Article Dans Une Revue International Mathematics Research Notices Année : 2022

Components in Meandric Systems and the Infinite Noodle

Valentin Féray
Paul Thévenin

Résumé

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 $.
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hal-03797902 , version 1 (06-01-2025)

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Valentin Féray, Paul Thévenin. Components in Meandric Systems and the Infinite Noodle. International Mathematics Research Notices, 2022, 2023 (14), pp.12538-12560. ⟨10.1093/imrn/rnac156⟩. ⟨hal-03797902⟩
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