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Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz;  Balázs Ráth;  Artëm Sapozhnikov — 2014

Annales de l'I.H.P. Probabilités et statistiques

Random interlacements at level u is a one parameter family of connected random subsets of ℤ d , d ≥ 3 ( (2010) 2039–2087). Its complement, the vacant set at level u , exhibits a non-trivial percolation phase transition in u ( (2009) 831–858; (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique ( (2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...

Cycle structure of percolation on high-dimensional tori

Remco van der Hofstad;  Artëm Sapozhnikov — 2014

Annales de l'I.H.P. Probabilités et statistiques

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdős–Rényi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the of the largest critical components...

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