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Transactions of the American Mathematical Society

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The emergence of the electrostatic field as a Feynman sum in random tilings with holes

Author: Mihai Ciucu
Journal: Trans. Amer. Math. Soc. 362 (2010), 4921-4954
MSC (2000): Primary 82B23, 82D99; Secondary 05A16, 60F99
Published electronically: April 28, 2010
MathSciNet review: 2645056
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Abstract: We consider random lozenge tilings on the triangular lattice with holes $ Q_{1},\dots ,Q_{n}$ in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge covering it. We show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole $ Q_{i}$ as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside $ Q_{i}$. This is then restated in terms of random surfaces, yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is, in the scaling limit, a sum of helicoids.

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  • 1. M. Ciucu, Rotational invariance of quadromer correlations on the hexagonal lattice, Adv. in Math. 191 (2005), 46-77. MR 2102843 (2006j:52023)
  • 2. M. Ciucu, A random tiling model for two dimensional electrostatics, Mem. Amer. Math. Soc. 178 (2005), no. 839. MR 2172582 (2007c:82022)
  • 3. M. Ciucu, The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions, Mem. Amer. Math. Soc. 199 (2009), no. 935, 1-100. arXiv:math-ph/0501071.
  • 4. M. Ciucu, Dimer packings with gaps and electrostatics, Proc. Natl. Acad. Sci. USA 105 (2008), 2766-2772. MR 2383565 (2009a:82012)
  • 5. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), 117-166. MR 1412441 (97k:52026)
  • 6. H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), 297-346. MR 1815214 (2002k:82038)
  • 7. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. of Math. 4 (1998), 137-165. MR 1641839 (99j:60011)
  • 8. T. H. Colding and W. P. Minicozzi II, Disks that are double spiral staircases, Notices Amer. Math. Soc. 50 (2003), 327-339. MR 1954009 (2004i:53010)
  • 9. R. P. Feynman, ``QED: The strange theory of light and matter,'' Princeton University Press, Princeton, New Jersey, 1985.
  • 10. M. E. Fisher and J. Stephenson, Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers, Phys. Rev. (2) 132 (1963), 1411-1431. MR 0158705 (28:1928)
  • 11. J. D. Jackson, ``Classical Electrodynamics,'' Third Edition, Wiley, New York, 1998. MR 0436782 (55:9721)
  • 12. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 591-618. MR 1473567 (99b:82039)
  • 13. R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and Amoebae, Ann. of Math. 163 (2006), 1019-1056. MR 2215138 (2007f:60014)
  • 14. S. Sheffield, Random Surfaces, Astérisque, 2005, no. 304. MR 2251117 (2007g:82021)
  • 15. W. P. Thurston, Conway's tiling groups, Amer. Math. Monthly 97 (1990), 757-773. MR 1072815 (91k:52028)

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Additional Information

Mihai Ciucu
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Received by editor(s): January 13, 2009
Received by editor(s) in revised form: April 15, 2009
Published electronically: April 28, 2010
Additional Notes: This research was supported in part by NSF grants DMS 0500616 and 0801625.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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