Expansion Records

The number of trajectories to go to an element of the R_k+dk,o+do subclass coming from the R_k,o subclass is :

And the total number of trajectories of dk steps is 2^dk

Each element of that subclass is such that :

So, an number of the R_k,o subclass has a probility :

to have an expansion :

If we consider that that event of probability p occurs for x around 1/p :

We can build a model for maximum expansion by :

Let us plot this model and real expansion records at dk, number of steps to reach the Maximum Height

Figure 1 : Maximum Expansion Model