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Dimer models and limit shapes from different perspectives
Vu, Hieu Trung
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https://hdl.handle.net/2142/129438
Description
- Title
- Dimer models and limit shapes from different perspectives
- Author(s)
- Vu, Hieu Trung
- Issue Date
- 2025-05-01
- Director of Research (if dissertation) or Advisor (if thesis)
- Di Francesco, Philippe
- Kedem, Rinat
- Doctoral Committee Chair(s)
- Yong, Alexander
- Committee Member(s)
- Russkikh, Marianna
- Department of Study
- Mathematics
- Discipline
- Mathematics
- Degree Granting Institution
- University of Illinois Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- Dimer models
- limit shapes
- arctic curves
- statistical mechanics
- discrete complex analysis
- Abstract
- This thesis considers two projects and some progress toward the theory of dimer models in statistical mechanics. The two questions we are trying to answer are the limit shapes of the dimer model and the height function fluctuation in the scaling limit. The first project considers the asymptotic and arctic curves of dimer models arising from a special set of initial data of the octahedron recurrence. This is a joint work with Di Francesco. This approach of ours bypassed the classical approach of constructing a probability measure from the inverse of the Kasteleyn matrix, but with the cost of no direct access to the fluctuation of height function in the liquid regions. The second project is an ongoing project with David Keating. We study a recently discovered structure known as perfect t-embedding on the complex plane, which would resolve the limitation of the previous project approach and complete the analysis of the Gaussian fluctuation of the height function in the liquid region of the dimer model. The existence of such planar embedding structures for dimer models implies that the fluctuations of the height function are governed by the Gaussian Free Field, which is a conjecture by Kenyon and Okounkov in 2007.
- Graduation Semester
- 2025-05
- Type of Resource
- Thesis
- Handle URL
- https://hdl.handle.net/2142/129438
- Copyright and License Information
- Copyright 2025 Hieu Trung Vu
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Graduate Dissertations and Theses at Illinois PRIMARY
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