Revisiting the Lensed Fraction of High-Redshift Quasars
  The observed lensed fraction of high-redshift quasars $(\sim0.2\%)$ is
significantly lower than previous theoretical predictions $(\gtrsim4\%)$. We
revisit the lensed fraction of high-redshift quasars predicted by theoretical
models, where we adopt recent measurements of galaxy velocity dispersion
functions (VDFs) and explore a wide range of quasar luminosity function (QLF)
parameters. We use both analytical methods and mock catalogs which give
consistent results. For ordinary QLF parameters and the depth of current
high-redshift quasar surveys $(m_z\lesssim22)$, our model suggests a
multiply-imaged fraction of $F_\text{multi}\sim 0.4\%-0.8\%$. The predicted
lensed fraction is $\sim1\%-6\%$ for the brightest $z_s\sim6$ quasars
$(m_z\lesssim19)$, depending on the QLF. The systematic uncertainties of the
predicted lensed fraction in previous models can be as large as $2-4$ times and
are dominated by the VDF. Applying VDFs from recent measurements decreases the
predicted lensed fraction and relieves the tension between observations and
theoretical models. Given the depth of current imaging surveys, there are
$\sim15$ lensed quasars at $z_s>5.5$ detectable over the sky. Upcoming sky
surveys like the LSST survey and the {\em Euclid} survey will find several tens
of lensed quasars at this redshift range.
