sites $n \pm \hat{\mu}$ after $b$ steps (see Fig. \[fig:rw\]). The lattice spacing $a$ serves here as a microscopic cutoff, so that $z_b(a)$ rather than $z_b(0)$ appears when closing open paths. Since a worldline can start at any site on the spacetime lattice, the right hand contains a factor $L^d$, denoting the total number of lattice sites, and $\ln Z_0$ is extensive.

\[t\]\[t\]\[.8\]\[0\][$n$]{} ![A Brownian random walk on a square lattice returning to a site adjacent to its starting position $n$. As indicated, the particle took 27 steps in total. Observe that various bonds are traversed more than once. \[fig:rw\]](rw.eps "fig:"){width="50.00000%"}

The factor $1/2b$ appearing in Eq. (\[Z0\]) reflects the possibility to trace out a closed path starting from any site along the path and going in either direction round it. The factor $N$ reflects the degeneracy associated with a closed worldline. Written as a sum over *single* loops of arbitrary length and shape, Eq. (\[Z0\]) reads $$\label{lnZcomp}
\ln Z_0 = N \sum_{\substack{\mathrm{single} \\ \mathrm{loops}}} K^b,$$ where the loops are considered to have no orientation, whence the absence of the factor $\frac{1}{2}$, and no longer rooted, whence the absence of the factor $L^d$.

For a noninteracting theory, $z_b(a)$ can be 