Dimer Cooling 1 (Random dimer tiling)
thomas fernique
2012-02-10
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For detailed explanations, see http://arxiv.org/abs/1111.7297
A dimer tiling is a perfect matching of the cells of a domain of the triangular grid (each cell is grouped with a unique cell to form a rhombi called dimer).
A dimer tiling can easily be seen as piles of cubes. A simple local transformation, called flip, which is known to be ergodic on the set of dimer tilings of a domain is removing/adding a cube (that is, turning by 120° a hexagon formed by three rhombi).
Now, let us call "error" in a dimer tiling two rhombi identical up to a translation which are adjacent along an edge. Then, draw uniformly at random one of the many possible dimer tilings of a hexagonal domain and run the following discrete time homogeneous Markov chain : draw uniformly at random a vertex and, if it is possible to perform a flip around this vertex, do it if and only if it does not increase the total number of errors in the dimer tiling.
Can we bound the hitting time of an error-free dimer tiling?
A dimer tiling is a perfect matching of the cells of a domain of the triangular grid (each cell is grouped with a unique cell to form a rhombi called dimer).
A dimer tiling can easily be seen as piles of cubes. A simple local transformation, called flip, which is known to be ergodic on the set of dimer tilings of a domain is removing/adding a cube (that is, turning by 120° a hexagon formed by three rhombi).
Now, let us call "error" in a dimer tiling two rhombi identical up to a translation which are adjacent along an edge. Then, draw uniformly at random one of the many possible dimer tilings of a hexagonal domain and run the following discrete time homogeneous Markov chain : draw uniformly at random a vertex and, if it is possible to perform a flip around this vertex, do it if and only if it does not increase the total number of errors in the dimer tiling.
Can we bound the hitting time of an error-free dimer tiling?
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