Branching number and percolation
María P. Fernández de Córdoba

The branching number of a rooted tree represents the average number of branches per vertex. This number is strongly related with Bernoulli percolation process, which involves removing edges at random on the tree and whose goal is to study the nature of the resulting clusters.

We extend these notions to any measurable graphed pseudogroup of finite type on a probability space. Concerning to the branching number, we prove that if this number is equal to 1, then the action of the pseudogroup is amenable. In fact, when the measure is harmonic, the action is Liouvillian.

Regarding Bernoulli percolation on graphed pseudogroups, we remove edges at random and we define a lower and an upper critical percolation. We study the influence of the number of ends of the orbits on the critical percolation.

Finally, we define a new type of percolation for graphed (pseudo)group actions on probability spaces. The relative percolation studies what happens when we consider a Borel set A of positive measure, we keep the edges whose endpoints belong to A and we remove the others. We use again the number of ends in order
to achieve information about the new clusters obtained by relative percolation.