Vacant Set of Random Interlacements and Percolation
By A. Sznitman. 2007
273 citations recorded by the source corpus.
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite, connected, translation invariant random subset of Z^d. We introduce a critical value such that the vacant set percolates for u below the critical value, and does not percolate for u above the critical value. Our main results show that the critical value is finite when d is bigger or equal to 3, and strictly positive when d is bigger or equal to 7.
Subfields: Condensed Matter Physics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/303ad61a723a257aca197d7caad307391edac040
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In Condensed Matter Physics
In Mathematical Physics
Citations in the corpus
- On the uniqueness of the infinite cluster of the vacant set of random interlacements (2008, unknown): “3 of [8], which is proved using similar ideas to those of the present paper,”
- Topics in Occupation Times and Gaussian Free Fields (2012, background): “More recently such Poisson gases of Markovian loops have reappeared in the context of the “Brownian loop soup” of Lawler and Werner [16] and are related to the so-called “random interlacements”, see Sznitman [27].”
- Connectivity properties of random interlacement and intersection of random walks (2010, unknown): “21) in [16], which states that the random interlacement I is a connected subgraph of Zd.”
- Gumbel fluctuations for cover times in the discrete torus (2012, unknown): “Random interlacements are intimitaly related to random walk in the torus; intuitively speaking, the trace of Y· in a “local box” of the torus, when run up to time uNd, in some sense “looks like” Iu intersected with a box (see [26], [23]).”
- On the fragmentation of a torus by random walk (2010, unknown): “6) It is proved by Sznitman in [21] and by Sidoravicius and Sznitman in [17] that indeed u⋆ ∈ (0,∞), for all d ≥ 3.”
- Soft local times and decoupling of random interlacements (2012, unknown): “One such method is called “Poissonization” and appears in various works, for instance [23], [22], [32].”