Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model
Mihai Ciucu
DOI: 10.1023/A:1018652320257
Abstract
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 2d + 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 2d + 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, we construct d-uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2d + 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 2d + 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.Pages: 281–293
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References
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2. M. Ciucu, “Perfect matchings of cellular graphs,” J. Alg. Combin. 5 (1996), 87-103.
3. M. Ciucu, “A complementation theorem for perfect matchings of graphs having a cellular completion,” J. Combin. Theory Ser. A 81 (1998), 34-68.
4. M. Ciucu, “An improved upper bound for the three dimensional dimer problem,” Duke Math. J. 94 (1998), 1-11.
5. N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, “Alternating-sign matrices and domino tilings (Part II),” J. Alg. Combin. 1 (1992), 219-234.
6. E.H. Lieb, “Residual entropy of square ice,” Phys. Rev. 162 (1967), 162-172.
7. E.H. Lieb, “Exact solution of the F -model of an antiferroelectric,” Phys. Rev. Lett. 18 (1967), 1046-1048.
8. E.H. Lieb, “Exact solution of the two-dimensional Slater KDP model of a ferroelectric,” Phys. Rev. Lett. 19 (1967), 108-110.
9. H. Minc, Nonnegative Matrices, Wiley, New York, 1988.
10. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
11. B. Sutherland, “Exact solution of a two-dimensional model for hydrogen-bonded crystals,” Phys. Rev. Lett. 19 (1967), 103-104.