  Quantum fluctuations of the vacuum stress-energy tensor are highly
non-Gaussian, and can have unexpectedly large effects on spacetime geometry. In
this paper, we study a two-dimensional dilaton gravity model coupled to a
conformal field, in which the distribution of vacuum fluctuations is well
understood. In this model, the fluctuations of the matter field are responsible
for the fluctuations of the geometry itself. By analyzing the geodesic
deviation in this model, we show that a pencil of massive particles propagating
on this fuzzy spacetime eventually converges and collapses. This is consistent
with our earlier analysis of null geodesics in [Phys. Rev. Lett.\ 107, 021303
(2011)].
