The recurrence function of a random Sturmian word

TL;DR

This paper analyzes the probabilistic behavior of the recurrence function in random Sturmian words, focusing on the distribution and density of waiting times to discover all factors of a given length.

Contribution

It provides a probabilistic analysis of the recurrence function in random Sturmian words, including limit distribution and density results for the waiting times.

Findings

Derived the limit distribution of the recurrence function

Established the limit density of waiting times

Analyzed the probabilistic behavior of factors in Sturmian words

Abstract

This paper describes the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which can be viewed as a waiting time to discover all the factors of length $n$ of the Sturmian word. This parameter is central to combinatorics of words. Having fixed a possible length $n$ for the factors, we let $\alpha$ to be drawn uniformly from the unit interval $[0,1]$, thus defining a random Sturmian word of slope $\alpha$. Thus the waiting time for these factors becomes a random variable, for which we study the limit distribution and the limit density.