The nature of low-luminosity AGNs discovered by JWST based on clustering analysis: progenitors of low-$z$ quasars?

James Webb Space Telescope (JWST) has discovered many faint AGNs at high-$z$ by detecting their broad Balmer lines. However, their high number density, lack of X-ray emission, and overly high black hole masses with respect to their host stellar masses suggest that they are a distinct population from general type-1 quasars. Here, we present clustering analysis of 27 low-luminosity broad-line AGNs found by JWST (JWST AGNs) at $5<z<6$ based on cross-correlation analysis with 679 photometrically-selected galaxies to characterize their host dark matter halo (DMH) masses. From the angular and projected cross-correlation functions, we find that their typical DMH mass is $\log (M_\mathrm{halo}/h^{-1}\mathrm{M}_\odot) = 11.46_{-0.25}^{+0.19},$ and $11.53_{-0.20}^{+0.15}$, respectively. This result implies that the host DMHs of these AGNs are $\sim1$ dex smaller than those of luminous quasars. The DMHs of the JWST AGNs at $5<z<6$ are predicted to grow to $10^{12\unicode{x2013}13}\,h^{-1}\mathrm{M}_\odot$ at $z\lesssim3$, which is comparable to that of a more luminous quasar at the same epoch. Applying the empirical stellar-to-halo mass ratio to the measured DMH mass, we evaluate their host stellar mass as $\log(M_*/\mathrm{M}_\odot)=9.48_{-0.41}^{+0.31},$ and $9.60_{-0.33}^{+0.24}$, which are higher than some of those estimated by the SED fitting. We also evaluate their duty cycle as $f_\mathrm{duty}=0.37_{-0.15}^{+0.19}$ per cent, corresponding to $\sim4\times10^6$ yr as the lifetime of the JWST AGNs. While we cannot exclude the possibility that the JWST AGNs are simply low-mass type-1 quasars, these results suggest that the JWST AGNs are a different population from type-1 quasars and the progenitors of quasars at $z\lesssim3$.