Delocalisation and Continuity in 2D: Loop  , Six-Vertex, and Random-Cluster Models.

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Probability Theory

Background:

• The study addresses a long-standing conjecture regarding critical points in statistical mechanics models.
• Existing proofs for phase transition continuity often rely on complex integrability tools or specific theories.
• The loop $$ and six-vertex models are fundamental in understanding phase transitions.

Purpose of the Study:

• To prove the existence of macroscopic loops in the loop $$ model and delocalization in the six-vertex model.
• To provide a new, more general proof for the continuity of phase transitions in 2D random-cluster and Potts models.
• To investigate the critical behavior and scaling limits of these models.

Main Methods:

• Development of a novel FKG (Fortuin-Kasteleyn-Ginibre) property for the non-coexistence theorem.
• Application of the $$-circuit argument for the six-vertex model.
• Extension of existing renormalization inequalities to quantify delocalization.

Main Results:

• Existence of macroscopic loops in the loop $$ model and delocalization of the associated Lipschitz function.
• Proof of delocalization in the six-vertex model with $$.
• New proofs for the continuity of phase transitions in 2D random-cluster and Potts models for $$.
• Quantification of delocalization as logarithmic in specific regimes, consistent with Gaussian free field conjectures.

Conclusions:

• The findings confirm key aspects of the Fan, Domany, and Nienhuis conjecture.
• The novel approach bypasses the need for integrability tools or Russo-Seymour-Welsh theory.
• The results provide a deeper understanding of phase transitions and critical phenomena in statistical mechanics.