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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
DOI: https://doi.org/10.1090/S0002-9947-10-05087-7
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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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-10-05087-7
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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