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: $40.15<\log (L_{\mathrm{L}\mathrm{y}\alpha }\mathrm{/}[{\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{s}}^{-1}])<43.35$. 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, $M_{\mathrm{m}\mathrm{i}\mathrm{n}}$, and (on average) one satellite LAE, $M_1$, increase weakly with an increasing line luminosity. The satellite fractions are $\lesssim 10$% ($\lesssim 20$%) 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 $L_{\mathrm{L}\mathrm{y}\alpha }$. We report a strong ($8\sigma$) clustering dependence on $L_{\mathrm{L}\mathrm{y}\alpha }$, where the highest luminosity LAE subsample ($\log (L_{\mathrm{L}\mathrm{y}\alpha }\mathrm{/}[{\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{s}}^{-1}])\approx 42.53$) clusters more strongly ($b_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}=3.13_{-0.15}^{+0.08}$) and resides in more massive DMHs ($\log (M_{\mathrm{h}}\mathrm{/}[h^{-1}{\mathrm{M}}_{\odot }])=11.43_{-0.10}^{+0.04}$) than the lowest luminosity one ($\log (L_{\mathrm{L}\mathrm{y}\alpha }\mathrm{/}[{\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{s}}^{-1}])\approx 40.97$), which presents a bias of $b_{\mathrm{l}\mathrm{o}\mathrm{w}}=1.79_{-0.06}^{+0.08}$ and occupies $\log (M_{\mathrm{h}}\mathrm{/}[h^{-1}{\mathrm{M}}_{\odot }])=10.00_{-0.09}^{+0.12}$ 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.