论文标题
曲折系统和无限面的组件
Components in meandric systems and the infinite noodle
论文作者
论文摘要
我们在这里研究大型典型曲折系统的渐近行为。更准确地说，我们将$ m_n $ on 2n $上的随机统一系统$ m_n $（如$ n \ rightarrow \ infty $）与Curien，Kozma，Kozma，Sidoravicius和Tournier和Tournier（{ 2019）。结果，用$ cc（m_n）$表示$ m_n $的连接组件的数量，我们证明了$ cc（m_n）/n $的概率的融合到某个常数$κ$，回答了一个由Goulden-nica-nica-nica-nica-puder（{\ emint。nnic.nest。nest。nnic.not。nest。nest。nes.nest。 Kargin（{\ EM统计物理学杂志}，181（6）：2322---2345，2020）。该结果还提供了有关非交叉分区晶格的Hasse图的渐近几何形状的信息。最后，我们获得了常数$κ$的表达方式，作为蜿蜒的无限总和，这使我们能够计算$κ$的上和下近似值。
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 $n \rightarrow \infty$, 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 $κ$, 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 $κ$ as infinite sums over meanders, which allows us to compute upper and lower approximations of $κ$.
