引力分配中的相变
Phase Transitions in Gravitational Allocation
摘要：
给定d ≥ 3维单位质量（"星“）的泊松点过程，牛顿引力将空间划分为等体积的吸引域（单元）。在早期的工作中，我们发现这些细胞的直径具有指数尾。在这里，我们分析了定量几何的细胞，并表明它们的大偏差发生在拉伸指数尺度。更准确地说，细胞中的质量exp（-R^gamma）移动距离R的概率像exp（-R^f_d（gamma））那样衰减，其中我们精确地识别函数f_d。这些函数是分段光滑的，f_d'的不连续性代表相变。在d=3维中，大偏差是由于一个"遥远的吸引星系“，但当f_3（伽马）=1时发生相变（在那一点上，由于单个恒星的波动占主导地位）。当d>=5时，大偏差是由于一个细管（一个"虫洞“），沿着这个细管，恒星密度单调增加，直到点f_d（伽马）=1（在那里，由于单个恒星的波动再次占主导地位）。在4维中，我们发现了一个双相变，在低维行为（吸引星系）和高维行为（虫洞）之间的过渡发生在伽马=4/3。作为后果，我们确定的尾部行为的距离从一个明星在其细胞中的一个统一的点，并证明了一个尖锐的下界细胞的直径的尾部概率，匹配我们的早期上界。
英文摘要：
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.
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