The vacant set of the random walk on the torus is known to undergo a percolation phase transition at Poissonian time scales in dimensions 3 and higher. The two talks will discuss recent progress regarding the nature of the transition, both for this model and its Gibbsian limit, the vacant set of random interlacements. In particular, the results yield the long purported equality of various critical parameters naturally associated to this phase transition. Various challenges encountered along the way call for completely new tools, built from scratch, which are of independent interest.
Based on joint works with H. Duminil-Copin, F. Severo and A. Teixeira.