ENUMERATION of MEANDERS and MASUR-VEECH VOLUMES

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ENUMERATION of MEANDERS and MASUR-VEECH VOLUMES. / Delecroix, Vincent; Goujard, Élise; Zograf, Peter; Zorich, Anton.

In: Forum of Mathematics, Pi, Vol. 8, e4, 2020.

Research output: Contribution to journal › Article › peer-review

Author

Delecroix, Vincent ; Goujard, Élise ; Zograf, Peter ; Zorich, Anton. / ENUMERATION of MEANDERS and MASUR-VEECH VOLUMES. In: Forum of Mathematics, Pi. 2020 ; Vol. 8.

BibTeX

@article{be53ce0631f147ddade025bb6e2e423e,

title = "ENUMERATION of MEANDERS and MASUR-VEECH VOLUMES",

abstract = "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. Poincar{\'e} 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 crossings grows exponentially when 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 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. ",

author = "Vincent Delecroix and {\'E}lise Goujard and Peter Zograf and Anton Zorich",

year = "2020",

doi = "10.1017/fmp.2020.2",

language = "English",

volume = "8",

journal = "Forum of Mathematics, Pi",

issn = "2050-5086",

publisher = "Wiley-Blackwell",

}

RIS

TY - JOUR

T1 - ENUMERATION of MEANDERS and MASUR-VEECH VOLUMES

AU - Delecroix, Vincent

AU - Goujard, Élise

AU - Zograf, Peter

AU - Zorich, Anton

PY - 2020

Y1 - 2020

N2 - 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 crossings grows exponentially when 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 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.

AB - 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 crossings grows exponentially when 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 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.

UR - http://www.scopus.com/inward/record.url?scp=85082197804&partnerID=8YFLogxK

U2 - 10.1017/fmp.2020.2

DO - 10.1017/fmp.2020.2

M3 - Article

AN - SCOPUS:85082197804

VL - 8

JO - Forum of Mathematics, Pi

JF - Forum of Mathematics, Pi

SN - 2050-5086

M1 - e4

ER -

ID: 98425988