[Paper Review] Fully packed loop models on finite geometries

Jan de Gier|ArXiv.org|Jan 26, 2009

Theoretical and Computational Physics82 references3 citations

TL;DR

This paper studies fully packed loop (FPL) models on finite square and rectangular lattices with alternating boundary conditions, establishing exact relations to the six-vertex model and alternating sign matrices. It derives a non-Gaussian cubic exponential scaling form for the nest distribution function at criticality and identifies a crossover exponent φ = 2/3 for boundary phase transitions, with closed-form expressions for average nest counts under symmetry constraints.

ABSTRACT

Fully packed loop models describe the statistics of closely packed nested polygons on the square lattice. Many exact results can be obtained for these models, even for finite geometries, using their close relationship to alternating-sign matrices and the solvable six-vertex and O(n=1) lattice models. Some results for the exact partition function of fully packed loop models on various finite geometries are briefly reviewed, as well as the well-known order-disorder bulk phase transition present in these models. A detailed study is presented of the distribution of boundary nests of polygons in fully packed loop models with mirror or rotational symmetry. The probability distribution function of such nests, as well as the average number of nests, are obtained analytically, albeit conjecturally. It is further shown that fully packed loop models undergo another phase transition as a function of the boundary nest fugacity. At criticality, we derive a scaling form for the nest distribution function which displays an unusual non-Gaussian cubic exponential behaviour.

Motivation & Objective

• To analyze fully packed loop configurations on finite square and rectangular lattices with alternating boundary conditions.
• To establish connections between FPL models, the six-vertex model, and alternating sign matrices (ASMs) via height configurations.
• To derive exact expressions for partition functions of symmetric FPL subsets (horizontal and half-turn symmetry) and compute average nest counts.
• To investigate boundary phase transitions via the nest fugacity and determine critical scaling behavior of the nest distribution function.

Proposed method

• Utilizes a bijection between FPL configurations, six-vertex model states, and alternating sign matrices (ASMs), with height configurations defined via cumulative sums of ASM entries.
• Employs local height update operators G_ij acting on even and odd sublattices to generate FPL configurations and study their symmetries.
• Applies hypergeometric summation identities to compute the average number of nests in symmetric FPL configurations, particularly under horizontal and half-turn symmetry.
• Derives asymptotic behavior of the nest distribution function using solutions to hypergeometric differential equations for different regimes of the nest fugacity z.
• Analyzes critical behavior by solving the hypergeometric equation in the z=1 regime, leading to a scaling form with cubic exponential decay.
• Uses numerical evaluation and analytical continuation to confirm the scaling function f(x) = b x e^{-b x^3 / 3} at criticality.

Experimental results

Research questions

• RQ1 What is the exact scaling form of the nest distribution function at the critical point of the boundary phase transition?
• RQ2 How does the average number of polygon nests behave asymptotically under horizontal or half-turn symmetry constraints?
• RQ3 What is the crossover exponent φ governing the transition between different regimes of the nest fugacity z?
• RQ4 How are fully packed loop models on finite geometries related to the six-vertex model and alternating sign matrices?
• RQ5 Can closed-form expressions be derived for the partition functions of symmetric FPL subsets on finite lattices?

Key findings

• At criticality (z=1), the nest distribution function exhibits a non-Gaussian cubic exponential scaling form f(x) = b x e^{-b x^3 / 3}, confirmed numerically for L=600.
• The crossover exponent φ = 2/3 governs the transition between regimes of the nest fugacity z, observed in both horizontally symmetric and half-turn symmetric FPL models.
• For half-turn symmetric FPLs, the average number of nests grows asymptotically as ⟨1+m⟩ ≈ (Γ(5/6)/√π) L^{2/3} + O(1) at z=1, consistent with the φ=2/3 scaling.
• For z<1, the average number of nests grows polynomially as ⟨1+m⟩ ≈ (2+z)/(2(1−z)) + O(1), while for z>1, it scales linearly as ⟨1+m⟩ ≈ √((z−1)/(4z−1)) L + O(1).
• Closed-form expressions for the partition functions of horizontally symmetric and half-turn symmetric FPL models are derived experimentally and remain conjectural.
• The model exhibits a bulk phase transition at τ=2, corresponding to the anisotropy parameter of straight loop segments, with critical behavior linked to the O(n=1) loop model.