Fully Packed Loop Models on Finite Geometries

Abstract

A fully packed loop (FPL) model on the square lattice is the statistical ensemble of all loop configurations, where loops are drawn on the bonds of the lattice, and each loop visits every site once [4, 18]. On finite geometries, loops either connect external terminals on the boundary, or form closed circuits, see for example Fig. 13.1. In this chapter we shall be mainly concerned with FPL models on squares and rectangles with an alternating boundary condition where every other boundary terminal is covered by a loop segment, see Fig. 13.1. An FPL model thus describes the statistics of closely packed polygons on a finite geometry. Polygons may be nested, corresponding to punctures studied in Chapter 8. FPL models can be generalised to include weights. In particular we will study FPL models where a weight τ is given to each straight local loop segment. The partition function of an FPL model on various geometries can be computed exactly using its relation to the solvable six-vertex lattice model. It is well known that the model undergoes a bulk phase transition at τ = 2. We furthermore study nests of polygons connected to the boundary. In the case of FPL models with mirror or rotational symmetry, the probability distribution function of such nests is known analytically, albeit conjecturally. FPL 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. The purpose of this chapter is to collect and discuss known results for FPL models which may be relevant to polygon models. For that reason we have not put an emphasis on derivations, many of which are well-documented in the existing literature, but rather on interpretations of results.