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ISRN Discrete Mathematics
Volume 2012 (2012), Article ID 758721, 6 pages
http://dx.doi.org/10.5402/2012/758721
Research Article

Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice

Institute for Cyber Security, University of Texas at San Antonio, San Antonio, TX 78249, USA

Received 5 July 2012; Accepted 16 August 2012

Academic Editors: U. A. Rozikov and X. Yong

Copyright © 2012 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a long-range percolation in the hierarchical lattice Ξ© 𝑁 of order 𝑁 where probability of connection between two nodes separated by distance π‘˜ is of the form min { 𝛼 𝛽 βˆ’ π‘˜ , 1 } , 𝛼 β‰₯ 0 and 𝛽 > 0 . We show the uniqueness of the infinite component for this model.