  Given a Poisson point process of unit masses (``stars'') in dimension d>=3,
Newtonian gravity partitions space into domains of attraction (cells) of equal
volume. In earlier work, we showed the diameters of these cells have
exponential tails. Here we analyze the quantitative geometry of the cells and
show that their large deviations occur at the stretched-exponential scale. More
precisely, the probability that mass exp(-R^gamma) in a cell travels distance R
decays like exp(-R^f_d(gamma)) where we identify the functions f_d exactly.
These functions are piecewise smooth and the discontinuities of f_d' represent
phase transitions. In dimension d=3, the large deviation is due to a ``distant
attracting galaxy'' but a phase transition occurs when f_3(gamma)=1 (at that
point, the fluctuations due to individual stars dominate). When d>=5, the large
deviation is due to a thin tube (a ``wormhole'') along which the star density
increases monotonically, until the point f_d(gamma)=1 (where again fluctuations
due to individual stars dominate). In dimension 4 we find a double phase
transition, where the transition between low-dimensional behavior (attracting
galaxy) and high-dimensional behavior (wormhole) occurs at gamma=4/3.
  As consequences, we determine the tail behavior of the distance from a star
to a uniform point in its cell, and prove a sharp lower bound for the tail
probability of the cell's diameter, matching our earlier upper bound.
