Consider a 4 dimensional lattice with spacing $\Delta$. Want to get expression of
$$\sum_\text{all paths} e^{-k(\Delta) L} $$ in the limit of $\Delta \rightarrow 0$
where $L$ is the number of "links" in one path. k depends on spacing.
I don't know how to start.
The quantity in the question is $$ \sum_{\rm all\ paths}e^{-k(\Delta)L}=\sum_{L=0}^{\infty} e^{-k(\Delta)L}2^{4L}=\frac{1}{1-16e^{-k(\Delta)}}\ . $$ At each step one has $2^4$ choices for a direction to go, so the number of paths of length $L$, say all starting from the origin, is $2^{4L}$. For the limit $\Delta\rightarrow 0$, one would need to know something about $k$ as a function of $\Delta$.