Some aspects of probability distribution for percolation of several fluids on the hexagonal lattice

TL;DR

This paper investigates the probability distribution of multiple fluids percolating on a hexagonal lattice, introducing a new observable and proving a central limit theorem for it, with conjectures supported by numerical experiments.

Contribution

It introduces a novel observable for multi-fluid percolation and establishes a central limit theorem for its distribution, extending percolation theory.

Findings

A new observable for percolation of multiple fluids is defined.

A central limit theorem analogue is proved for this observable.

Numerical experiments support several conjectures about the model.

Abstract

We study random coloring of the hexagons of a honeycomb lattice into $2^{n-1}$ colors (that is the standard Potts model at infinite temperature). It may be considered as a generalization of percolation to $n$ pairwise independent, but mutually dependent liquids. We introduce a new observable that can be interpreted as the fraction of percolated liquids. An analogue of the central limit theorem for this observable is proved and several conjectures based on numeric experiments are proposed.