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Exact Asymptotics of Meander Numbers

In: Formal Power Series and Algebraic Combinatorics

Author

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  • P. Di Francesco

    (C.E.A. Saclay, Service de Physique Théorique)

Abstract

We address the meander problem “enumerate all topologically inequivalent configurations of a closed nonselfintersecting plane curve intersecting a given line through a fixed number of points”, known in many areas of mathematics, from the 16th Hilbert problem to the thory of knots and links, as well as in physics, from polymer folding to liquid crystals. We show that meander configurations may be viewed as the configurations of a suitable fully-packed loop two-dimensional statistical model defined on a random surface. Using standard physics results relating critical singularities of a lattice model to its gravitational version on random surfaces, we predict the meander configuration exponent % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % ypa0ZaaSGaaeaacaGGOaGaaGOmaiaaiMdacqGHRaWkdaGcaaqaaiaa % igdacaaI0aGaaGynaaWcbeaakiaacMcaaeaacaaIXaGaaGOmaaaaaa % a!4039! $$\alpha = {\raise0.7ex\hbox{${(29 + \sqrt {145} )}$} \!\mathord{\left/ {\vphantom {{(29 + \sqrt {145} )} {12}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${12}$}}$$ and many other meandric exponents.

Suggested Citation

  • P. Di Francesco, 2000. "Exact Asymptotics of Meander Numbers," Springer Books, in: Daniel Krob & Alexander A. Mikhalev & Alexander V. Mikhalev (ed.), Formal Power Series and Algebraic Combinatorics, pages 3-14, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04166-6_1
    DOI: 10.1007/978-3-662-04166-6_1
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