Branching Processes That Grow Faster Than Binary Splitting

Abstract

A drastic qualitative change in a physical system occurring after an arbitrarily small change of a parameter (like temperature) is usually called a critical phenomenon. One of the most thoroughly studied mathematical models of such phenomena is that of percolation which paraphrases the passage of a fluid through a porous medium consisting of very many small channels ([2]). The qualitative property of interest is whether or not the fluid will percolate the medium macroscopically if it passes each small channel with probability p (the parameter of the process). Typically this has probability zero for all values of p smaller than a critical value pc, and strictly positive probability for p larger than pc. For most percolation models it is very difficult to determine the exact value of Pc, and even more difficult to describe the behavior of the probability of percolation near pc. These difficulties are mainly caused by the interconnectivity structure of the small channels in the medium. In this note we present a model which has the same flavor as percolation models (where the interconnectivity structure is that of a tree), in which the above mentioned quantities can be exactly computed in an elementary way. The relation to existing models is in fact very close: our model has been derived from the analysis of Mandelbrot's percolation process in [1].