Phase Transitions in Gravitational Allocation

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(−Rγ) in a cell travels distance R decays like exp(−Rfd(γ)) where we identify the functions fd(·) 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 f3(γ) = 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 fd(γ) = 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 γ = 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. U.C. Berkeley. Supported by NSF grant DMS-0707054 and a Sloan Research Fellowship. New York University. Partially completed during stay at the Institut Henri Poincare Centre Emile Borel. Research supported by NSF Grant OISE 0730136. Microsoft Research. Hebrew University of Jerusalem. Supported by the Israel Science Foundation (ISF) grant number 1051/08.