Abstract
Here we consider a lattice \({\mathcal {N}}\) with pure dimer coverings, as defined in Sect. 3.9.
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Notes
- 1.
As observed in Sect. 3.9, for this to be the case \(N\) must be even.
- 2.
- 3.
or \(-1\) according to whether 
is of even or odd parity, that is whether 
is obtained from 
by an even or odd number of transpositions. A transposition is an exchange of the position of two indices, with the order of the remaining indices unchanged. Any transposition can be effected by an odd number of transpositions of adjacent indices. - 4.
In order to achieve the positive sign it may be necessary to change the definition of \({\varvec{Z}}\) by multiplying the first row and column by \(-1\).
- 5.
Provided consistency is maintained, either anticlockwise or clockwise order can be used.
- 6.
This is proved by Montroll (1964), Sect. 4.8, using Ledermann’s theorem.
- 7.
It can be seen from Fig. 3.10 that similar conservation rules do not apply to the square lattice. The reason is that there the number of vacant edges in a row of vertical edges depends not only on the number of dimers in the row of horizontal edges immediately below, but also on the number of dimers in the row of vertical edges below that.
or 
is of even or odd parity, that is whether 
is obtained from 
by an even or odd number of transpositions. A transposition is an exchange of the position of two indices, with the order of the remaining indices unchanged. Any transposition can be effected by an odd number of transpositions of adjacent indices.Author information
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Lavis, D.A. (2015). Dimer Models. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_13
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DOI: https://doi.org/10.1007/978-94-017-9430-5_13
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