Abstract
The classic problem of counting monomer-dimer arrangements on a two-dimensional lattice is analyzed using techniques from theoretical computer science. Under a certain assumption, made precise in the text, it can be shown that the general problem is computationally intractable. This negative result contrasts with the special case of a system with monomer density zero, for which efficient solutions have been known for some time. A second, much easier result, obtained under the same assumption, is that the partition function of a three-dimensional Ising system is computationally intractable. Again, the negative result contrasts with known efficient techniques for evaluating the partition function of a two-dimensional system.
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Jerrum, M. Two-dimensional monomer-dimer systems are computationally intractable. J Stat Phys 48, 121–134 (1987). https://doi.org/10.1007/BF01010403
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DOI: https://doi.org/10.1007/BF01010403