function can be given for the partition function. Consider the partition function of the free theory: $$Z_0 = \prod_n \int \mathrm{d} \phi_n \; \mathrm{e}^{-S_0},$$ with the lattice action $S_0$ given by Eq. (\[Sl\]). This integral generalizes the standard Gaussian integral $$\label{gaussR}
\int \prod_{i=1}^n \mathrm{d} x_i \exp \left(-\frac{1}{2} \sum_{i,j=1}^n
x_i M_{i j} x_j \right) = (2 \pi)^{n/2} \, {\rm det}^{-1/2} (M),$$ with $M$ a symmetric positive-definite $n\times n$ matrix, and ${\rm
det} (M)$ its determinant. It can therefore be evaluated exactly, with the result $$\label{Z00}
\ln Z_0 = \ln \mathrm{Det}^{-N/2} (\Lambda) = - \frac{N}{2} \mathrm{Tr} \ln
\Lambda ,$$ where Tr denotes the functional trace. In deriving this, irrelevant prefactors are ignored and use is made of the identity $$\ln \mathrm{Det}(A) = \mathrm{Tr} \ln A,$$ relating the determinant and trace of an Hermitian operator $A$. The relation can be easily checked explicitly for a $n\times n$ matrix. With $\Lambda = I - K H$ as before, $\ln Z_0$ can be expanded in the hopping parameter $K$ as $$\label{Z0}
\ln Z_0 = \frac{N}{2} L^d \sum_{b=1}^{\infty} \frac{1}{b} z_b(a)  K^b$$ where $z_b(a)$ denotes the number of worldlines of $b$ steps, starting at a given lattice site $n$ and returning to one of its nearest neighbor 