Oriented chain on a Manhattan lattice Kasteleyns result

2 Oriented chain on a Manhattan lattice Kasteleyn s result 

A polymer liquid can be represented by a set of self-avoiding chains, drawn on a lattice so that each lattice site is on one of the chains but only one. Again, we may try to evaluate the number Z of configurations of the system or its entropy S = In Z. 

The problem has not exactly been solved, but Kasteleyn9 succeeded in handling a particularly interesting case. The lattice under consideration is an oriented lattice of the Manhattan type (see Fig. 3.14). Kasteleyn considers the configurations of the circuits which follow the one way lanes on the lattice and which cover the whole lattice without any overlap. 

Various results of graph theory enabled Kasteleyn to evaluate the number of configurations of the system in a rather simple manner (the origin of the chain is fixed once and for all in an arbitrary way). The calculation is simpler when the lattice is periodic. 

let us consider a rectangular Manhattan lattice consisting of M columns and N lines with cyclic boundary conditions (the lattice can be considered as drawn on a torus consequently, as the lattice is oriented, M and N are even). The number ZMi of configurations of the chain is given by the expression (Kasteleyn 1963).  

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