Percolation of arbitrary words in one dimension

TL;DR

This paper investigates the conditions under which arbitrary infinite binary words appear in one-dimensional iid Bernoulli sequences, providing complete results in some cases and partial insights in others.

Contribution

It offers new theoretical insights into the percolation of arbitrary words in one-dimensional sequences, resolving some cases and advancing understanding in the field.

Findings

Certain infinite words are shown to almost surely appear under specific conditions.

Partial results are provided for cases where the appearance criteria are not fully determined.

The work extends previous models of percolation to more general word patterns.

Abstract

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.