\beta}  \sum_{b=0}^{\infty} \, H^b_{n, n'}  K^b,$$ as can be read off from the action (\[Sl\]). From the definition of the hopping matrix $H$ it follows that $H_{n, n'}$ raised to the power $b$ gives the number of paths $z_b(n, n')$ of $b$ steps starting at site $n$ and ending at site $n'$. In this so-called *hopping expansion*,[@GFCM] the correlation function is therefore obtained by summing over all possible worldlines of arbitrary many steps joining the endpoints $n$ and $n'$:[@Ambjorn] $$\label{Sigma}
\left\langle \phi^\alpha_n \phi^\beta_{n'} \right\rangle = \frac{1}{2}
\delta_{\alpha, \beta} \sum_{b=0}^{\infty} z_b(n,n') K^b = \frac{1}{2}
\delta_{\alpha, \beta} \sum_\mathrm{worldlines} K^b .$$ Each path carries a weight $K^b$ according to the number of steps $b$ taken. This geometric representation of the correlation function as a sum over worldlines constitutes Feynman’s spacetime approach[@Feynman48] to fluctuating fields on the lattice. The equivalence of the two approaches is known as *particle-field duality*.

Because of translational invariance, the number of particle trajectories $z_b(n,n')$ depends only on the relative coordinate $z_b(n, n') = z_b(n-
n')$. It satisfies the recurrence relation $$\label{rec}
z_{b+1} (n) =  \sum_{\pm \mu} z_b (n+\hat{\mu}),$$ with the initial condition $z_0 (n) = \delta_{0,n}$. This relation is easily solved by going over to Fourier space $$z_b (n- 