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PKP Metadata Items |
Metadata for this Document |
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| 1. |
Title |
Title of document |
On Disagreement Percolation and Maximality of the Critical Value for iid Percolation |
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Creator |
Author's name, affiliation, country |
Johan Jonasson; Chalmers University of Technology |
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Subject |
Discipline(s) |
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Subject |
Keyword(s) |
coupling, Ising model, random-cluster model, transitive graph, planar graph |
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Subject |
Subject classification |
60K35, 82B20, 82B26, 82B43 |
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| 4. |
Description |
Abstract |
Two different problems are studied: - For an infinite locally finite connected graph $G$, let $p_c(G)$ be the critical value for the existence of an infinite cluster in iid bond percolation on $G$ and let $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G)<1\}$. Is $P_c<1$?
- Let $G$ be transitive with $p_c(G)<1$, take $p \in [0,1]$ and let $X$ and $Y$ be iid bond percolations on $G$ with retention parameters $(1+p)/2$ and $(1-p)/2$ respectively. Is there a $q<1$ such that $p > q$ implies that for any monotone coupling $(X',Y')$ of $X$ and $Y$ the edges for which $X'$ and $Y'$ disagree form infinite connected component(s) with positive probability? Let $p_d(G)$ be the infimum of such $q$'s (including $q=1$) and let $P_d = \sup\{p_d(G): G \text{ transitive }, p_c(G) < 1\}$. Is the stronger statement $P_d < 1$ true? On the other hand: Is it always true that $p_d(G) > p_c (G)$?
It is shown that if one restricts attention to biregular planar graphs then these two problems can be treated in a similar way and all the above questions are positively answered. We also give examples to show that if one drops the assumption of transitivity, then the answer to the above two questions is no. Furthermore it is shown that for any bounded-degree bipartite graph $G$ with $p_c(G) < 1$ one has $p_c(G) < p_d(G)$. Problem (2) arises naturally from [6] where an example is given of a coupling of the distinct plus- and minus measures for the Ising model on a quasi-transitive graph at super-critical inverse temperature. We give an example of such a coupling on the $r$-regular tree, ${\bf T}_r$, for $r > 1$. |
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Publisher |
Organizing agency, location |
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Contributor |
Sponsor(s) |
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Date |
(YYYY-MM-DD) |
2001-06-15
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Type |
Status & genre |
Peer-reviewed Article |
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Type |
Type |
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| 9. |
Format |
File format |
PDF |
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| 10. |
Identifier |
Uniform Resource Identifier |
http://ejp.ejpecp.org/article/view/88 |
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| 10. |
Identifier |
Digital Object Identifier |
10.1214/EJP.v6-88 |
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| 11. |
Source |
Journal/conference title; vol., no. (year) |
Electronic Journal of Probability; Vol 6 |
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Language |
English=en |
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Coverage |
Geo-spatial location, chronological period, research sample (gender, age, etc.) |
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Rights |
Copyright and permissions |
The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.
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You are free
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under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author. |