of this bond fugacity, long worldlines are exponentially suppressed as long as the mass term in the action is positive, $m^2>0$. Notice that in the last line of Eq. (\[discrete\]), the lower bound on the summation is replaced by $b=1$. This is justified because the $b=0$ term always vanishes for $n \neq n'$.

A more familiar expression for the correlation function is obtained from Eq. (\[Sigma\]) by first summing over the length $b$ of the particle trajectories. With $z_b(n,n')$ given in Eq. (\[prob\]), this leads to $$\begin{aligned}
\label{standard}
\left\langle \phi^\alpha_n \phi^\beta_{n'} \right\rangle &=&
  \delta_{\alpha, \beta} \frac{1}{2} \int_{-\pi}^{\pi}
  \frac{\mathrm{d}^d q}{(2 \pi)^d} \frac{\mathrm{e}^{\mathrm{i} q \cdot
  (n-n')}}{1-2 K \sum_\mu \cos (q_\mu)} \nonumber \\ &=&
  \delta_{\alpha, \beta} \frac{1}{2K} \int_{-\pi}^{\pi}
  \frac{\mathrm{d}^d q}{(2 \pi)^d} \frac{\mathrm{e}^{\mathrm{i} q \cdot
  (n-n')}}{2 \sum_\mu [1-\cos (q_\mu)] + m^2 a^2 }.\end{aligned}$$ For the original fields $\varphi_x$, which are connected to the rescaled fields $\phi_n$ through Eq. (\[K1\]), this translates into the standard form of the correlation function of a free field theory on a hypercubic lattice, $$\label{hyper}
\left\langle \varphi^\alpha_x \varphi^\beta_{x'} \right\rangle = \delta_{\alpha,
\beta} a^2 \int_{-\pi/a}^{\pi/a} \frac{\mathrm{d}^d k}{(2 \pi)^d}
\frac{\mathrm{e}^{\mathrm{i} k \cdot (x-x')}}{2 \sum_\mu [1-\cos
(k_\mu a)] + m^2 a^2 },$$ where the lattice sites are now labeled again by $x_\mu= n_\mu 