  We interpret recent measurements of the zero field muon relaxation rate in
the frustrated magnetic pyrochlore Dy$_2$Ti$_2$O$_7$ as resulting from the
quantum diffusion of muons in the substance. In this scenario, the plateau
observed at low temperature ($<7$ K) in the relaxation rate is due to coherent
tunneling of the muons through a spatially disordered spin state and not to any
magnetic fluctuations persisting at low temperature. Two further regimes either
side of a maximum relaxation rate at $T^* = 50$ K correspond to a crossover
between tunnelling and incoherent activated hopping motion of the muon. Our fit
of the experimental data is compared with the case of muonium diffusion in KCl.
