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Analytic results for the percolation transitions of the enhancedbinary tree
Umeå University, Faculty of Science and Technology, Department of Physics.
Umeå University, Faculty of Science and Technology, Department of Physics.
2010 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 82, 011113- p.Article in journal (Refereed) Published
Abstract [en]

Percolation for a planar lattice has a single percolation threshold, whereaspercolation for a negatively curved lattice displays two separate thresholds.The enhanced binary tree (EBT) can be viewed as aprototype model displaying two separate percolation thresholds.We present an analytic result for the EBT model which givestwo critical percolation threshold probabilities,$p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}$ and $p_{c2}=1/2$,and yields a size-scaling exponent $\Phi =\ln\left[\frac{p(1+p)}{1-p(1-p)}\right]/\ln 2$. It is inferred that the twothresholdvalues give exact upper limits and that $p_{c1}$ is furthermore exact. Inaddition, we argue that $p_{c2}$ is also exact. The physics of the model andthe results are described within themidpoint-percolation concept: Monte Carlo simulations are presented for thenumber of boundary points which are reached from the midpoint, and theresults are compared to the number of routes from the midpoint to theboundary given by the analytic solution. These comparisonsprovide a more precise physical picture of what happens at the transitions.Finally, the results are compared to related works, in particular, the Cayleytree and Monte Carlo results for hyperbolic lattices as well as earlierresults for the EBT model. It disproves a conjecture that the EBT has anexact relation to the thresholds of its dual lattice.

Place, publisher, year, edition, pages
2010. Vol. 82, 011113- p.
Keyword [en]
percolation, enhanced binary tree
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
URN: urn:nbn:se:umu:diva-35227DOI: 10.1103/PhysRevE.82.011113ISI: 000279819900002OAI: diva2:338027
Available from: 2010-08-13 Created: 2010-08-10 Last updated: 2010-12-01Bibliographically approved

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Minnhagen, PetterBaek, Seung Ki
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