Field representation of non-intersecting circuits covering a lattice

1 Field representation of non-intersecting circuits covering a lattice 

In what follows it will be assumed that HAB equals one if A and B are nearest neighbours and that otherwise HAB equals zero. Of course, this postulate is questionable, because, in this case, the matrix HAB is not positive definite, in contradiction to the assumptions we always make concerning Gaussian variables (see Appendix A). However, we shall show later on that this difficulty is not as serious as it looks, and for the time being, we shall ignore it. Now, let us introduce the quantity 

A straightforward application of Wick s theorem (see Appendix A) shows that 

Let us now come back to the definition of the Gaussian ensemble. The real symmetric operator H has eigenvectors 0m (with real components 0Am) corresponding to the real eigenvalues Em 

new variables q m can be associated with the variables pA by means of the orthogonal transformation which diagonalizes H 

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