Shlesinger-Hughes stochastic renormalization, iterated logarithmic tails and multiple hierarchical fractal aggregation

The multiple hierarchical fractal aggregation is used as an illustration for a new type of asymptotic behavior of fractal random processes defined in terms of a multiple Shlesinger- Hughes renormalization procedure. The joint probability distributions of aggregate sizes for a multiple association process are analyzed based on the following set of assumptions: (a) the system contains basic units of different types; however, each aggregate is made up of a single type of units; (b) the size of a single aggregate obeys a geometrical distribution of the Flory type; (c) the aggregation process occurs in a hierarchical way, i.e. each aggregate of a given type triggers the generation of a complex of another type. A chain of joint size probability distributions is introduced. We show that these probability distributions depend on the probabilities that a unit from a given aggregate is active, that is, that it may trigger the aggregation process of basic units of another type. These probabilities generate a chain of fractal exponents which characterize the asymptotic behavior of the size probability distributions. The probability distribution attached to a succession of q + 2 aggregation processes has a very long tail with a logarithmic shape: it is an inverse power of the qth iterate logarithm of size n, ln ln…ln n. This decay law is much slower than the inverse power laws characteristic for the usual statistical fractals.