Evaluating the origins of the secondary bias based on the correlation of
  halo properties with the linear density field
  Using two sets of large $N$-body simulations, we study the origin of the
correlations of halo assembly time ($z_{\rm f}$), concentration ($v_{\rm
max}/v_{\rm 200}$) and spin ($\lambda$) with the large-scale evolved density
field at given halo mass, i.e. the secondary bias. We find that the secondary
bias is the secondary effect of the correlations of halo properties with the
linear density estimated at the same comoving scale. Using the linear density
on different scales, we find two types of correlations. The internal
correlation, which reflects the correlation of halo properties with the mean
linear over-density $\delta_{\rm L}$ within the halo Lagrangian radius $R_{\rm
L}$, is positive for both $z_{\rm f}$ and $v_{\rm max}/v_{\rm 200}$, and
negative for $\lambda$. The external correlation, which describes the
correlation of halo properties with linear overdensity at $R>R_{\rm L}$ for
given $\delta_{\rm L}$, shows trends opposite to the internal correlation. Both
of the external and internal correlations depend only weakly on halo mass,
indicating a similar origin for halos of different masses. Our findings offer a
transparent perspective on the origin of the secondary bias. The secondary bias
can be largely explained by the competition of the external and internal
correlations together with the correlation of the linear density field on
different scales. These two types of correlations combined can establish the
complex halo-mass dependence of the secondary bias observed in the simulations.
