  Lyman-$\alpha$ emitters (LAEs) are a promising probe of the large-scale
structure at high redshift, $z\gtrsim 2$. In particular, the Hobby-Eberly
Telescope Dark Energy Experiment aims at observing LAEs at 1.9 $<z<$ 3.5 to
measure the Baryon Acoustic Oscillation (BAO) scale and the Redshift-Space
Distortion (RSD). However, Zheng et al. (2011) pointed out that the complicated
radiative transfer (RT) of the resonant Lyman-$\alpha$ emission line generates
an anisotropic selection bias in the LAE clustering on large scales, $s\gtrsim
10$ Mpc. This effect could potentially induce a systematic error in the BAO and
RSD measurements. Also, Croft et al. (2016) claims an observational evidence of
the effect in the Lyman-$\alpha$ intensity map, albeit statistically
insignificant. We aim at quantifying the impact of the Lyman-$\alpha$ RT on the
large-scale galaxy clustering in detail. For this purpose, we study the
correlations between the large-scale environment and the ratio of an apparent
Lyman-$\alpha$ luminosity to an intrinsic one, which we call the `observed
fraction', at $2<z<6$. We apply our Lyman-$\alpha$ RT code by post-processing
the full Illustris simulations. We simply assume that the intrinsic luminosity
of the Lyman-$\alpha$ emission is proportional to the star formation rate of
galaxies in Illustris, yielding a sufficiently large sample of LAEs to measure
the anisotropic selection bias. We find little correlations between large-scale
environment and the observed fraction induced by the RT, and hence a smaller
anisotropic selection bias than what was claimed by Zheng et al. (2011). We
argue that the anisotropy was overestimated in the previous work due to the
insufficient spatial resolution: it is important to keep the resolution such
that it resolves the high density region down to the scale of the interstellar
medium, $\sim1$ physical kpc. (abridged)

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics