Free probability aspect of irreducible meandric systems, and some related observations about meanders

Abstract

We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of [Formula: see text]. As a consequence, we show that the even generating function for irreducible meandric systems is the [Formula: see text]-transform of [Formula: see text], where [Formula: see text] and [Formula: see text] are classically (commuting) independent random variables, and each of [Formula: see text] has centred semicircular distribution of variance [Formula: see text]. Following this point of view, we make some observations about the symmetric linear functional on [Formula: see text] which has [Formula: see text]-transform given by the even generating function for meanders.