Percolation model with an additional source of disorder

TL;DR

This paper introduces a new percolation model incorporating variable transmission ranges influenced by environmental factors, analyzing how this additional disorder affects the critical percolation threshold and connectivity in a lattice network.

Contribution

It develops a generalized percolation framework with random radii for sites, exploring the impact of an arbitrary bond occupation rule on percolation thresholds and their continuous variation.

Findings

Percolation threshold increases with added disorder and varies continuously.

Threshold ranges from the ordinary percolation value to unity.

A specific case shows a transition controlled by the maximum radius R_0.

Abstract

The ranges of transmission of the mobiles in a Mobile Ad-hoc Network are not uniform in reality. They are affected by the temperature fluctuation in air, obstruction due to the solid objects, even the humidity difference in the environment, etc. How the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask. Here a new model of percolation phenomena, with an additional source of disorder, has been introduced for a theoretical understanding of this problem. As in ordinary percolation, sites of a square lattice are occupied randomly with the probability $p$. Each occupied site is then assigned a circular disc of random value $R$ for its radius. A bond is defined to be occupied if and only if the radii $R_1$ and $R_2$ of the discs centered at the ends satisfy certain pre-defined condition. In a very general formulation, one divides the $R_1 - R_2$ plane into two regions by an arbitrary closed curve. One defines that a point within one region represents an occupied bond, otherwise it is a vacant bond. Study of three different rules under this general formulation, indicates that the percolation threshold is always larger and varies continuously. This threshold has two limiting values, one is $p_c$(sq), the percolation threshold for the ordinary site percolation on the square lattice and the other being unity. The variation of the thresholds are characterized by exponents, which are not known in the literature. In a special case, all lattice sites are occupied by discs of random radii $R \in \{0,R_0\}$ and a percolation transition is observed with $R_0$ as the control variable, similar to the site occupation probability.