The vacant set of the random walk on the torus is known to undergo a percolation phase transition at Poissonian timescales in dimensions 3 and higher. The talk will discuss recent progress regarding the nature of the transition, both for this model, and its infinite-volume limit, the vacant set of random interlacements, introduced by Sznitman in Ann. Math., 171 (2010), 2039–2087. The discussion will lead up to recent progress regarding the long purported equality of several critical parameters naturally associated to this phase transition, and the anatomy of clusters in the vacant set. Based on joint works with H. Duminil-Copin, S. Goswami, F. Severo, A. Teixeira and Y. Shulzhenko.
