TY - JOUR

T1 - Phase transitions in gravitational allocation

AU - Chatterjee, Sourav

AU - Peled, Ron

AU - Peres, Yuval

AU - Romik, Dan

N1 - Funding Information:
Keywords and phrases: Phase transition, gravitation, fair allocation, Poisson process, translation-equivariant mapping 2010 Mathematics Subject Classification: 60D05 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.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Phase transition

KW - Poisson process

KW - fair allocation

KW - gravitation

KW - translation-equivariant mapping

UR - http://www.scopus.com/inward/record.url?scp=77958458405&partnerID=8YFLogxK

U2 - 10.1007/s00039-010-0090-7

DO - 10.1007/s00039-010-0090-7

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AN - SCOPUS:77958458405

SN - 1016-443X

VL - 20

SP - 870

EP - 917

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 4

ER -