Encoding Scheme for Infinite Set of Symbols: The Percolation Process on Infinite Perfect Binary Trees

TL;DR

This paper demonstrates that the percolation process on infinite perfect binary trees can serve as an encoding scheme for an infinite set of symbols, with finite average codeword length and entropy within certain percolation density ranges.

Contribution

It introduces a novel encoding scheme based on percolation clusters on infinite binary trees, linking percolation theory with information encoding.

Findings

Finite average codeword length for 1/2 ≤ p < 4^(-1/3)

Finite entropy within the same percolation density range

Percolation clusters encode an infinite symbol set

Abstract

It is shown here that the percolation cluster that emerges from the percolation process on infinite perfect binary trees, is genuinely an encoding scheme for an infinite set of symbols. The average codeword length and the entropy of such an encoding scheme are still finite as long as the percolation density $p$ is between $1/2\leq p < \sqrt[3]{1/4}$.