A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

中文翻译：

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曲折和 MASUR-VEECH 卷的枚举

一种蜿蜒是直线和平面内的简单闭合曲线（或 2 球面上的一对简单闭合曲线）横向相交的拓扑配置。曲折可以追溯到 H. Poincaré，自然出现在数学、理论物理学和计算生物学的各个领域（特别是，它们提供了聚合物折叠的模型）。曲折的枚举是一个重要的开放性问题。蜿蜒曲折的数量 $2N$ 当交叉口呈指数增长时 $N$ 增长，但长期存在的精确渐近问题仍然遥不可及。我们表明，如果另外固定曲流的拓扑类型（或最小弧的总数），情况会变得更容易处理。然后我们可以推导出曲折数的简单渐近公式为 $N$ 趋于无穷大。我们还通过识别沿共同赤道的两个弧系统（两个半球各一个）的端点来计算在球体上获得简单闭合曲线的渐近概率。我们带来的新工具基于将曲折解释为具有一个水平和一个垂直圆柱体的方形瓷砖表面。证明将最近关于零属亚纯二次微分模空间的 Masur-Veech 体积的结果与我们的新观察结果相结合，即整数二次微分的水平和垂直分离图是渐近不相关的。我们在本文中施加的附加组合约束产生了显式多项式渐近。