Local Group analogues in a cosmological context -- I. Relating velocity structure to the cosmic web

Our Local Group, dominated in mass by the Milky Way (MW) and M31, provides a unique laboratory for testing $\Lambda$CDM cosmology on small scales owing to its proximity. However, its connection to the surrounding large-scale environment, which is essential for interpreting its properties, is inadequately understood. In this work, we explore the connection between Local Group analogues (LGAs) and their surrounding large-scale environments using the ABACUSSUMMIT simulation suite, highlighting the key role of the coupling energy of the MW-M31 orbit, $E_{\rm coupling}$. We find that LGAs with high $E_{\rm coupling}$ preferentially reside in denser regions, whereas those with low $E_{\rm coupling}$ tend to occupy low-density environments. Furthermore, LGAs with low $E_{\rm coupling}$ exhibit strong alignment with cosmic filaments, manifested as a pronounced polar anisotropy in the distribution of tracer haloes. By contrast, LGAs with high $E_{\rm coupling}$ show a weaker polar anisotropy but an enhanced azimuthal anisotropy, with large-scale tracer haloes preferentially lying in the plane spanned by the halo pair and the orbital spin vector. Within this framework, our Local Group is characterised by typical $E_{\rm coupling}$ residing in a relatively under-dense environment, yet it remains consistent with the 95\% range of analogue systems identified in the simulation.