The correlation among the gene genealogies at different loci is crucial in biology, yet challenging to understand because such correlation depends on many factors including genetic linkage, recombination, natural selection and population structure. Based on a diploid Wright-Fisher model with a single mating type and partial selfing for a constant large population with size $N$, we quantify the combined effect of genetic drift and two competing factors, recombination and selfing, on the correlation of coalescence times at two linked loci for samples of size two. Recombination decouples the genealogies at different loci and decreases the correlation while selfing increases the correlation. We obtain explicit asymptotic formulas for the correlation for four scaling scenarios that depend on whether the selfing probability and the recombination probability are of order $O(1/N)$ or $O(1)$ as $N$ tends to infinity. Our analytical results confirm that the asymptotic lower bound in [King, Wakeley, Carmi (Theor. Popul. Biol. 2018)] is sharp when the loci are unlinked and when there is no selfing, and provide a number of new formulas for other scaling scenarios that have not been considered before. We present asymptotic results for the variance of Tajima's estimator of the population mutation rate for infinitely many loci as $N$ tends to infinity. When the selfing probability is of order $O(1)$ and is equal to a positive constant $s$ for all $N$ and if the samples at both loci are in the same individual, then the variance of the Tajima's estimator tends to $s/2$ (hence remains positive) even when the recombination rate, the number of loci and the population size all tend to infinity.

翻译：不同位点基因谱系间的相关性在生物学中至关重要，但由于这种相关性受遗传连锁、重组、自然选择和群体结构等多种因素影响，使其难以理解。基于一个具有单一交配类型和部分自交的二倍体Wright-Fisher模型（群体大小恒定为$N$的大群体），我们量化了遗传漂变与两个竞争因素——重组和自交——对两个连锁位点（样本量为2）融合时间相关性的联合效应。重组使不同位点的谱系解耦并降低相关性，而自交则增加相关性。针对四种缩放场景（根据自交概率和重组概率在$N$趋于无穷时是$O(1/N)$阶还是$O(1)$阶），我们获得了相关性的显式渐近公式。分析结果证实，当位点不连锁且无自交时，[King, Wakeley, Carmi (Theor. Popul. Biol. 2018)]中的渐近下界是紧致的，并为先前未考虑的其他缩放场景提供了多个新公式。我们展示了当$N$趋于无穷时，无限多位点条件下Tajima群体突变率估计量方差的渐近结果。若对于所有$N$，自交概率为$O(1)$阶且等于正常数$s$，并且两个位点的样本来自同一个体，则即使重组率、位点数量和群体大小均趋于无穷，Tajima估计量的方差仍趋近于$s/2$（因此保持正值）。