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 $.

中文翻译：

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Meandric 系统和无限面条中的组件

我们在这里研究一个大型的典型弯曲系统的渐近行为。更准确地说，我们展示了一个随机均匀的中间系统 $\boldsymbol {M}_n$ 在 $2n$ 点上的淬火局部收敛，如 $n \rightarrow \infty $，朝向由 Curien 等人引入的无限面条。[3]。因此，用 $cc( \boldsymbol {M}_n)$ 表示 $\boldsymbol {M}_n$ 的连通分量的数量，我们证明了 $cc(\boldsymbol {M}_n)/ 的概率收敛性n$ 到某个常数 $\kappa $，回答了 Goulden–Nica–Puder [8] 和 Kargin [12] 独立提出的问题。该结果还提供了有关非交叉分区晶格的 Hasse 图的渐近几何信息。最后，我们得到常数 $\kappa $ 的表达式为曲折上的无限和，