Central‐limit theorems on groups
Central‐limit theorems on groups
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群的中心极限定理
DOI：
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发表时间：
1986
期刊：
影响因子：
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通讯作者：
P. A. Mello
中科院分区：
文献类型：
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作者：
P. A. Mello
在极限 N → ∞ 下研究群中 N 个统计独立（且同分布，每个概率密度为 p1）元素乘积的概率密度 pN。结果表明，对于紧群 R(2) 和 R(3)，pN→1 为 N→∞，与 p1 无关。其他紧凑群体也会出现类似的行为，这似乎是合理的。对于非紧群，我们研究了无序导体物理学中感兴趣的 SU(1,1) 情况。详细分析了 p1 各向同性（即与相位无关）的情况。当 p1 固定并且 N≫1 时，可以找到适当变量的高斯分布。当原始变量按 1/N 重新缩放并取极限 N→∞ 时，保持导体长度与局部化长度的比率固定，就会找到所得概率密度的显式积分表示。还表明后者满足群流形上的“扩散”方程。
The probability density pN of the product of N statistically independent (and identically distributed, each with probability density p1) elements of a group is studied in the limit N→∞. It is shown, for the compact groups R(2) and R(3), that pN→1 as N→∞, independently of p1. It is made plausible that a similar behavior is to be expected for other compact groups. For noncompact groups, the case of SU(1,1)which is of interest to the physics of disordered conductors, is studied. The case in which p1 is isotropic, i.e., independent of the phases, is analyzed in detail. When p1 is fixed and N≫1, a Gaussian distribution in the appropriate variable is found. When the original variables are rescaled by 1/N and the limit N→∞ is taken, keeping the ratio of the length of the conductor to the localization length fixed, an explicit integral representation for the resulting probability density is found. It is also exhibited that the latter satisfies a ‘‘diffusion’’ equation on the group manifold.