DELECROIX, Vincent; GOUJARD, Elise; ZOGRAF, Peter; ZORICH, Anton

doi:10.1017/fmp.2020.2

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DELECROIX, Vincent
Laboratoire Bordelais de Recherche en Informatique [LaBRI]

GOUJARD, Elise
Institut de Mathématiques de Bordeaux [IMB]

ZOGRAF, Peter

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Forum of Mathematics, Pi. 2020, vol. 8, p. e4

Cambridge Univ Press

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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. In physics, meanders provide a model of polymer folding, and their enumeration is directly related to the entropy of the associated dynamical systems. We combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero and our previous result that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated to derive two applications to asymptotic enumeration of meanders. First, we get simple asymptotic formulae for the number of pairs of transverse simple closed curves on a sphere and for the number of closed meanders of fixed combinatorial type when the number of crossings 2N goes to infinity. Second, we 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. Here the total number of minimal arcs of the two arc systems is considered as a fixed parameter while the number of all arcs (same for each of the two hemispheres) grows. The number of all meanders with 2N crossings grows exponentially when N grows. However, the additional combinatorial constraints we impose in this article yield polynomial asymptotics.< Réduire

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Enumeration of meanders and Masur-Veech volumes

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