Abstract
We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer r, we express the generating function of meandric systems on
2n
points with
n \to r
loops in terms of a finite (the size depends on
r
) subclass of irreducible meandric systems, via the moment-cumulant formula from free probability theory. We show that the generating function, after an appropriate change of variable, is a rational function, and we bound its degree. Exact expressions for the generating functions are obtained for
r \leq 6
, as well as the asymptotic behavior of the meandric numbers for general
r
.