form $$\label{Sl}
S_0 = \sum_{n, n'} \phi_n \cdot \Lambda_{n, n'} \phi_{n'} ,$$ with $$\label{Lambda} 
\Lambda_{n, n'} =\delta_{n, n'} - K \sum_{\pm \mu} \delta_{n, n' +
\hat{\mu} }$$ where the sum $\sum_{\pm \mu}$ extends over the positive as well as the negative directions. In matrix notation $$\Lambda =  I - K \, H,$$ where $I$ is the identity matrix and $H$ denotes the so-called hopping matrix whose elements $H_{n, n'}$ are unity if the two lattice sites $n$ and $n'$ are nearest neighbors, and zero otherwise. Physically, $H$ describes the hopping, or propagation of a particle from one lattice site, representing a spacetime cell, to an adjacent one, with each hop carrying a weight $K$. Since the theory is noninteracting, a particle is free to hop at random to any of its nearest neighbors without restriction. It can and will in general revisit sites previously visited and can even hop onto a site already occupied by another particle. The particle trajectories are therefore *phantom* worldlines that can freely intersect and share bonds of the lattice without energy penalty.

The inverse of $\Lambda$ determines the lattice correlation function: $$\label{lgreen}
\left\langle \phi^\alpha_n \phi^\beta_{n'} \right\rangle = 
\frac{1}{2} \delta_{\alpha, \beta}  \Lambda^{-1}_{n, n'} = 
\frac{1}{2} \delta_{\alpha, 