A Sharp Deconfinement Transition for Potts Lattice Gauge Theory in Codimension Two

Area of Science:

• Statistical Mechanics
• Percolation Theory
• Lattice Gauge Theory

Background:

• The seminal 1983 work by Aizenman et al. established a sharp phase transition for 2D Bernoulli plaquette percolation.
• This transition relates to the boundedness of large rectangular loops by surfaces of plaquettes in 3D lattices.

Purpose of the Study:

• To generalize the sharp phase transition result to higher-dimensional lattices.
• To investigate phase transitions in dependent percolation models, specifically the plaquette random-cluster model.
• To establish sharp phase transitions for Wilson loop expectations in dual Potts lattice gauge theories.

Main Methods:

• Extension of the original proof techniques to (d-1)-dimensional plaquette percolation in d-dimensional lattices.
• Application of these methods to the plaquette random-cluster model, a dependent percolation model.
• Leveraging the duality between the random-cluster model and Potts lattice gauge theory.

Main Results:

• Demonstrated sharp phase transitions for (d-1)-dimensional plaquette percolation in Z^d.
• Proved sharp phase transitions for Wilson loop expectations in (d-2)-dimensional q-state Potts hyperlattice gauge theory.
• Developed the general theory of the i-plaquette random cluster model and its connection to (i-1)-dimensional Potts lattice gauge theory.

Conclusions:

• The sharp phase transition phenomenon in percolation theory is robust and extendable to higher dimensions and dependent models.
• The results provide a rigorous foundation for understanding phase transitions in related lattice gauge theories.
• The study highlights the power of random-cluster models in analyzing complex statistical physics systems.