Abstract
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 γ) in a cell travels distance R decays like exp\({\left(-R^{f_d(\gamma)}\right)}\) where we identify the functions f d (·) exactly. These functions are piecewise smooth and the discontinuities of \({f^{\prime}_d}\) 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(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 f d (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 highdimensional 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.
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References
Arnol’d V.I.: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York (1989)
S. Chatterjee, R. Peled, Y. Peres, D. Romik, Gravitational allocation to Poisson points, Ann. Math., to appear.
Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Amer. Stat. Soc. 58, 13–30 (1963)
Nazarov F., Sodin M., Volberg A.: Transportation to random zeroes by the gradient flow, Geom. Funct. Anal. 17, 887–935 (2007)
R. Peled, Simple universal bounds for Chebyshev-type quadratures, preprint; arXiv:0903.4625
Sodin M., Tsirelson B.: Random complex zeroes II: Perturbed lattice. Israel J. Math. 152, 105–124 (2006)
Acknowledgments
We thank Nir Lev, for explaining the relevance of oscillatory integrals to the proof of Theorem 6.1; Boris Tsirelson and Mikhail Sodin, for several useful conversations, in particular concerning approximation of continuous measures with discrete ones; Greg Kuperberg and Sasha Sodin, for useful discussions on cubatures. We also thank the referee for a very careful reading of the paper and many helpful comments.
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S.C. supported by NSF grant DMS-0707054 and a Sloan Research Fellowship. R.P. partially completed during stay at the Institut Henri Poincaré - Centre Emile Borel. Research supported by NSF Grant OISE 0730136. D.R. supported by the Israel Science Foundation (ISF) grant number 1051/08.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Chatterjee, S., Peled, R., Peres, Y. et al. Phase Transitions in Gravitational Allocation. Geom. Funct. Anal. 20, 870–917 (2010). https://doi.org/10.1007/s00039-010-0090-7
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DOI: https://doi.org/10.1007/s00039-010-0090-7