Recurrence Function on Sturmian Words: A Probabilistic Study

This paper is a first attempt to describe the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which provides precise information on the structure of such a word. With each Sturmian word of slope α, we associate particular sequences of factor lengths which have a given “position” with respect to the sequence of continuants of α, we then let α to be uniformly drawn inside the unit interval [0, 1]. This probabilistic model is well-adapted to better understand the role of the position in the recurrence properties.