Percolation of words on $\Z^d$ with long range connections

TL;DR

This paper studies a percolation model on integer lattices with long-range connections, demonstrating that all binary sequences can be observed almost surely with bounded connection lengths, and explores how this bound scales as the connection probability decreases.

Contribution

It introduces a novel percolation model allowing long-range connections and proves the existence of a finite bound for observing all binary sequences almost surely.

Findings

Existence of a finite K(p) for observing all words at any p

Scaling behavior of K(p) as p approaches zero

Results on the probability of observing almost all words

Abstract

Consider an independent site percolation model on $\Z^d$, with parameter $p \in (0,1)$, where all long range connections in the axes directions are allowed. In this work we show that given any parameter $p$, there exists and integer $K(p)$ such that all binary sequences (words) $\xi \in \{0,1\}^{\N}$ can be seen simultaneously, almost surely, even if all connections whose length is bigger than $K(p)$ are suppressed. We also show some results concerning the question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related results are also obtained for the question of whether or not almost all words are seen.