Banachraum

We study the behavior of harmonic functions on the infinite cluster in supercritical Bernoulli bond percolation. Most of the results which have been proved on these functions assert that they behave similarly to the harmonic functions on the lattice Zd (a typical example is the Liouville theorem on the percolation cluster). In this talk, we will be interested in the differences between harmonic functions on the lattice and on the percolation cluster through some specific properties. In this direction, we prove that there cannot exist a Lipschitz harmonic function on the percolation cluster. This property is false on the lattice as the function x -> x_1 is both 1-Lipschitz and harmonic. We will review some of the known results on harmonic functions on the percolation cluster and discuss some ideas of the proof. This is joint work with A. Bou-Rabee and W. Cooperman.

Paul Dario

Paris