The emergence of the electrostatic field as a Feynman sum in random tilings with holes
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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.References
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Additional Information
- Mihai Ciucu
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 605457
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2645056