Components of meandric systems and the infinite noodle
A meandric system of size $n$ is a non-intersecting collection of closed loops in the plane crossing the real line in exactly $2n$ points (up to continuous deformation). Connected meandric systems are called meanders, and their enumeration is a notoriously hard problem in enumerative combinatorics. In this talk, we discuss a different question, raised independently by Goulden–Nica–Puder and Kargin: what is the number of connected components $cc(M_n)$ of a uniform random meandric system of size $2n$? We prove that this number grows linear with $n$, and concentrates around its mean value, in the sense that $cc(M_n)/n$ converges in probability to a constant. Our main tool is the definition of a notion of local convergence for meandric systems, and the identification of the ''quenched Benjamini–Schramm'' limit of $M_n$. The latter is the so-called infinite noodle, a largely not understood percolation model recently introduced by Curien, Kozma, Sidoravicius and Tournier.
Our main result has also a geometric interpretation, regarding the Hasse diagram $H_n$ of the non-crossing partition lattice $NC(n)$: informally, our result implies that, in $H_n$, almost all pairs of vertices are asymptotically at the same distance from each other. We use here a connection between $H_n$ and meandric systems discovered by Goulden, Nica and Puder.
This talk is based on joint work with Paul Thevenin of Uppsala University.