A stochastic derivation of the geodesic rule

We argue that the geodesic rule, for global defects, is a consequence of the randomness of the values of the Goldstone field $\phi$ in each causally connected volume. As these volumes collide and coalescence, $\phi$ evolves by performing a random walk on the vacuum manifold $\mathcal{M}$. We derive a Fokker-Planck equation that describes the continuum limit of this process. Its fundamental solution is the heat kernel on $\mathcal{M}$, whose leading asymptotic behavior establishes the geodesic rule.