Colored Percolation

TL;DR

The paper introduces and analyzes the 'Colored Percolation' model on two-dimensional lattices, exploring how site coloring and bond connections influence percolation thresholds and phase diagrams, with various generalizations and preferences affecting critical behavior.

Contribution

It presents a novel multi-color percolation model with multiple generalizations, including color preferences and independent bond parameters, expanding understanding of percolation phenomena.

Findings

Percolation threshold $p_c(n)$ converges to $p_c$ as $1/n$.

Threshold $p_c(q,m)$ varies non-trivially with $q$, minimized at $q_{min} = m/n$.

Phase diagrams reveal distinct percolating and non-percolating phases.

Abstract

A model named `Colored Percolation' has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability $p$ and are then colored by one of the $n$ distinct colors using uniform probability $q = 1/n$. Denoting different colors by the letters of the Roman alphabet, we have studied different versions of the model like $AB, ABC, ABCD, ABCDE, ...$ etc. Here, only those lattice bonds having two different colored atoms at the ends are defined as connected. The percolation thresholds $p_c(n)$ asymptotically converges to its limiting value of $p_c$ as $1/n$. The model has been generalized by introducing a preference towards a subset of colors when $m$ out of $n$ colors are selected with probability $q/m$ each and rest of the colors are selected with probability $(1 - q)/(n - m)$. It has been observed that $p_c(q,m)$ depends non-trivially on $q$ and has a minimum at $q_{min} = m/n$. In another generalization the fractions of bonds between similar and dissimilar colored atoms have been treated as independent parameters. Phase diagrams in this parameter space have been drawn exhibiting percolating and non-percolating phases.