Abstract. This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.

Elliott Lieb’s ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Q equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara-Miwa models.

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.

. Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d- uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2 d + 2-vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2 d + 2-vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs. 1. Introduction In [5] there are introduced the Aztec diamond graphs, which can be defined as follows. Consider a (2n+1) \Theta (2n+1) chessboard with black corners. The graph whose vertices are the white squares and whose edges connect precisely...

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 Figure 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 Figure 1. Fig. 1. Fully packed loops inside a square with alternating boundary condition. 2 Jan de Gier 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

Fractal structure of the six-vertex model is introduced with the use of the IFS (Iterated Function Systems). The fractal dimension satisfies an equation written by the free energy of the six-vertex model. It is pointed out that the transfer matrix method and the n-equivalence relation introduced in lattice theories have also been introduced in the area of fractal geometry. All the results can be generalized for the models suitable to the transfer matrix treatment, and hence this gives general relation between solvable lattice models and fractal geometry.