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 $M_n$ on $2n$ points, as $n\to \infty$, towards the infinite noodle introduced by Curien, Kozma, Sidoravicius and Tournier ({\em Ann. Inst. Henri Poincaré D}, {6}(2):221--238, 2019). As a consequence, denoting by $cc(M_n)$ the number of connected components of $M_n$, we prove the convergence in probability of $cc(M_n)/n$ to some constant $\kappa$, answering a question raised independently by Goulden--Nica--Puder ({\em Int. Math. Res. Not.}, 2020(4):983--1034, 2020) and Kargin ({\em Journal of Statistical Physics}, 181(6):2322--2345, 2020). 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 $\kappa$.

22 pages, 7 figures. v2: addition of an erratum

Cited by in corpus (2)

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