Dimer model

In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.

For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the vertices of the grid. For instance, draw a chessboard, fix a node ${\displaystyle A_{0}}$ with height 0, then for any node there is a path from ${\displaystyle A_{0}}$ to it. On this path define the height of each node ${\displaystyle A_{n+1}}$ (i.e. corners of the squares) to be the height of the previous node ${\displaystyle A_{n}}$ plus one if the square on the right of the path from ${\displaystyle A_{n}}$ to ${\displaystyle A_{n+1}}$ is black, and minus one otherwise.

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