Tracing the galaxy-halo connection with galaxy clustering in COSMOS-Web from z = 0.1 to z ~ 12

We explore the evolving relationship between galaxies and their dark matter halos from $z \sim 0.1$ to $z \sim 12$ using mass-limited angular clustering measurements in the 0.54 deg$^2$ of the COSMOS-Web survey. This study provides the first measurements of the mass-limited two-point correlation function at $z \ge 10$ and a consistent analysis spanning 13.4 Gyr of cosmic history, setting new benchmarks for future simulations and models. Using a halo occupation distribution (HOD) framework, we derive characteristic halo masses and the stellar-to-halo mass relationship (SHMR) across redshifts and stellar mass bins. Our results first indicate that HOD models fit data at $z \ge 2.5$ best when incorporating a non-linear scale-dependent halo bias, boosting clustering at non-linear scales (r = 10-100 kpc). We find that galaxies at z > 10.5 with $\log(M_\star / M_\odot) \ge 8.85$ are hosted by halos with $M_{\rm h} \sim 10^{10.5}\,M_\odot$, achieving a star formation efficiency (SFE) $M_\star / (f_b M_{\rm h}) $ up to 1 dex higher than at $z \le 1$. The high galaxy bias at $z \ge 8$ suggests that these galaxies reside in massive halos with intrinsic high SFE. Our SHMR evolves significantly with redshift, starting high at $z \ge 10.5$, decreasing until $z \sim 2 - 3$, then increasing again until the present. Current simulations fail to reproduce both massive high-$z$ galaxies and this evolution, while semi-empirical models linking SFE to halo mass, accretion rates, and redshift align with our findings. We propose that $z > 8$ galaxies experience bursty star formation without significant feedback altering their growth, driving the rapid growth of massive galaxies observed by JWST. Over time, increasing feedback efficiency and exponential halo growth suppress star formation. At $z \sim 2 - 3$ and after, halo growth slows down while star formation continues, supported by gas reservoirs in halos.