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Percolation on uniform infinite planar maps


 
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1. Title Title of document Percolation on uniform infinite planar maps
 
2. Creator Author's name, affiliation, country Laurent Ménard; Université Paris Ouest; France
 
2. Creator Author's name, affiliation, country Pierre Nolin; ETH Zürich; Switzerland
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random map; UIPQ; percolation threshold; peeling process
 
3. Subject Subject classification 05C80; 82B43
 
4. Description Abstract We construct the uniform infinite planar map (UIPM), obtained as the $n \to \infty$ local limit of planar maps with $n$ edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are $p^{\textrm{bond}}_c=1/2$ and $p^{\textrm{site}}_c=2/3$ respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is $p^{\textrm{bond}}_c=1/3$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) US National Science Foundation
 
7. Date (YYYY-MM-DD) 2014-09-02
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2675
 
10. Identifier Digital Object Identifier 10.1214/EJP.v19-2675
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 19
 
12. Language English=en en
 
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