  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.
