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Given a Poisson point process of unit masses (``stars'') in dimension d>=3,Newtonian gravity partitions space into domains of attraction (cells) of equalvolume. In earlier work, we showed the diameters of these cells haveexponential tails. Here we analyze the quantitative geometry of the cells andshow that their large deviations occur at the stretched-exponential scale. Moreprecisely, the probability that mass exp(-R^gamma) in a cell travels distance Rdecays 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' representphase transitions. In dimension d=3, the large deviation is due to a ``distantattracting galaxy'' but a phase transition occurs when f_3(gamma)=1 (at thatpoint, the fluctuations due to individual stars dominate). When d>=5, the largedeviation is due to a thin tube (a ``wormhole'') along which the star densityincreases monotonically, until the point f_d(gamma)=1 (where again fluctuationsdue to individual stars dominate). In dimension 4 we find a double phasetransition, where the transition between low-dimensional behavior (attractinggalaxy) and high-dimensional behavior (wormhole) occurs at gamma=4/3.  As consequences, we determine the tail behavior of the distance from a starto a uniform point in its cell, and prove a sharp lower bound for the tailprobability of the cell's diameter, matching our earlier upper bound.

Phase Transitions in Gravitational Allocation