Phase Transitions in Gravitational Allocation

Details Phase Transitions in Gravitational Allocation Phase Transitions in Gravitational Allocation

2009-03-26

arxiv.org/abs/0903.4647v1

Mathematics

Probability

66 pages. 9 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Phase Transitions in Gravitational Allocation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Phase Transitions in Gravitational Allocation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Phase Transitions in Gravitational Allocation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-65156