Loop models and their critical limits

This page is for summarizing known results about two-dimensional loop models and their critical limits. There are many different loop models, depending on the lattice, on whether loops can cross, on whether the model is dilute, dense, fully packed, etc. In the critical limit, there are not that many different universality classes.

Results that are sufficiently well-established and well-covered in the literature could be summarized in Wikipedia. Here, we can have more recent or speculative results.

Lattice statistical models

For the moment we are collecting useful information and reference. Ultimately we want to organize these models into something resembling a classification.

• 6 vertex model on a square lattice, giving rise to a free boson in the critical limit?
• Nonintersecting loops on a honeycomb lattice. (Honeycomb model.) (4 vertex model.)
• Fully-packed loops on a honeycomb lattice . (3 vertex model.)
• Completely packed loops on a square lattice. (Model T.) (2 vertex model.)
• Potts' random cluster model. (2 bond model.)

What about RSOS?

Conformal field theories

• O(n) model.
• U(n) model.
• Potts model.
• Loop CFT.
• Compactified free boson.

Near-critical limits

Critical percolation occurs for the value $p={\frac {1}{2}}$ of the probability $p$ that determines the values of sites or bonds. Near-critical percolation is defined by taking $p\to {\frac {1}{2}}$ as the lattice size becomes large, such that the limit exists and is neither criticial nor non-critical. This gives rise to a massive field theory, which is invariant under rotations but not under rescalings.

Relevant Wikipedia articles

• w:Potts model
• w:N-vector_model
• w:Ising model
• w:Two-dimensional_critical_Ising_model

See also

• Particle approach to loop models