Delocalization of Planar Integer-Valued Height Functions and O(2)-Invariant Spin Models

Explore a 57-minute lecture from the USC Probability and Statistics Seminar featuring Matan Harel discussing the delocalization of planar integer-valued height functions. Delve into the relationship between two-dimensional lattice models, focusing on O(2)-invariant spin models and integer-valued height function models. Learn how delocalization in height function models implies power-law decay in associated O(2)-invariant spin models. Discover the extension of Lammers' work showing delocalization of certain integer-valued height functions on doubly periodic graphs. Gain new insights into the Berezinksii-Kosterlitz-Thouless phase transition for two-dimensional O(2)-invariant lattice models. Follow the comprehensive syllabus covering topics such as the Ising Model, XY Model, continuous symmetry, Mermin-Wagner theorem, Peierls vs. Spin Wave, integer-valued random surfaces, localization and delocalization, duality relations, lattice monotonicity, and the connection between level lines and slow decay.

Intro
The Ising Model and Long-Range Order
The XY Model and Continuous Symmetry
Continuous Symmetry and Mermin-Wagner
Peierls vs. The Spin Wave
The BKT Phase Transition
Integer-valued Random Surfaces
Localization and Delocalization
Duality Relations
Lattice Monotonicity for the ZGF
Surgery for the ZGF with Lattice Monotonicity
Surgery with Lattice Monotonicity
Defect lines and level lines
From level lines to slow decay