Séminaire Probabilités et Statistiques
Percolation and the phase transition for the vacant set of random walk
March 2023
Intervenant : Pierre-Francois Rodriguez
Institution : Department of Mathematics, Imperial College London
Heure : 14h00 - 15h00
Lieu : 3L15

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.

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