On the lattice Z^d the function x -> x_1 is harmonic. It also happens to be Lipschitz and integer-valued. Can a non-constant harmonic function with either of these properties exist on the infinite cluster in supercritical Bernoulli bond percolation? One of our main results answers this question in the negative.
In the first part of this talk, we will give an overview of our results, the main ideas involved, and open questions stemming from it. We also describe our motivation — broadly, we would like to understand how randomness in the graph can affect the qualitative behavior of the graph Laplacian. Some of our results also have implications for the Abelian sandpile model on the infinite cluster.
In the second part, we will give a detailed sketch of the proofs.
The talk is based on https://arxiv.org/abs/2303.04736