Dimer models and limit shapes from different perspectives

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Vu, Hieu Trung

Contributors dc:contributor

• Di Francesco, Philippe
• Kedem, Rinat
• Yong, Alexander
• Russkikh, Marianna

• Dimer models
• limit shapes
• arctic curves
• statistical mechanics
• discrete complex analysis

Statement dc:rights

• Copyright 2025 Hieu Trung Vu

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129438

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129438