2026-09-22T10:14:26Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129438 2025-10-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms The student, Hieu Trung Vu, accepted the attached license on 2025-04-23 at 19:43. The student, Hieu Trung Vu, submitted this Dissertation for approval on 2025-04-23 at 19:50. This Dissertation was approved for publication on 2025-05-01 at 08:49. DSpace SAF Submission Ingestion Package generated from Vireo submission #21933 on 2025-10-19 at 18:18:55 Dimer models and limit shapes from different perspectives Vu, Hieu Trung 2025-05-01 Di Francesco, Philippe Kedem, Rinat Yong, Alexander Russkikh, Marianna Dimer models limit shapes arctic curves statistical mechanics discrete complex analysis eng 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. 2025-05 Thesis https://hdl.handle.net/2142/129438 Copyright 2025 Hieu Trung Vu Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation