  Single-file diffusion refers to the motion in narrow channels of particles
which cannot bypass each other. These strong correlations between particles
lead to tracer subdiffusion, which has been observed in contexts as varied as
transport in porous media, zeolites or confined colloidal suspensions, and
theoretically studied in numerous works. Most approaches to this celebrated
many-body problem were restricted to the description of the tracer only, whose
essential properties, such as large deviation functions or two-time correlation
functions, were determined only recently. Here, we go beyond this standard
description by introducing and determining analytically generalised density
profiles (GDPs) in the frame of the tracer. In addition to controlling the
statistical properties of the tracer, these quantities fully characterise the
correlations between the tracer position and the bath particles density.
Considering the hydrodynamic limit of the problem, we unveil universal scaling
properties of the GDPs with space and time, and a non-monotonic dependence with
the distance to the tracer despite the absence of any asymmetry. Our analytical
approach provides exact results for the GDPs of paradigmatic models of
single-file diffusion, such as Brownian particles with hardcore repulsion, the
Symmetric Exclusion Process and the Random Average Process. The range of
applicability of our approach is further illustrated by considering extensions
to general interactions between particles and out-of-equilibrium situations.
