  We study the single particle dynamics of a mobile non-Abelian anyon hopping
around many pinned anyons on a surface. The dynamics is modelled by a discrete
time quantum walk and the spatial degree of freedom of the mobile anyon becomes
entangled with the fusion degrees of freedom of the collective system. Each
quantum trajectory makes a closed braid on the world lines of the particles
establishing a direct connection between statistical dynamics and quantum link
invariants. We find that asymptotically a mobile Ising anyon becomes so
entangled with its environment that its statistical dynamics reduces to a
classical random walk with linear dispersion in contrast to particles with
Abelian statistics which have quadratic dispersion.
