  Single-file systems, in which particles diffuse in narrow channels while not
overtaking each other, is a fundamental model for the tracer subdiffusion
observed in confined geometries, such as in zeolites or carbon nanotubes.
Twenty years ago, the mean squared displacement of a tracer was determined at
large times, for any diffusive single-file system. Since then, for a general
single-file system, even the determination of the fourth cumulant, which probes
the deviation from Gaussianity, has remained an open question. Here, we fill
this gap and provide an explicit formula for the fourth cumulant of an
arbitrary single-file system. Our approach also allows us to quantify the
perturbation induced by the tracer on its environment, encoded in the
correlation profiles. These explicit results constitute a first step towards
obtaining a closed equation for the correlation profiles for arbitrary
single-file systems.
