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

2007/10/24 by Mihai Ciucu, Ciucu, Mihai
Computer Science · Materials Science · Physics and Astronomy · #05 #60 #82 #Advanced Mathematical Theories and Applications #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties #Statistical Mechanics (cond-mat.stat-mech)

paper · pdf · doi:10.48550/arxiv.0710.4490

openalex publication_date 2007/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider random lozenge tilings on the triangular lattice with holes Q₁,...,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.

Citations

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