We perform Monte-Carlo simulations to analyse the structure and microscopic dynamics of a viscous Lennard-Jones liquid coupled to a quenched reference configuration of the same liquid. The coupling between the two replicas is introduced via a field E conjugate to the overlap Q between the two particle configurations. This allows us to study the evolution of various static and dynamic correlation functions across the (E, T) equilibrium phase diagram. As the temperature is decreased, we identify increasingly marked precursors of a first-order phase transition between a low-Q and a high-Q phase induced by the field E. We show in particular that both static and dynamic susceptibilities have a maximum at a temperature-dependent value of the coupling field, which defines a Widom line'. We also show that, in the high-overlap regime, diffusion and structural relaxation are strongly decoupled because single-particle motion mostly occurs via discrete hopping on the sites defined by the reference configuration. These results, obtained using conventional numerical tools, provide encouraging signs that an equilibrium phase transition exists in coupled viscous liquids, but also demonstrate that important numerical challenges must be overcome to obtain more conclusive numerical evidence.
