Components in Meandric Systems and the Infinite Noodle

Abstract

We investigate here the asymptotic behaviour of a large, typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system |$\boldsymbol {M}_n$| on |$2n$| points, as |$n \rightarrow \infty $|⁠, towards the infinite noodle introduced by Curien et al. [3]. As a consequence, denoting by |$cc( \boldsymbol {M}_n)$| the number of connected components of |$\boldsymbol {M}_n$|⁠, we prove the convergence in probability of |$cc(\boldsymbol {M}_n)/n$| to some constant |$\kappa $|⁠, answering a question raised independently by Goulden–Nica–Puder [8] and Kargin [12]. This result also provides information on the asymptotic geometry of the Hasse diagram of the lattice of non-crossing partitions. Finally, we obtain expressions of the constant |$\kappa $| as infinite sums over meanders, which allows us to compute upper and lower approximations of |$\kappa $|⁠.