Random walks and Green's functions on supercritical percolation cluster

We study the continuous-time random walk in a random degenerate environment: the infinite supercritical percolation cluster. This subject has been a topic of interest over the past decades and a number of results pertaining to the random walker have been established: Gaussian heat kernel bounds for the transition kernel, quenched invariance principle for the walk, local limit theorem etc. An important ingredient of the proofs is to implement a renormalization structure for the infinite cluster. In this talk, we will present some of the results aforementioned, will introduce a renormalization structure for the infinite cluster which was used to adapt the recently developed theory of quantitative stochastic homogenization to this degenerate setting, and will present some new results which can be derived from this construction. This is joint work with S.Armstrong and C.Gu.