Ergodicity and indistinguishability in percolation theory

We strengthen the indistinguishability theorem of Lyons and Schramm in the case of Bernoulli percolation: infinite clusters do not only agree on properties that are constant on clusters, they also agree asymptotically on properties that are asymptotically constant on clusters. This is made possible by a detour in the realm of orbit equivalence. We also show that some natural model of coalescing random walks on the free group with two generators defines a percolation that is indistinguishable but not strongly indistinguishable (it is not a Bernoulli percolation).

L'Enseignement Mathématique, 61(2):285-319, 2015

If you are familiar with both percolation and orbit equivalence, you may want to have a look at this old poster.