\pm \hat{\mu},
n')$. That is, $P_b$ satisfies the recurrence relation: $$\label{recu}
P_b(n, n') = \frac{1}{2d} \sum_{\pm \mu}  P_{b-1}(n+ \hat{\mu}, n')$$ where the factor $2d$ denotes the number of nearest neighbors on a $d$-dimensional square lattice. When $P_{b-1}(n, n')$ is subtracted from both sides of Eq. (\[recu\]), the difference equation $$P_b(n, n') - P_{b-1}(n, n') = \frac{1}{2d} \sum_{\pm \mu}
\left[P_{b-1}(n + \hat{\mu}, n') - P_{b-1}(n, n') \right]$$ follows, with the boundary condition $P_0 (n, n') = \delta_{n,n'}$. As before, the continuum limit is taken by simultaneously letting $a \to 0$ and $b \to \infty$, such that $b a^2 \to \mbox{const}$. The difference equation then turns into the differential equation $$\label{diff}
\partial_b P_b(n, n') = \frac{a^2}{2d} \nabla^2 P_b(n, n'),$$ with the celebrated solution $$\label{celebrate}
P_b(n,n') = \left( \frac{d}{2 \pi b} \right)^{d/2} \exp \left[-
\frac{d}{2} \frac{(n - n')^2}{b}\right].$$ For the probability density $p_b(x, x') \equiv P_b(n,n')/a^d$ this gives the Gaussian distribution $$\label{celebrated}
p_b(x,x') = \left( \frac{d}{2 \pi b a^2} \right)^{d/2} \exp \left[-
\frac{d}{2} \frac{(x - x')^2}{b a^2}\right].$$ Each dimension the particle is free to roam contributes a factor $\sqrt{d/2 \pi b a^2}$ to the prefactor.

### Loop Gas {#sec:loopgas}

A similar geometric representation in terms of particle trajectories as found for the correlation 