Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model

  • Authors:

  • Mihai Ciucu

  • Affiliations:

  • Georgia Institute of Technology, School of Mathematics, Atlanta, 6A 30332-0160. ciucu@math.gatech.edu

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1999

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Abstract

Motivated by the close relationship between the number of perfectmatchings of the Aztec diamond graph introduced in [5] andthe free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d ≥ 1, we constructd-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 givenby 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-vertexmodel exists and we obtain bounds for it. As a consequence, weobtain an especially good asymptotical approximation for thenumber of matchings of our hypergraphs.