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The Interaction of a Gap with a Free Boundary in a Two Dimensional Dimer System

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Abstract

Let ℓ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let \({\mathcal{H}}\) be the half plane to the left of ℓ. We consider lozenge tilings of \({\mathcal{H}}\) that have a triangular gap of side-length two and in which ℓ is a free boundary — i.e., tiles are allowed to protrude out half-way across ℓ. We prove that the correlation function of this gap near the free boundary has asymptotics \({\frac{1}{4\pi r}}\), r → ∞, where r is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.

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References

  1. Andrews, G.E.: Plane partitions I: The MacMahon conjecture. In: Studies in foundations and combinatorics, G.-C. Rota, ed., Adv. in Math. Suppl. Studies, Vol. 1, New York London: Academic Press, pp. 131–150, 1978

  2. Andrews, G.E., Askey, R.A., Roy, R.: Special functions. In: Encyclopedia of Math. And Its Applications 71, Cambridge: Cambridge University Press, 1999

  3. Baik, J., Kriecherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete orthogonal polynomials. In: Asymptotics and applications Ann. Math. Studies, Princeton, NJ: Princeton University Press, 2007

  4. Bailey W.N.: Generalized hypergeometric series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  5. Ciucu M.: Rotational invariance of quadromer correlations on the hexagonal lattice. Adv. in Math. 191, 46–77 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ciucu M.: A random tiling model for two dimensional electrostatics. Mem. Amer. Math. Soc. 178(839), 1–106 (2005)

    MathSciNet  Google Scholar 

  7. Ciucu M.: Dimer packings with gaps and electrostatics. Proc. Natl. Acad. Sci. USA 105, 2766–2772 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Ciucu M.: The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions. Mem. Amer. Math. Soc. 199(935), 1–100 (2009)

    MathSciNet  Google Scholar 

  9. Ciucu M.: The emergence of the electrostatic field as a Feynman sum in random tilings with holes. Trans. Amer. Math. Soc. 362, 4921–4954 (2010)

    MATH  MathSciNet  Google Scholar 

  10. Ciucu M., Krattenthaler C.: The number of centered lozenge tilings of a symmetric hexagon. J. Combin. Theory Ser. A 86, 103–126 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cohn H., Larsen M., Propp J.: The shape of a typical boxed plane partition. New York J. of Math. 4, 137–165 (1998)

    MATH  MathSciNet  Google Scholar 

  12. Di Francesco, P., Reshetikhin, N.: Asymptotic shapes with free boundaries. preprint; http://arxiv.org/abs/0908.1630v1 [mathph], 2009

  13. Feynman, R.P.: The Feynman Lectures on Physics, vol. II, Reading, MA: Addison-Wesley, 1963

  14. Fischer I.: Another refinement of the Bender–Knuth (ex-)conjecture. Eur. J. Combin. 27, 290–321 (2006)

    Article  MATH  Google Scholar 

  15. Fisher M.E., Stephenson J.: Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. 132(2), 1411–1431 (1963)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Gessel, I.M., Viennot, X.: Determinants, paths, and plane partitions. Preprint, 1989, available at: http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf (1989)

  17. Gordon B.: A proof of the Bender–Knuth conjecture. Pac. J. Math. 108, 99–113 (1983)

    MATH  Google Scholar 

  18. Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. Addison-Wesley, Reading, MA (1989)

    MATH  Google Scholar 

  19. Ishikawa M., Wakayama M.: Minor summation formula for pfaffians. Linear and Multilinear Algebra 39, 285–305 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kenyon R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33, 591–618 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Kenyon R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185, 239–286 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kenyon R., Okounkov A., Sheffield S.: Dimers and amoebae. Ann. of Math. 163, 1019–1056 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Krattenthaler, C.: The major counting of nonintersecting lattice paths and generating functions for tableaux. Mem. Amer. Math. Soc. 115(552), (1995)

  24. Krattenthaler, C.: An alternative evaluation of the Andrews–Burge determinant. In: Mathematical Essays in Honor of Gian-Carlo Rota, B. E. Sagan, R. P. Stanley eds., Progress in Math., Vol. 161, Boston: Birkhäuser, 1998, pp. 263–270

  25. Krattenthaler, C.: Advanced determinant calculus. Séminaire Lotharingien Combin. 42 (“The Andrews Festschrift”) (1999), Article B42q, 67 pp

  26. Lindström B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85–90 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  27. Macdonald I.G.: Symmetric Functions and Hall Polynomials. Second edition. Oxford University Press, New York-London (1995)

    MATH  Google Scholar 

  28. MacMahon, P.A.: Combinatory Analysis. Vol. 2, Cambridge: Cambridge University Press, 1916; reprinted. New York: Chelsea, 1960

  29. Mehta M.L., Wang R.: Calculation of a certain determinant. Commun. Math. Phys. 214, 227–232 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Olver, F.W.J.: Asymptotics and special functions. Reprint of the 1974 original [New York: Academic Press] Wellesley, MA: A K Peters, Ltd., 1997

  31. Proctor R.A.: Bruhat lattices, plane partitions generating functions, and minuscule representations. Europ. J. Combin. 5, 331–350 (1984)

    MATH  MathSciNet  Google Scholar 

  32. Sheffield, S.: Random surfaces. Astérisque, Vol. 304, Paris: Soc. Math. France, 2005

  33. Slater L.J.: Generalized hypergeometric functions. University Press Cambridge, Cambridge (1966)

    MATH  Google Scholar 

  34. Stembridge J.R.: Nonintersecting paths, pfaffians and plane partitions. Adv. in Math. 83, 96–131 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  35. Vella D., Mahadevan L.: The “Cheerios effect”. Amer. J. Phys. 73, 817–825 (2005)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405-5701, USA

    M. Ciucu

  2. Fakultät für Mathematik der Universität Wien, Nordbergstraße 15, A-1090, Wien, Austria

    C. Krattenthaler

Authors
  1. M. Ciucu
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  2. C. Krattenthaler
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Correspondence to M. Ciucu.

Additional information

Communicated by H. Spohn

Research partially supported by NSF grant DMS-0500616.

Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory.”

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Ciucu, M., Krattenthaler, C. The Interaction of a Gap with a Free Boundary in a Two Dimensional Dimer System. Commun. Math. Phys. 302, 253–289 (2011). https://doi.org/10.1007/s00220-010-1186-5

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