Percolation for 2D Classical Heisenberg Model and Exit Sets of Vector Valued GFF

Area of Science:

• Statistical Physics
• Probability Theory
• Mathematical Physics

Background:

• Investigates the geometric properties of exit sets associated with 2D vector-valued Gaussian Free Fields (GFF).
• Examines the percolation properties of level sets of GFF and their relation to spin models.
• Addresses the massiveness of spin models, particularly the XY model, and its dependence on temperature and disorder.

Purpose of the Study:

• To analyze the geometry of exit sets for 2D vector-valued GFF and their behavior based on the field's dimension.
• To understand the impact of random conductances on spin O(N) models, using GFF exit sets for geometric description.
• To rigorously revisit and counter-argue predictions regarding the massiveness of spin models at different temperatures.

Main Methods:

• Geometric analysis of exit sets for 2D vector-valued Gaussian Free Fields.
• Application of exit set geometry to describe quenched disorder in spin O(N) models.
• Rigorous mathematical proofs to establish properties of spin models and GFF fluctuations.

Main Results:

• Exit sets are shown to be degenerate for d < 2 and conjectured to be fractal for d > 2.
• A counter-example is constructed: an XY model with high conductances and disconnected low-conductance regions remains massive.
• Fluctuations of the classical Heisenberg model at high d are proven to be a d-dimensional vectorial GFF.

Conclusions:

• The study provides a rigorous geometric description of disorder in spin models.
• It challenges established predictions on spin model massiveness, offering new insights into phase transitions.
• Establishes a rigorous proof for GFF fluctuations in the Heisenberg model and connects correlation functions to percolation events.