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

TL;DR

This paper explores the connection between irreducible meander systems and free probability theory, showing that their generating functions relate to the R-transform of free semicircular variables, and offers new insights into their algebraic structure.

Contribution

It establishes a novel link between irreducible meander systems and free probability, specifically relating their generating functions to the R-transform of free semicircular variables.

Findings

The even generating function for irreducible meandric systems equals the R-transform of XY.

X and Y are independent semicircular variables with mean zero and variance 1.

Provides observations on the symmetric linear functional with R-transform given by meander generating functions.

Abstract

We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of NC(n). As a consequence, we show that the even generating function for irreducible meandric systems is the R-transform of XY, where X and Y are classically (commuting) independent random variables, and each of X,Y has centred semicircular distribution of variance 1. Following this point of view, we make some observations about the symmetric linear functional on polynomials which has R-transform given by the even generating function for meanders.