Research Statement of Marek Biskup

The main result is the proof that the simple random walk on a.e. supercritical percolation cluster scales to Brownian motion under the usual diffusive scaling of space and time. Sidoravicius and Sznitman [47] previously proved this in dimensions d ≥ 4 by comparing the quenched and annealed path distributions; their estimates used the fact that two random walk paths in high-enough dimensions are unlikely to meet at a large number of points. Much work has been done in recent years on estimates of the heat kernel (Mathieu-Remy [35], Barlow [4]). The approach of [A] is different from that of Sidoravicius and Sznitman and is based on the notion of harmonic deformation of the infinite cluster. This is an embedding of the percolation graph on which the random walk is a martingale. The construction of the deformation invokes the so called corrector, which has been a standard tool in homogenization theory, and it follows the line of e.g. Kipnis and Varadhan’s paper [31]. The hard part is the proof that the deformation grows sublinearly with the distance which is needed in order to show that the paths of the walk on the natural embedding and the harmonic embedding stay sufficiently close to each other. This is achieved by combining facts about percolation (comparison of the graph distance and the Euclidean distance, a’la Antal-Pisztora [3]) and ergodic theory. The proof in d = 2 does not need anything beyond that; in d ≥ 3 we also need rather sophisticated heat-kernel estimates proved recently by Barlow [4]. An independent proof of this result was simultaneously given by Mathieu-Piatnitski [34] who employ more traditional tools of homogenization theory.