论文标题
帕累托取样中的相互对称性破裂
Reciprocal symmetry breaking in Pareto sampling
论文作者
论文摘要
令$ w_1，\ ldots，w_n $为$ \ mathrm {pareto}（α）$随机变量按其总和进行标准化的样本，因此$ \ sum_i w_i = 1 $。 $ w_i $可以代表自旋玻璃中山谷的重量（如果$ 0 <α<1 $），或者是家谱中不同谱系（家庭）的频率。本文考虑了有$ n $个人在$ \ mathrm {pareto}（α）$ fordspring-number分布（$ 1 <α<2 $）的种群中。两个随机选择的个体是兄弟姐妹的概率，$ y_2 = \ sum_i w_i^2 $，给出了家庭归一化大小的样本平均值，其倒数为人群中的有效家庭（或复制谱系）在$ n _ {\ nathrm {e}}} = 1/y__2 = 1/y__2 = 1/y__2 = 1/y__2。典型的样本平均值与所有可能样本的平均值大不相同，即$ y_2 $不是自动化数量。典型的$ y_2 $及其倒数不会以相反的方式随$ n $而变化。非自我平均效应对于理解大量产卵物种（例如海洋鱼类）的遗传多样性至关重要。
Let $W_1,\ldots,W_N$ be a sample of $\mathrm{Pareto}(α)$ random variables normalized by their sum, such that $\sum_i W_i=1$. The $W_i$ may represent the weights of valleys in a spin glass (if $0<α<1$), or the frequency of different lineages (families) in a genealogy. This paper considers a population in which there are $N$ individuals reproducing with $\mathrm{Pareto}(α)$ offspring-number distribution ($1<α<2$). The probability of two randomly-chosen individuals being siblings, $Y_2=\sum_i W_i^2$, gives the sample mean of the normalized size of families, and its reciprocal gives the effective number of families (or reproducing lineages) in the population, $N_{\mathrm{e}}=1/Y_2$. The typical sample mean is very different from the average over all possible samples, i.e. $Y_2$ is not a self-averaging quantity. The typical $Y_2$ and its reciprocal do not vary with $N$ in opposite ways. Non-self-averaging effects are crucial in understanding genetic diversity in mass spawning species such as marine fishes.
