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The planar dimer model boundary: A survey. (English) Zbl 1026.82007
Baake, Michael (ed.) et al., Directions in mathematical quasicrystals. Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 307-328 (2000).
From the introduction: A dimer covering of a finite graph is a perfect matching of the graph, that is, a set of edges with the property that every vertex is contained in a unique edge. The dimer model is the statistical model dealing with the set of all dimer coverings of a graph. {\it P. W. Kasteleyn} [Physica 27, 1209-1225 (1961)] and {\it H. N. V. Temperley} and {\it M. E. Fisher} [Philos. Mag. (8) 6, 1061-1063 (1961; Zbl 0126.25102)] initiated the study of the dimer model by showing how to count exactly the number of perfect matchings of an $m\times n$ grid (when at least one of $m$ and $n$ is even). The number is the square root of the determinant of an $mn\times mn$ matrix, and is explicitly: $$\Biggl[ \prod_{j=1}^m \prod_{k=1}^n \Biggl( 2\cos \biggl( \frac{\pi j}{m+1} \biggr)+ 2i\cos \biggl( \frac{\pi k}{n+1} \biggr)\Biggr) \Biggr]^{1/2}.$$ Later, {\it P. W. Kasteleyn} [J. Math. Phys. 4, 287-293 (1963)] showed how to efficiently count the number of perfect matchings of any finite planar graph: the number is the square root of the determinant of a matrix closely related to the adjacency matrix of the graph. Since that time a great deal of work has been done in the physical aspects of the dimer model. In the early 1990’s some intriguing combinatorial results brought a renewed mathematical interest in the model. These results touched on the influence of the boundary on a random dimer covering of a bounded graph. In this paper we give a short survey of some of these new results and concentrate on the mathematical point of view. The first section deals with purely combinatorial questions of the type: which boundary conditions allow the existence of a tiling? Section 3 discusses Kasteleyn’s method of computing the partition function $Z$ in each of our examples. In Section 4 we discuss how to compute densities of local patterns. In Section 5 we discuss how the boundary of a region influences the local entropies and local densities in a region. In Section 6 we discuss in the domino model how the boundary, even when it does not have an influence on the local statistics, can still affect the long-range properties of the limiting measure. For the entire collection see [Zbl 0955.00025].

MSC:
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82-02Research monographs (statistical mechanics)
60K35Interacting random processes; statistical mechanics type models; percolation theory