  Single-file diffusion is a one-dimensional interacting infinite-particle
system in which the order of particles never changes. An intriguing feature of
single-file diffusion is that the mean-square displacement of a tagged particle
exhibits an anomalously slow sub-diffusive growth. We study the full statistics
of the displacement using a macroscopic fluctuation theory. For the simplest
single-file system of impenetrable Brownian particles we compute the large
deviation function and provide an independent verification using an exact
solution based on the microscopic dynamics. For an arbitrary single-file
system, we apply perturbation techniques and derive an explicit formula for the
variance in terms of the transport coefficients. The same method also allows us
to compute the fourth cumulant of the tagged particle displacement for the
symmetric exclusion process.
