|Department of Mathematics, Imperial College London
|14h00 - 15h00
Consider the set of points visited by the random walk on the discrete d-dimensional torus of side length N, for d ≥ 3, at times of order uN^d, for a parameter u > 0 and in the large-N limit. Its complement (the vacant set of the walk) is believed to undergo an abrupt percolation phase transition across a non-degenerate critical value u_∗ = u_∗(d), in the following sense: for all u < u_∗, the vacant set contains a giant connected component with high probability, which has a non-vanishing asymptotic density. In stark contrast, for all u > u_∗ the vacant set scatters into tiny connected components. I will survey existing results regarding the above conjecture, both for this model and the related vacant set of random interlacements, introduced by Sznitman in Ann. Math., 171 (2010), 2039–2087, which corresponds to its Gibbsian limit. The discussion will lead up to recent progress regarding the long purported equality of various critical parameters naturally associated to this phase transition, and the nature of its associated universality class.