Dimer models and limit shapes from different perspectives

Vu, Hieu Trung

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

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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