The effect of small quenched noise on connectivity properties of random interlacements
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | The effect of small quenched noise on connectivity properties of random interlacements |
| 2. | Creator | Author's name, affiliation, country | Balázs Ráth; The University of British Columbia; Canada |
| 2. | Creator | Author's name, affiliation, country | Artëm Sapozhnikov; Max-Planck Institute for Mathematics in the Sciences; Germany |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Random interlacements; Bernoulli percolation; slab; vacant set; quenched noise; long-range correlations; transience |
| 3. | Subject | Subject classification | 60K35; 82B43 |
| 4. | Description | Abstract | Random interlacements (at level $u$) is a one parameter family of random subsets of $\mathbb{Z}^d$ introduced by Sznitman. The vacant set at level $u$ is the complement of the random interlacement at level $u$. While the random interlacement induces a connected subgraph of $\mathbb{Z}^d$ for all levels $u$, the vacant set has a non-trivial phase transition in $u$. In this paper, we study the effect of small quenched noise on connectivity properties of the random interlacement and the vacant set. For a positive $\varepsilon$, we allow each vertex of the random interlacement (referred to as occupied) to become vacant, and each vertex of the vacant set to become occupied with probability $\varepsilon$, independently of the randomness of the interlacement, and independently for different vertices. We prove that for any $d\geq 3$ and $u>0$, almost surely, the perturbed random interlacement percolates for small enough noise parameter $\varepsilon$. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is transient. As for the vacant set, we show that for any $d\geq 3$, there is still a non trivial phase transition in $u$ when the noise parameter $\varepsilon$ is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when $\varepsilon\to 0$. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | ERC-2009-AdG 245728-RWPERCRI |
| 7. | Date | (YYYY-MM-DD) | 2013-01-07 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/2122 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v18-2122 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 18 |
| 12. | Language | English=en | en |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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