Clustering dependence on Lyman-$\alpha$ luminosity from MUSE surveys at $3<z<6$

[Abbreviated] We investigate the dependence of Lyman-$\alpha$ emitter (LAE) clustering on Lyman-$\alpha$ luminosity. We use 1030 LAEs from the MUSE-Wide survey, 679 LAEs from MUSE-Deep, and 367 LAEs from the to-date deepest ever spectroscopic survey, the MUSE Extremely Deep Field. All objects have spectroscopic redshifts of $3<z<6$ and cover a large dynamic range of Ly$\alpha$ luminosities: $40.15<\log (L_{\rm{Ly}\alpha}/[\rm{erg \:s}^{-1}])<43.35$. We apply the Adelberger et al. K-estimator as the clustering statistic and fit the measurements with state-of-the-art halo occupation distribution (HOD) models. From the three main data sets, we find that the large-scale bias factor, the minimum mass to host one central LAE, $M_{\rm{min}}$, and (on average) one satellite LAE, $M_1$, increase weakly with an increasing line luminosity. The satellite fractions are $\lesssim10$% ($\lesssim20$%) at $1\sigma$ ($3\sigma$) confidence level, supporting a scenario in which DMHs typically host one single LAE. We next bisected the three main samples into disjoint subsets to thoroughly explore the dependence of the clustering properties on $L_{\rm{Ly}\alpha}$. We report a strong ($8\sigma$) clustering dependence on $L_{\rm{Ly}\alpha}$, where the highest luminosity LAE subsample ($\log(L_{\rm{Ly}\alpha}/[\rm{erg \:s}^{-1}])\approx42.53$) clusters more strongly ($b_{\rm{high}}=3.13^{+0.08}_{-0.15}$) and resides in more massive DMHs ($\log(M_{\rm{h}}/[h^{-1}\rm{M}_{\odot}])=11.43^{+0.04}_{-0.10}$) than the lowest luminosity one ($\log(L_{\rm{Ly}\alpha}/[\rm{erg \:s}^{-1}])\approx40.97$), which presents a bias of $b_{\rm{low}}=1.79^{+0.08}_{-0.06}$ and occupies $\log(M_{\rm{h}}/[h^{-1}\rm{M}_{\odot}])=10.00^{+0.12}_{-0.09}$ halos. We discuss the implications of these results for evolving Ly$\alpha$ luminosity functions, halo mass dependent Ly$\alpha$ escape fractions, and incomplete reionization signatures.