(1 - beta). The new formalism is of interest for describing the lacu nary structures corresponding to the final stages of chemical processe s in low dimensional systems and for the statistics of rare events.