Z. Naturforsch. **68a,** 475 – 478
(2013)

doi:10.5560/ZNA.2013-0015

A Note on the Perturbation of Mixed Percolation on the Hierarchical Group

Yilun
Shang

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

Received December 10, 2012 / revised February 12, 2013 / published online April 10, 2013

We study mixed percolation on the hierarchical group of order *N* where each node is open with probability 1 − *γ*, 0 ≤ *γ* ≤ 1, and the probability of connection between two open nodes separated by a distance *k* is of the form 1 − exp (− *αβ*^{−k}), *α* ≥ 0, and *β* > 0. The parameters *α* and *γ* are the percolation parameters, while *β* describes the long-range nature of the model. In terms of parameters *α*, *β*, and *γ*, we show some perturbation results for the percolation function
*θ*(*α*, *β*, *γ*), which is the probability of existing an infinite component containing a prescribed node.

*Key words:* Mixed Percolation; Hierarchical Group; Percolation Function.

*PACS numbers:* 64.60.ah; 02.50.Cw; 02.10.Ox