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Law of large numbers for critical first-passage percolation on the triangular lattice


 
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1. Title Title of document Law of large numbers for critical first-passage percolation on the triangular lattice
 
2. Creator Author's name, affiliation, country Chang-Long Yao; Academy of Mathematics and Systems Science, CAS; China
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) critical percolation; first-passage percolation; scaling limit; conformal loop ensemble; law of large numbers
 
3. Subject Subject classification 60K35;82B43
 
4. Description Abstract We study the site version of (independent) first-passage percolation on the triangular lattice $T$.  Denote the passage time of the site $v$ in $T$ by $t(v)$, and assume that $\mathbb{P}(t(v)=0)=\mathbb{P}(t(v)=1)=1/2$.  Denote by $a_{0,n}$ the passage time from 0 to (n,0), and by b_{0,n} the passage time from 0 to the halfplane $\{(x,y) : x\geq n\}$.  We prove that there exists a constant $0<\mu<\infty$ such that as $n\rightarrow\infty$, $a_{0,n}/\log n\rightarrow \mu$ in probability and $b_{0,n}/\log n\rightarrow \mu/2$ almost surely.  This result confirms a prediction of Kesten and Zhang.  The proof relies on the existence of the full scaling limit of critical site percolation on $T$, established by Camia and Newman.
 
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7. Date (YYYY-MM-DD) 2014-03-15
 
8. Type Status & genre Peer-reviewed Article
 
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9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/3268
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-3268
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
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