Phase transitions in gravitational allocation

Sourav Chatterjee, Ron Peled, Yuval Peres, Dan Romik*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 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.

Original languageEnglish
Pages (from-to)870-917
Number of pages48
JournalGeometric and Functional Analysis
Issue number4
StatePublished - 2010
Externally publishedYes


  • Phase transition
  • Poisson process
  • fair allocation
  • gravitation
  • translation-equivariant mapping


Dive into the research topics of 'Phase transitions in gravitational allocation'. Together they form a unique fingerprint.

Cite this