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

Evaluating the origins of the secondary bias based on the correlation of halo properties with the linear density field